#72272080

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#511811 ⋊ C₂
Sylow	C₃ ⋊ #72216366

Subgroups

	subgroup	G/H
center	C₂	#72271204
commutator, cosocle	C₂²×C₃×D₄	C₂⁵
Frattini	C₂²×D₄	C₂⁴×D₃
Fitting	C₃×#511811	C₂
socle	C₂×C₃	#6628896
hypercenter	#511811	D₃
hypocenter	C₃	#72216366
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #511811

At least 7446 conjugacy classes of subgroups; only the 1131 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|b^2=j,e^2=i,g^2=j,b^a=bf,c^b=cf,d^a=di,d^b=dh,e^a=ei,e^b=eg,e^c=ei,f^e=fi,g^a=gi,g^b=gj,g^c=gi,g^f=gj,h^a=hj,i^b=ij,k^c=k^2

Conjugacy classes & character table

153 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.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);