#72252256

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#534 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#3026

Subgroups

	subgroup	G/H
center	C₂	C₂×#10595278
commutator, cosocle	C₂³×C₃²	C₂⁵
Frattini	C₂³	C₂³×D₃²
Fitting	C₃²×#534	C₂²
socle	C₂×C₃²	C₂²×#120
hypercenter	#534	D₃²
hypocenter	C₃²	C₂×#3026
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #534

11628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^8,3^2|a^2=h,d^2=h,d^a=dg,d^b=dh,d^c=df,f^a=fh,g^c=gh,i^a=i^2,i^e=i^2,j^a=i^2j,j^e=j^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252356
maximal normal subgroup of	#72252356
maximal quotient of	#72252284

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.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
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);