#72259018

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×C₃×D₄ ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×S₄
commutator, cosocle	C₂²×C₃×A₄	C₂⁴
Fitting	C₂³×C₃×D₄	C₂×D₃
socle	C₂⁴×C₃	C₂³×D₃
hypercenter	C₂×D₄	D₃×S₄
hypocenter	C₃×A₄	C₂×#120
LCS	G ▷ C₂²×C₃×A₄ ▷ C₃×A₄
UCS	C₁ ◁ C₂² ◁ C₂×D₄

8750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

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

Conjugacy classes & character table

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