#72227358

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×#108 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#2957

Subgroups

	subgroup	G/H
center	C₄	C₂×D₃²×D₄
commutator	C₃²×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂³×D₃²
Fitting	C₂×C₃²×#108	C₂²
socle	C₂×C₃²	C₂²×C₄×D₄
prime radical	C₂×#10594684	C₂
cosocle	C₂×C₃²×C₄	C₂⁵
hypercenter	C₂×#108	D₃²
hypocenter	C₃²	C₂×#2957
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

5524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 38 points:

(1,4,5,18,6,19,12,27)(2,9,10,25,8,16,21,30)(3,13,15,17,14,29,20,31)(7,23,11,24,22,32,26,28)(33,34,35)(36,37,38);(34,35)(37,38);(2,8)(7,22)(9,16)(10,21)(11,26)(23,32)(24,28)(25,30);(1,3,12,15,6,14,5,20)(2,7,21,11,8,22,10,26)(4,17,27,29,19,31,18,13)(9,24,30,32,16,28,25,23);(1,2)(3,11,14,26)(4,16)(5,10)(6,8)(7,15,22,20)(9,19)(12,21)(13,28,29,24)(17,23,31,32)(18,30)(25,27)(37,38)

Polycyclic presentation:

2^8,3^2|a^2=f,b^2=fgh,e^2=g,g^2=h,b^a=bg,c^a=ch,e^a=eh,e^b=eg,f^b=fh,g^b=gh,i^a=i^2,i^d=i^2,j^a=i^2j,j^d=j^2

Conjugacy classes & character table

204 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, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.7*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);