#72272089

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#23882 ⋊ C₂
Sylow	C₃ ⋊ #72214173

Subgroups

	subgroup	G/H
center	C₂²	D₃×#2403
commutator, cosocle	C₂³×C₃×C₄	C₂⁵
Frattini	C₂³×C₄	C₂⁴×D₃
Fitting	C₂×C₃×#23882	C₂
socle	C₂²×C₃	C₂×#2403
hypercenter	C₂×#23882	D₃
hypocenter	C₃	#72214173
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#23882

At least 10154 conjugacy classes of subgroups; only the 1635 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|b^2=f,f^2=j,b^a=bg,c^b=cfj,d^b=dh,e^a=ehi,f^c=fj,g^e=gj,i^b=ij,k^d=k^2

Conjugacy classes & character table

204 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, 2, 2, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 4, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);