diff --git a/src/AlgAss/AlgGrp.jl b/src/AlgAss/AlgGrp.jl
index 1e1517f7ef..5569d2be57 100644
--- a/src/AlgAss/AlgGrp.jl
+++ b/src/AlgAss/AlgGrp.jl
@@ -32,8 +32,8 @@ group(A::GroupAlgebra) = A.group
 
 has_one(A::GroupAlgebra) = true
 
-function (A::GroupAlgebra{T, S, R})(c::Vector{T}; copy::Bool = false) where {T, S, R}
-  length(c) != dim(A) && error("Dimensions don't match.")
+function (A::GroupAlgebra{T, S, R})(c::Union{Vector{T}, SRow{T}}; copy::Bool = false) where {T, S, R}
+  c isa Vector && length(c) != dim(A) && error("Dimensions don't match.")
   return GroupAlgebraElem{T, typeof(A)}(A, copy ? deepcopy(c) : c)
 end
 
@@ -64,12 +64,12 @@ end
 Returns the group ring $K[G]$.
 $G$ may be any set and `op` a group operation on $G$.
 """
-group_algebra(K::Ring, G; op = *) = GroupAlgebra(K, G, op = op)
+group_algebra(K::Ring, G; op = *, cached::Bool = true, sparse::Bool = false) = GroupAlgebra(K, G; op, sparse, cached)
 
-group_algebra(K::Ring, G::FinGenAbGroup) = GroupAlgebra(K, G)
+group_algebra(K::Ring, G::FinGenAbGroup; cached::Bool = true, sparse::Bool = false) = GroupAlgebra(K, G; sparse, cached)
 
-function group_algebra(K::Field, G; op = *)
-  A = GroupAlgebra(K, G, op = op)
+function group_algebra(K::Field, G; op = *, sparse::Bool = false, cached::Bool = true)
+  A = GroupAlgebra(K, G; op, sparse, cached)
   if iszero(characteristic(K))
     A.issemisimple = 1
   else
diff --git a/src/AlgAss/Elem.jl b/src/AlgAss/Elem.jl
index 976dbfc966..712d87c90a 100644
--- a/src/AlgAss/Elem.jl
+++ b/src/AlgAss/Elem.jl
@@ -4,6 +4,10 @@
 #
 ################################################################################
 
+_is_sparse(a::AbstractAssociativeAlgebraElem) = false
+
+_is_sparse(a::GroupAlgebraElem) = parent(a).sparse
+
 function AbstractAlgebra.promote_rule(U::Type{<:AbstractAssociativeAlgebraElem{T}}, ::Type{S}) where {T, S}
   if AbstractAlgebra.promote_rule(T, S) === T
     return U
@@ -90,7 +94,11 @@ function one(A::AbstractAssociativeAlgebra)
   if !has_one(A)
     error("Algebra does not have a one")
   end
-  return A(deepcopy(A.one)) # deepcopy needed by mul!
+  if !A.sparse
+    return A(deepcopy(A.one)) # deepcopy needed by mul!
+  else
+    return A(deepcopy(A.sparse_one))
+  end
 end
 
 ################################################################################
@@ -151,20 +159,29 @@ end
 
 function +(a::AbstractAssociativeAlgebraElem{T}, b::AbstractAssociativeAlgebraElem{T}) where {T}
   parent(a) != parent(b) && error("Parents don't match.")
-  v = Vector{T}(undef, dim(parent(a)))
-  for i = 1:dim(parent(a))
-    v[i] = coefficients(a, copy = false)[i] + coefficients(b, copy = false)[i]
+  if !_is_sparse(a)
+    v = Vector{T}(undef, dim(parent(a)))
+    for i = 1:dim(parent(a))
+      v[i] = coefficients(a, copy = false)[i] + coefficients(b, copy = false)[i]
+    end
+    return parent(a)(v)
+  else
+    vv = a.coeffs_sparse + b.coeffs_sparse
+    return parent(a)(vv)
   end
-  return parent(a)(v)
 end
 
 function -(a::AbstractAssociativeAlgebraElem{T}, b::AbstractAssociativeAlgebraElem{T}) where {T}
   parent(a) != parent(b) && error("Parents don't match.")
-  v = Vector{T}(undef, dim(parent(a)))
-  for i = 1:dim(parent(a))
-    v[i] = coefficients(a, copy = false)[i] - coefficients(b, copy = false)[i]
+  if _is_sparse(a)
+    return parent(a)(a.coeffs_sparse - b.coeffs_sparse)
+  else
+    v = Vector{T}(undef, dim(parent(a)))
+    for i = 1:dim(parent(a))
+      v[i] = coefficients(a, copy = false)[i] - coefficients(b, copy = false)[i]
+    end
+    return parent(a)(v)
   end
-  return parent(a)(v)
 end
 
 function *(a::AssociativeAlgebraElem{T}, b::AssociativeAlgebraElem{T}) where {T}
@@ -200,25 +217,52 @@ end
 function *(a::GroupAlgebraElem{T, S}, b::GroupAlgebraElem{T, S}) where {T, S}
   parent(a) != parent(b) && error("Parents don't match.")
   A = parent(a)
-  d = dim(A)
-  v = Vector{T}(undef, d)
-  for i in 1:d
-    v[i] = zero(base_ring(A))
-  end
-  t = zero(base_ring(A))
-  mt = multiplication_table(A, copy = false)
-  acoeff = coefficients(a, copy = false)
-  bcoeff = coefficients(b, copy = false)
-  for i in 1:d
-    if iszero(acoeff[i])
-      continue
+  if !_is_sparse(a)
+    d = dim(A)
+    v = Vector{T}(undef, d)
+    for i in 1:d
+      v[i] = zero(base_ring(A))
     end
-    for j in 1:d
-      k = mt[i, j]
-      v[k] = addmul!(v[k], acoeff[i], bcoeff[j], t)
+    t = zero(base_ring(A))
+    mt = multiplication_table(A, copy = false)
+    acoeff = coefficients(a, copy = false)
+    bcoeff = coefficients(b, copy = false)
+    for i in 1:d
+      if iszero(acoeff[i])
+        continue
+      end
+      for j in 1:d
+        k = mt[i, j]
+        v[k] = addmul!(v[k], acoeff[i], bcoeff[j], t)
+      end
+    end
+    return A(v)
+  else
+    s = sparse_row(base_ring(A))
+    for (i, ci) in a.coeffs_sparse
+      r = sparse_row(base_ring(A), [(__elem_index(A, A.base_to_group[i] * A.base_to_group[j]), ci * cj) for (j, cj) in b.coeffs_sparse])
+      s += r
     end
+    return A(s)
+  end
+end
+
+################################################################################
+#
+#  Getindex for group algebra elements
+#
+################################################################################
+
+function getindex(a::GroupAlgebraElem{S, GroupAlgebra{S, T, U}}, g::U) where {S, T, U}
+  if _is_sparse(a)
+    if !haskey(parent(a).group_to_base, g)
+      return zero(base_ring(parent(a)))
+    else
+      return a.coeffs_sparse[parent(a).group_to_base[g]]
+    end
+  else
+    return a.coeffs[parent(a).group_to_base[g]]
   end
-  return A(v)
 end
 
 ################################################################################
@@ -303,6 +347,9 @@ mul!(c::AbstractAssociativeAlgebraElem{T}, a::Union{ Int, ZZRingElem }, b::Abstr
 
 function mul!(c::GroupAlgebraElem{T, S}, a::GroupAlgebraElem{T, S}, b::GroupAlgebraElem{T, S}) where {T, S}
   parent(a) != parent(b) && error("Parents don't match.")
+  if _is_sparse(a)
+    return a * b
+  end
   A = parent(a)
   d = dim(A)
 
@@ -449,7 +496,11 @@ divexact_left(a::AbstractAssociativeAlgebraElem, b::AbstractAssociativeAlgebraEl
 ################################################################################
 
 function *(a::AbstractAssociativeAlgebraElem{S}, b::S) where {S <: RingElem}
-  return typeof(a)(parent(a), coefficients(a, copy = false).* Ref(b))
+  if !_is_sparse(a)
+    return typeof(a)(parent(a), coefficients(a, copy = false).* Ref(b))
+  else
+    return typeof(a)(parent(a), a.coeffs_sparse * b)
+  end
 end
 
 *(b::S, a::AbstractAssociativeAlgebraElem{S}) where {S <: RingElem} = a*b
@@ -676,10 +727,19 @@ function show(io::IO, a::AbstractAssociativeAlgebraElem)
   if get(io, :compact, false)
     print(io, coefficients(a, copy = false))
   else
-    print(io, "Element of ")
-    print(io, parent(a))
-    print(io, " with coefficients ")
-    print(io, coefficients(a, copy = false))
+    if a isa GroupAlgebraElem && parent(a).sparse
+      sum = Expr(:call, :+)
+      if !iszero(a)
+        for (i, ci) in a.coeffs_sparse
+          push!(sum.args,
+                Expr(:call, :*, AbstractAlgebra.expressify(ci, context = io),
+                                AbstractAlgebra.expressify(parent(a).base_to_group[i], context = io)))
+        end
+      end
+      print(io, AbstractAlgebra.expr_to_string(AbstractAlgebra.canonicalize(sum)))
+    else
+      print(io, coefficients(a, copy = false))
+    end
   end
 end
 
@@ -963,6 +1023,9 @@ end
 isone(a::AbstractAssociativeAlgebraElem) = a == one(parent(a))
 
 function iszero(a::AbstractAssociativeAlgebraElem)
+  if _is_sparse(a)
+    return length(a.coeffs_sparse) == 0
+  end
   return all(i -> iszero(i), coefficients(a, copy = false))
 end
 
diff --git a/src/AlgAss/Types.jl b/src/AlgAss/Types.jl
index d69398e353..254d402a0f 100644
--- a/src/AlgAss/Types.jl
+++ b/src/AlgAss/Types.jl
@@ -145,6 +145,9 @@ end
   center
   maps_to_numberfields
   maximal_order
+  sparse::Bool
+  ind::Int
+  sparse_one
 
   function GroupAlgebra(K::Ring, G::FinGenAbGroup, cached::Bool = true)
     A = GroupAlgebra(K, G, op = +, cached = cached)
@@ -152,9 +155,11 @@ end
     return A
   end
 
-  function GroupAlgebra(K::Ring, G; op = *, cached = true)
+  function GroupAlgebra(K::Ring, G; op = *, cached = true, sparse::Bool = false)
     return get_cached!(GroupAlgebraID, (K, G, op), cached) do
       A = new{elem_type(K), typeof(G), elem_type(G)}()
+      A.sparse = sparse
+      A.ind = -1
       A.is_commutative = 0
       A.is_simple = 0
       A.issemisimple = 0
@@ -163,36 +168,50 @@ end
       d = Int(order(G))
       A.group_to_base = Dict{elem_type(G), Int}()
       A.base_to_group = Vector{elem_type(G)}(undef, d)
-      A.mult_table = zeros(Int, d, d)
-
-      i = 2
-      for g in collect(G)
-        if isone(g)
-          A.group_to_base[deepcopy(g)] = 1
-          A.base_to_group[1] = deepcopy(g)
-          continue
+
+      if A.sparse
+        if G isa FinGenAbGroup
+          el = zero(G)
+        else
+          el = one(G)
         end
-        A.group_to_base[deepcopy(g)] = i
-        A.base_to_group[i] = deepcopy(g)
-        i += 1
+        A.group_to_base[el] = 1
+        A.base_to_group[1] = el
+        A.sparse_one = sparse_row(K, [1], [one(K)])
+        A.ind = 1 # index - 1 for the next group element
       end
 
-      v = Vector{elem_type(K)}(undef, d)
-      for i in 1:d
-        v[i] = zero(K)
-      end
-      v[1] = one(K)
+      if !A.sparse
+        A.mult_table = zeros(Int, d, d)
+        i = 2
+        for g in collect(G)
+          if isone(g)
+            A.group_to_base[deepcopy(g)] = 1
+            A.base_to_group[1] = deepcopy(g)
+            continue
+          end
+          A.group_to_base[deepcopy(g)] = i
+          A.base_to_group[i] = deepcopy(g)
+          i += 1
+        end
 
-      A.one = v
+        v = Vector{elem_type(K)}(undef, d)
+        for i in 1:d
+          v[i] = zero(K)
+        end
+        v[1] = one(K)
+
+        A.one = v
 
-      for i = 1:d
-        for j = 1:d
-          l = op(A.base_to_group[i], A.base_to_group[j])
-          A.mult_table[i, j] = A.group_to_base[l]
+        for i = 1:d
+          for j = 1:d
+            l = op(A.base_to_group[i], A.base_to_group[j])
+            A.mult_table[i, j] = A.group_to_base[l]
+          end
         end
-      end
 
-      @assert all(A.mult_table[1, i] == i for i in 1:dim(A))
+        @assert all(A.mult_table[1, i] == i for i in 1:dim(A))
+      end
 
       return A
     end::GroupAlgebra{elem_type(K), typeof(G), elem_type(G)}
@@ -204,19 +223,35 @@ const GroupAlgebraID = AbstractAlgebra.CacheDictType{Tuple{Ring, Any, Any}, Grou
 mutable struct GroupAlgebraElem{T, S} <: AbstractAssociativeAlgebraElem{T}
   parent::S
   coeffs::Vector{T}
+  coeffs_sparse
 
   function GroupAlgebraElem{T, S}(A::S) where {T, S}
     z = new{T, S}()
     z.parent = A
-    z.coeffs = Vector{T}(undef, size(A.mult_table, 1))
-    for i = 1:length(z.coeffs)
-      z.coeffs[i] = A.base_ring()
+    if !A.sparse
+      z.coeffs = Vector{T}(undef, size(A.mult_table, 1))
+      for i = 1:length(z.coeffs)
+        z.coeffs[i] = A.base_ring()
+      end
+    else
+      z.coeffs_sparse = sparse_row(base_ring(A))
     end
     return z
   end
 
   function GroupAlgebraElem{T, S}(A::S, g::U) where {T, S, U}
-    return A[A.group_to_base[g]]
+    if A.sparse
+      i = get!(A.group_to_base, g) do
+        A.ind += 1
+        A.base_to_group[A.ind] = g
+        return A.ind
+      end
+      a = GroupAlgebraElem{T, S}(A)
+      a.coeffs_sparse = sparse_row(base_ring(A), [i], [one(base_ring(A))])
+      return a
+    else
+      return A[A.group_to_base[g]]
+    end
   end
 
   # This does not make a copy of coeffs
@@ -226,8 +261,22 @@ mutable struct GroupAlgebraElem{T, S} <: AbstractAssociativeAlgebraElem{T}
     z.coeffs = coeffs
     return z
   end
+
+  function GroupAlgebraElem{T, S}(A::S, coeffs::SRow{T}) where {T, S}
+    z = new{T, S}()
+    z.parent = A
+    z.coeffs_sparse = coeffs
+    return z
+  end
 end
 
+__elem_index(A, g) = get!(A.group_to_base, g) do
+        A.ind += 1
+        A.base_to_group[A.ind] = g
+        return A.ind
+      end
+
+
 ################################################################################
 #
 #  AbsAlgAssIdl