#72253282

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#32878 ⋊ C₂
Sylow	C₃ ⋊ #6628842

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃×D₄
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#32878	C₂
socle	C₂³×C₃	C₂³×D₄
hypercenter	#32878	D₃
hypocenter	C₃	#6628842
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ #32878

59586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|g^2=i,b^a=bh,c^a=cg,d^a=df,e^a=ei,g^a=gi,g^c=gi,j^e=j^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6628842
maximal normal subgroup of	D₃×#6628842

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