#72228652

order	2916
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₃)² ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₃²	#66472
commutator, Fitting, hypocenter	H(3,𝔽₃)²	C₄
prime radical, cosocle	#13091502	C₂
LCS	G ▷ H(3,𝔽₃)²
UCS	C₁ ◁ C₃²

781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,8,9,15)(4,11)(5,7)(6,10,13,16)(12,17,18,14)(19,20)(21,24,26,31)(22,28,27,32)(23,25)(29,33,35,36)(30,34);(1,12,9)(2,13,14)(3,16,5)(4,10,18)(6,11,8)(7,17,15)(19,23,30)(20,32,33)(22,26,29)(24,28,34)(25,36,31);(19,23,30)(20,32,33)(22,26,29)(24,28,34)(25,36,31)

Polycyclic presentation:

2^2,3^6|a^2=b,c^a=ce,c^b=c^2,d^a=df,d^b=d^2,e^a=ce^2h^2,e^b=e^2h,e^c=eh,e^d=eh,f^a=df^2g^2,f^b=f^2g,f^c=fh,f^d=fg

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228655, #72228680
maximal normal subgroup of	#72228655, #72228680

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.3*F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.4*F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);