#72259031

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×C₃×D₄ ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃×S₄
commutator, cosocle	C₂×C₃×A₄	C₂⁵
Fitting	C₂³×C₃×D₄	C₂×D₃
socle	C₂³×C₃	C₂⁴×D₃
hypercenter	C₂×D₄	D₃×S₄
hypocenter	C₃×A₄	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×A₄ ▷ C₃×A₄
UCS	C₁ ◁ C₂ ◁ C₂×D₄

12370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

2^6,3^2,2^2|b^a=bf,c^a=cf,d^b=df,g^a=g^2,g^e=g^2,h^a=g^2h,h^e=h^2,i^e=j,i^h=j,j^e=i,j^h=ij

Conjugacy classes & character table

129 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, 3, 3, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 5, F.9);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);