#72222733

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×M₃₂ ⋊ C₂²
Sylow	C₃² ⋊ C₂×#3407

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂³×D₃²
commutator, socle	C₂×C₃²	C₂⁴×C₈
Fitting	C₂×C₃²×M₃₂	C₂²
prime radical	C₂×#64306	C₄
cosocle	C₃²×C₈	C₂⁵
hypercenter	C₂×M₃₂	D₃²
hypocenter	C₃²	C₂×#3407
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

4348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=f,f^2=g,g^2=h,c^a=ch,c^b=ch,e^a=eh,i^b=i^2,i^d=i^2,j^b=i^2j,j^d=j^2

Conjugacy classes & character table

312 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228223, #72228226
maximal normal subgroup of	#72228223, #72228226

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