Properties

Label 648.531
Order \( 2^{3} \cdot 3^{4} \)
Exponent \( 2^{2} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 3 \)
$\card{Z(G)}$ \( 3 \)
$\card{\Aut(G)}$ \( 2^{4} \cdot 3^{4} \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3 \)
Perm deg. $81$
Trans deg. $81$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(648, 531);
 
Copy content gap:G := SmallGroup(648, 531);
 
Copy content sage:G = PermutationGroup(['(1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52)', '(4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76)', '(4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81)', '(4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76)', '(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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78)', '(1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70)', '(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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80)'])
 
Copy content sage_gap:G = libgap.SmallGroup(648, 531)
 
Copy content oscar:G = small_group(648, 531)
 

Group information

Description:$\He_3.\SL(2,3)$
Order: \(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
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: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
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:$C_3^3:\GL(2,3)$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
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 3, $C_3$ 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:$5$
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 9 12 18
Elements 1 9 26 54 18 216 108 216 648
Conjugacy classes   1 1 3 1 2 8 2 6 24
Divisions 1 1 2 1 1 2 1 1 10
Autjugacy classes 1 1 2 1 1 2 1 1 10

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 9 16 18 36
Irr. complex chars.   3 3 7 0 6 3 2 0 0 0 24
Irr. rational chars. 1 2 1 1 0 1 0 1 2 1 10

Minimal presentations

Permutation degree:$81$
Transitive degree:$81$
Rank: $2$
Inequivalent generating pairs: $192$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 3 6 18
Arbitrary 3 6 18

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: ${\langle a, b, c, d, e, f \mid c^{4}=d^{3}=e^{3}=f^{3}=[a,f]=[b,f]=[c,f]= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([7, -3, -2, 2, -2, -3, 3, -3, 3024, 253, 64, 443, 135, 58, 12604, 7011, 1698, 1425, 16133, 8748, 1867, 530, 915]); a,b,c,d,e,f := Explode([G.1, G.2, G.3, G.5, G.6, G.7]); AssignNames(~G, ["a", "b", "c", "c2", "d", "e", "f"]);
 
Copy content gap:G := PcGroupCode(79270167234778603226629307883506561358102677644231,648); a := G.1; b := G.2; c := G.3; d := G.5; e := G.6; f := G.7;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(79270167234778603226629307883506561358102677644231,648)'); a = G.1; b = G.2; c = G.3; d = G.5; e = G.6; f = G.7;
 
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(79270167234778603226629307883506561358102677644231,648)'); a = G.1; b = G.2; c = G.3; d = G.5; e = G.6; f = G.7;
 
Copy content oscar:# This uses Oscar's interface to GAP, as Oscar (currently) has no native support for PC groups G = PcGroup(GAP.evalstr("PcGroupCode(79270167234778603226629307883506561358102677644231,648)")); a = gen(G, 1); b = gen(G, 2); c = gen(G, 3); d = gen(G, 5); e = gen(G, 6); f = gen(G, 7);
 
Permutation group:Degree $81$ $\langle(1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 81 | (1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52), (4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76), (4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81), (4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76), (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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78), (1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70), (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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80) >;
 
Copy content gap:G := Group( (1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52), (4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76), (4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81), (4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76), (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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78), (1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70), (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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80) );
 
Copy content sage:G = PermutationGroup(['(1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52)', '(4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76)', '(4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81)', '(4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76)', '(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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78)', '(1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70)', '(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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80)'])
 
Copy content sage_gap:G = gap.new('Group( (1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52), (4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76), (4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81), (4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76), (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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78), (1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70), (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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80) )')
 
Copy content oscar:G = @permutation_group(81, (1,56,29,2,57,30,3,55,28)(4,59,32,5,60,33,6,58,31)(7,62,35,8,63,36,9,61,34)(10,65,38,11,66,39,12,64,37)(13,68,41,14,69,42,15,67,40)(16,71,44,17,72,45,18,70,43)(19,74,47,20,75,48,21,73,46)(22,77,50,23,78,51,24,76,49)(25,80,53,26,81,54,27,79,52), (4,17,8,24)(5,18,9,22)(6,16,7,23)(10,13,21,25)(11,14,19,26)(12,15,20,27)(31,46,35,38)(32,47,36,39)(33,48,34,37)(40,50,52,43)(41,51,53,44)(42,49,54,45)(58,80,62,68)(59,81,63,69)(60,79,61,67)(64,70,75,77)(65,71,73,78)(66,72,74,76), (4,19,8,11)(5,20,9,12)(6,21,7,10)(13,23,25,16)(14,24,26,17)(15,22,27,18)(31,53,35,41)(32,54,36,42)(33,52,34,40)(37,43,48,50)(38,44,46,51)(39,45,47,49)(58,71,62,78)(59,72,63,76)(60,70,61,77)(64,67,75,79)(65,68,73,80)(66,69,74,81), (4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(31,35)(32,36)(33,34)(37,48)(38,46)(39,47)(40,52)(41,53)(42,54)(43,50)(44,51)(45,49)(58,62)(59,63)(60,61)(64,75)(65,73)(66,74)(67,79)(68,80)(69,81)(70,77)(71,78)(72,76), (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)(28,35,33)(29,36,31)(30,34,32)(37,43,40)(38,44,41)(39,45,42)(46,54,50)(47,52,51)(48,53,49)(55,68,79)(56,69,80)(57,67,81)(58,71,73)(59,72,74)(60,70,75)(61,65,76)(62,66,77)(63,64,78), (1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23)(28,40,53)(29,41,54)(30,42,52)(31,43,47)(32,44,48)(33,45,46)(34,37,50)(35,38,51)(36,39,49)(55,75,65)(56,73,66)(57,74,64)(58,78,68)(59,76,69)(60,77,67)(61,81,71)(62,79,72)(63,80,70), (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)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,51,50)(52,54,53)(55,57,56)(58,60,59)(61,63,62)(64,66,65)(67,69,68)(70,72,71)(73,75,74)(76,78,77)(79,81,80))
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $\SU(3,2)$ . $C_3$ $C_3$ . $\PU(3,2)$ $\He_3$ . $\SL(2,3)$ $(C_3^2:S_3)$ . $A_4$ more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization: $C_{3} $
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 252 subgroups in 26 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_3$ $G/Z \simeq$ $\PU(3,2)$
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: $G' \simeq$ $\SU(3,2)$ $G/G' \simeq$ $C_3$
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: $\Phi \simeq$ $C_3$ $G/\Phi \simeq$ $\PU(3,2)$
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: $\operatorname{Fit} \simeq$ $\He_3$ $G/\operatorname{Fit} \simeq$ $\SL(2,3)$
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: $R \simeq$ $\He_3.\SL(2,3)$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_3$ $G/\operatorname{soc} \simeq$ $\PU(3,2)$
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: $P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3.\He_3$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\He_3.\SL(2,3)$ $\rhd$ $\SU(3,2)$ $\rhd$ $C_3^2:S_3$ $\rhd$ $\He_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $\He_3.\SL(2,3)$ $\rhd$ $\SU(3,2)$ $\rhd$ $C_3^2:S_3$ $\rhd$ $\He_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $\He_3.\SL(2,3)$ $\rhd$ $\SU(3,2)$
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 $C_1$ $\lhd$ $C_3$
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 6 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 $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 3B 4A 6A 9A 9B 12A 18A
Size 1 9 2 24 54 18 72 144 108 216
2 P 1A 1A 3A 3B 2A 3A 9A 9B 6A 9A
3 P 1A 2A 1A 1A 4A 2A 3A 3A 4A 6A
Schur
648.531.1a 1 1 1 1 1 1 1 1 1 1
648.531.1b 2 2 2 2 2 2 −1 −1 2 −1
648.531.2a 2 2 −2 2 2 0 −2 −1 −1 0 1
648.531.2b 4 −4 4 4 0 −4 1 1 0 −1
648.531.3a 3 3 3 3 −1 3 0 0 −1 0
648.531.3b 18 −6 −9 0 6 3 0 0 −3 0
648.531.6a 36 12 −18 0 0 −6 0 0 0 0
648.531.8a 8 0 8 −1 0 0 2 −1 0 0
648.531.8b 16 0 16 −2 0 0 −2 1 0 0
648.531.9a 18 −6 −9 0 −2 3 0 0 1 0