#72253127

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#85486
commutator	C₂×#41⋊C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#3573	D₃
prime radical	C₂×#5945	C₂²
cosocle	C₂²×#41⋊C₃	C₂³
hypercenter	C₂×#131	S₄
hypocenter	SL(2,𝔽₃)	#380
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

1883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

(5,22,31,7,28,51,12,52,29,23,11,49)(6,36,43,14,44,62,18,59,41,37,17,60)(8,48,54,20,27,65,9,46,53,19,25,66)(10,50,30,21)(13,56,58,32,40,68,15,64,57,38,34,67)(16,61,42,35)(24,45,26,47)(33,63,39,55);(5,6,8,13,12,18,9,15)(7,14,20,32,23,37,19,38)(10,16,24,33,30,42,26,39)(11,17,25,34,28,44,27,40)(21,35,47,55,50,61,45,63)(22,36,48,56,52,59,46,64)(29,41,53,57,31,43,54,58)(49,60,66,67,51,62,65,68);(1,2,3,4)(6,13)(7,19)(8,9)(10,28)(11,30)(15,18)(16,40)(17,39)(20,23)(21,48)(22,47)(24,25)(26,27)(29,31)(32,38)(33,44)(34,42)(35,59)(36,61)(41,58)(43,57)(45,52)(46,50)(49,66)(51,65)(55,56)(60,62)(63,64)

Polycyclic presentation:

2^6,3,2^3|a^2=d,b^2=e,c^2=j,e^2=j,f^2=j,h^2=j,i^2=j,b^a=be,c^a=cf,e^a=ej,f^a=fj,f^c=fj,g^a=g^2,h^a=ij,h^g=hi,i^a=hj,i^g=h,i^h=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72253130

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.8);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);