#72252255

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×#120 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#2846

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃²
commutator, socle, cosocle	C₂³×C₃²	C₂⁵
Fitting	C₂×C₃²×#120	C₂²
hypercenter	C₂×#120	D₃²
hypocenter	C₃²	C₂×#2846
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×#120

68052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^8,3^2|b^2=f,b^a=bh,c^b=cg,d^b=df,i^a=i^2,i^e=i^2,j^a=i^2j,j^e=j^2

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72252284, #72252285

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