#72259880

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

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

8818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Polycyclic presentation:

2^4,3^6|a^2=d,b^a=bd,e^c=e^2,f^a=f^2,f^c=f^2,f^e=fi,g^a=f^2g,g^c=g^2,g^e=gj,h^c=h^2,h^f=hi^2,h^g=hi,i^a=i^2,j^a=i^2j

Conjugacy classes & character table

220 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, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);