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Properties

Label 1296.2891
Order \( 2^{4} \cdot 3^{4} \)
Exponent \( 2^{3} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{4} \cdot 3^{5} \)
$\card{\mathrm{Out}(G)}$ \( 3 \)
Perm deg. $27$
Trans deg. $27$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(1296, 2891);
 
Copy content gap:G := SmallGroup(1296, 2891);
 
Copy content sage:G = PermutationGroup(['(2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20)', '(1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23)', '(4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23)', '(4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18)', '(4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23)', '(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)', '(1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])
 
Copy content sage_gap:G = libgap.SmallGroup(1296, 2891)
 
Copy content oscar:G = small_group(1296, 2891)
 

Group information

Description:$\He_3:\GL(2,3)$
Order: \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Copy content oscar:order(G)
 
Exponent: \(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Copy content oscar:exponent(G)
 
Automorphism group:$\SU(3,2).C_3.C_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$C_2$ x 4, $C_3$ x 4
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Copy content oscar:composition_series(G)
 
Derived length:$6$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

This group is nonabelian and solvable.

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content oscar:is_abelian(G)
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content oscar:is_nilpotent(G)
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content oscar:is_solvable(G)
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;
 
Copy content gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");
 
Copy content sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 
Copy content sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 
Copy content oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))
 

Order 1 2 3 4 6 8 9 12
Elements 1 117 98 54 450 324 144 108 1296
Conjugacy classes   1 2 5 1 5 2 1 1 18
Divisions 1 2 5 1 5 1 1 1 17
Autjugacy classes 1 2 3 1 3 2 1 1 14

Copy content comment:Compute statistics about the characters of G
 
Copy content magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]
 
Copy content sage_gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i G.CharacterDegrees()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 2 3 4 6 8 12 16 18
Irr. complex chars.   2 3 2 1 3 2 3 1 1 18
Irr. rational chars. 2 1 2 2 3 2 3 1 1 17

Minimal presentations

Permutation degree:$27$
Transitive degree:$27$
Rank: $2$
Inequivalent generating pairs: $144$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 6 6 6
Arbitrary 6 6 6

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: ${\langle a, b, c, d, e, f, g \mid a^{2}=b^{3}=d^{4}=e^{3}=f^{3}=g^{3}=[b,e]= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([8, -2, -3, -2, 2, -2, -3, 3, -3, 65, 1010, 226, 114, 1155, 107, 211, 91, 23045, 8469, 2333, 1957, 16134, 17486, 3158, 2494, 710, 718, 55303]); a,b,c,d,e,f,g := Explode([G.1, G.2, G.3, G.4, G.6, G.7, G.8]); AssignNames(~G, ["a", "b", "c", "d", "d2", "e", "f", "g"]);
 
Copy content gap:G := PcGroupCode(937965433026430725280738121754529260739305841104431172945407333076039,1296); a := G.1; b := G.2; c := G.3; d := G.4; e := G.6; f := G.7; g := G.8;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(937965433026430725280738121754529260739305841104431172945407333076039,1296)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.6; f = G.7; g = G.8;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(937965433026430725280738121754529260739305841104431172945407333076039,1296)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.6; f = G.7; g = G.8;
 
Permutation group:Degree $27$ $\langle(2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 27 | (2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20), (1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23), (4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18), (4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) >;
 
Copy content gap:G := Group( (2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20), (1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23), (4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18), (4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) );
 
Copy content sage:G = PermutationGroup(['(2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20)', '(1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23)', '(4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23)', '(4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18)', '(4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23)', '(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)', '(1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])
 
Copy content sage_gap:G = gap.new('Group( (2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20), (1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23), (4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18), (4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) )')
 
Copy content oscar:G = @permutation_group(27, (2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20), (1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23), (4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18), (4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26))
 
Matrix group:$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 1 & 0 & 0 & -1 \\ 1 & 0 & 1 & 0 & 1 & -1 \\ -1 & 1 & 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & -1 & 0 & -1 & 0 & -1 \\ -1 & 0 & -1 & -1 & -1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \\ 1 & -1 & 0 & 0 & 1 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Copy content comment:Define the group as a matrix group with coefficients in Z
 
Copy content magma:G := MatrixGroup< 6, Integers() | [[0, 1, 1, 0, 0, -1, 1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [1, 0, 0, 0, 1, -1, 1, 0, 0, 1, 1, 0, 1, -1, 0, -1, 0, -1, -1, 0, -1, -1, -1, 1, 0, 0, 1, 1, 1, 0, 1, -1, 0, 0, 1, -1]] >;
 
Copy content gap:G := Group([[[0, 1, 1, 0, 0, -1], [1, 0, 1, 0, 1, -1], [-1, 1, 0, 0, -1, 1], [0, 0, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0]], [[1, 0, 0, 0, 1, -1], [1, 0, 0, 1, 1, 0], [1, -1, 0, -1, 0, -1], [-1, 0, -1, -1, -1, 1], [0, 0, 1, 1, 1, 0], [1, -1, 0, 0, 1, -1]]]);
 
Copy content sage:MS = MatrixSpace(Integers(), 6, 6) G = MatrixGroup([MS([[0, 1, 1, 0, 0, -1], [1, 0, 1, 0, 1, -1], [-1, 1, 0, 0, -1, 1], [0, 0, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0]]), MS([[1, 0, 0, 0, 1, -1], [1, 0, 0, 1, 1, 0], [1, -1, 0, -1, 0, -1], [-1, 0, -1, -1, -1, 1], [0, 0, 1, 1, 1, 0], [1, -1, 0, 0, 1, -1]])])
 
Copy content sage_gap:G = gap.new('Group([[[0, 1, 1, 0, 0, -1], [1, 0, 1, 0, 1, -1], [-1, 1, 0, 0, -1, 1], [0, 0, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0]], [[1, 0, 0, 0, 1, -1], [1, 0, 0, 1, 1, 0], [1, -1, 0, -1, 0, -1], [-1, 0, -1, -1, -1, 1], [0, 0, 1, 1, 1, 0], [1, -1, 0, 0, 1, -1]]])')
 
Copy content oscar:G = matrix_group([matrix(ZZ, [[0, 1, 1, 0, 0, -1], [1, 0, 1, 0, 1, -1], [-1, 1, 0, 0, -1, 1], [0, 0, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0]]), matrix(ZZ, [[1, 0, 0, 0, 1, -1], [1, 0, 0, 1, 1, 0], [1, -1, 0, -1, 0, -1], [-1, 0, -1, -1, -1, 1], [0, 0, 1, 1, 1, 0], [1, -1, 0, 0, 1, -1]])])
 
$\left\langle \left(\begin{array}{rrrr} 2 & 2 & 2 & 0 \\ 2 & 1 & 0 & 2 \\ 2 & 2 & 0 & 1 \\ 2 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ 2 & 1 & 2 & 1 \\ 2 & 0 & 1 & 2 \\ 1 & 2 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 1 & 2 & 2 & 2 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 2 & 1 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 1 & 0 \\ 1 & 0 & 2 & 0 \\ 2 & 0 & 1 & 2 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 2 & 1 \\ 1 & 1 & 1 & 2 \\ 2 & 1 & 2 & 0 \\ 1 & 1 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 4, GF(3) | [[2, 2, 2, 0, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 1, 1], [0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 2, 1, 0], [2, 2, 0, 1, 0, 2, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0], [2, 0, 2, 0, 2, 1, 0, 2, 1, 0, 0, 0, 2, 0, 1, 1], [2, 2, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1], [0, 2, 1, 0, 1, 0, 2, 0, 2, 0, 1, 2, 1, 1, 2, 1], [2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0], [0, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 0, 1, 1, 1, 0]] >;
 
Copy content gap:G := Group([[[ Z(3), Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3)^0, Z(3)^0 ]], [[ 0*Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3)^0, Z(3) ], [ Z(3)^0, Z(3), Z(3)^0, 0*Z(3) ]], [[ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], [ Z(3)^0, Z(3), Z(3), Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ]], [[ Z(3), 0*Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ]], [[ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ]], [[ 0*Z(3), Z(3), Z(3)^0, 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0, Z(3) ], [ Z(3)^0, Z(3)^0, Z(3), Z(3)^0 ]], [[ Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ]], [[ 0*Z(3), Z(3), Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), Z(3)^0, Z(3), 0*Z(3) ], [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(3), 4, 4) G = MatrixGroup([MS([[2, 2, 2, 0], [2, 1, 0, 2], [2, 2, 0, 1], [2, 1, 1, 1]]), MS([[0, 1, 2, 1], [2, 1, 2, 1], [2, 0, 1, 2], [1, 2, 1, 0]]), MS([[2, 2, 0, 1], [0, 2, 0, 0], [1, 2, 2, 2], [2, 1, 0, 0]]), MS([[2, 0, 2, 0], [2, 1, 0, 2], [1, 0, 0, 0], [2, 0, 1, 1]]), MS([[2, 2, 0, 1], [0, 2, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]), MS([[0, 2, 1, 0], [1, 0, 2, 0], [2, 0, 1, 2], [1, 1, 2, 1]]), MS([[2, 0, 0, 1], [0, 1, 0, 0], [1, 0, 1, 1], [2, 0, 0, 0]]), MS([[0, 2, 2, 1], [1, 1, 1, 2], [2, 1, 2, 0], [1, 1, 1, 0]])])
 
Copy content sage_gap:G = gap.new('Group([[[ Z(3), Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3)^0, Z(3)^0 ]], [[ 0*Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3)^0, Z(3) ], [ Z(3)^0, Z(3), Z(3)^0, 0*Z(3) ]], [[ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], [ Z(3)^0, Z(3), Z(3), Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ]], [[ Z(3), 0*Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ]], [[ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ]], [[ 0*Z(3), Z(3), Z(3)^0, 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0, Z(3) ], [ Z(3)^0, Z(3)^0, Z(3), Z(3)^0 ]], [[ Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ]], [[ 0*Z(3), Z(3), Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), Z(3)^0, Z(3), 0*Z(3) ], [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ]]])')
 
Copy content oscar:G = matrix_group([matrix(GF(3), [[2, 2, 2, 0], [2, 1, 0, 2], [2, 2, 0, 1], [2, 1, 1, 1]]), matrix(GF(3), [[0, 1, 2, 1], [2, 1, 2, 1], [2, 0, 1, 2], [1, 2, 1, 0]]), matrix(GF(3), [[2, 2, 0, 1], [0, 2, 0, 0], [1, 2, 2, 2], [2, 1, 0, 0]]), matrix(GF(3), [[2, 0, 2, 0], [2, 1, 0, 2], [1, 0, 0, 0], [2, 0, 1, 1]]), matrix(GF(3), [[2, 2, 0, 1], [0, 2, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]), matrix(GF(3), [[0, 2, 1, 0], [1, 0, 2, 0], [2, 0, 1, 2], [1, 1, 2, 1]]), matrix(GF(3), [[2, 0, 0, 1], [0, 1, 0, 0], [1, 0, 1, 1], [2, 0, 0, 0]]), matrix(GF(3), [[0, 2, 2, 1], [1, 1, 1, 2], [2, 1, 2, 0], [1, 1, 1, 0]])])
 
Transitive group: 27T294 36T2302 more information
Copy content magma:G := TransitiveGroup(27, 294);
 
Copy content gap:G := TransitiveGroup(27, 294);
 
Copy content sage:G = TransitiveGroup(27, 294)
 
Copy content sage_gap:G = libgap.TransitiveGroup(27, 294)
 
Copy content oscar:G = transitive_group(27, 294)
 
Copy content magma:G := TransitiveGroup(36, 2302);
 
Copy content gap:G := TransitiveGroup(36, 2302);
 
Copy content sage:G = TransitiveGroup(36, 2302)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 2302)
 
Copy content oscar:G = transitive_group(36, 2302)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $\SU(3,2)$ $\,\rtimes\,$ $S_3$ $\He_3$ $\,\rtimes\,$ $\GL(2,3)$ $\Unitary(3,2)$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_3^2:S_3)$ . $S_4$ $C_3$ . $(C_3^2:\GL(2,3))$ more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

Abelianization: $C_{2} $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_1$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 
Copy content oscar:subgroups(G)
 

There are 2957 subgroups in 96 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $\He_3:\GL(2,3)$
Copy content comment:Center of the group
 
Copy content magma:Center(G);
 
Copy content gap:Center(G);
 
Copy content sage:G.center()
 
Copy content sage_gap:G.Center()
 
Copy content oscar:center(G)
 
Commutator: $G' \simeq$ $\Unitary(3,2)$ $G/G' \simeq$ $C_2$
Copy content comment:Commutator subgroup of the group G
 
Copy content magma:CommutatorSubgroup(G);
 
Copy content gap:DerivedSubgroup(G);
 
Copy content sage:G.commutator()
 
Copy content sage_gap:G.DerivedSubgroup()
 
Copy content oscar:derived_subgroup(G)
 
Frattini: $\Phi \simeq$ $C_3$ $G/\Phi \simeq$ $C_3^2:\GL(2,3)$
Copy content comment:Frattini subgroup of the group G
 
Copy content magma:FrattiniSubgroup(G);
 
Copy content gap:FrattiniSubgroup(G);
 
Copy content sage:G.frattini_subgroup()
 
Copy content sage_gap:G.FrattiniSubgroup()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $\He_3$ $G/\operatorname{Fit} \simeq$ $\GL(2,3)$
Copy content comment:Fitting subgroup of the group G
 
Copy content magma:FittingSubgroup(G);
 
Copy content gap:FittingSubgroup(G);
 
Copy content sage:G.fitting_subgroup()
 
Copy content sage_gap:G.FittingSubgroup()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $\He_3:\GL(2,3)$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Copy content oscar:solvable_radical(G)
 
Socle: $\operatorname{soc} \simeq$ $C_3$ $G/\operatorname{soc} \simeq$ $C_3^2:\GL(2,3)$
Copy content comment:Socle of the group G
 
Copy content magma:Socle(G);
 
Copy content gap:Socle(G);
 
Copy content sage:G.socle()
 
Copy content sage_gap:G.Socle()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3\wr C_3$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information

Click on a subgroup in the diagram to see information about it.

Series

Derived series $\He_3:\GL(2,3)$ $\rhd$ $\Unitary(3,2)$ $\rhd$ $\SU(3,2)$ $\rhd$ $C_3^2:S_3$ $\rhd$ $\He_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $\He_3:\GL(2,3)$ $\rhd$ $\Unitary(3,2)$ $\rhd$ $\SU(3,2)$ $\rhd$ $C_3^2:S_3$ $\rhd$ $\He_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $\He_3:\GL(2,3)$ $\rhd$ $\Unitary(3,2)$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 
Copy content oscar:upper_central_series(G)
 

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $18 \times 18$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.