#72259267

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#2403 ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×S₄
commutator, cosocle	C₂³×A₄	C₂⁵
Fitting	C₂²×#2403	D₃
socle	C₂⁵	C₂⁴×D₃
hypercenter	#2403	S₄
hypocenter	A₄	D₄×#120
LCS	G ▷ C₂³×A₄ ▷ A₄
UCS	C₁ ◁ C₂³ ◁ #2403

At least 13020 conjugacy classes of subgroups; only the 899 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3,2^2|b^2=f,b^a=bg,c^a=cg,c^b=cf,e^d=eh,i^a=i^2,i^d=i^2,j^a=k,j^d=k,j^i=k,k^a=j,k^d=j,k^i=jk

Conjugacy classes & character table

178 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 4, F.11);;
gap> SetConjugate(coll, 10, 9, F.11);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 4, F.10);;
gap> SetConjugate(coll, 11, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);