#72252279

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #32888

Subgroups

	subgroup	G/H
center	C₂²	C₂×#63977
commutator, cosocle	C₂²×C₃²×C₄	C₂⁴
Frattini	C₂²×C₄	C₂²×D₃²
Fitting	C₂³×C₃²×D₄	C₂²
socle	C₂²×C₃²	C₂×#120
hypercenter	C₂³×D₄	D₃²
hypocenter	C₃²	#32888
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×D₄

8814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^8,3^2|b^2=e,g^2=h,b^a=bg,c^b=cf,d^b=de,e^a=eh,g^a=gh,g^d=gh,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252285
maximal normal subgroup of	#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.5);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);