#72259680

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×H(3,𝔽₃)² ⋊ C₂²
Sylow	H(3,𝔽₃)² ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×(C₃⋊D₃)⋊C₃²
commutator, cosocle	C₂×C₃⁴	C₂³×C₃²
Frattini, socle	C₂×C₃²	C₂×C₃²×D₃²
Fitting	C₂²×H(3,𝔽₃)²	C₂²
hypercenter	C₂²	(C₃⋊D₃)⋊C₃²
hypocenter	C₃⁴	C₂×C₃²×D₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

4330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 22 points:

(6,8)(7,10)(9,11)(12,13)(14,15,18,19,17,20)(16,22);(2,4)(5,6,12)(7,9,11)(8,10,13)(15,19,20)(16,21,22);(1,2,3,4)(15,16)(17,18)(19,21)(20,22)

Polycyclic presentation:

2^3,3^2,2,3^4|a^2=f,b^a=bf,g^a=g^2,g^c=g^2,g^d=gi,g^e=gi,h^a=g^2h,h^c=h^2,h^d=hj,h^e=hi^2,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

274 conjugacy classes — view the full character table.

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, 3, 3, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);