#72259042

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#386 ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×S₄
commutator, cosocle	C₂³×A₄	C₂⁴
Fitting	C₂²×#386	D₃
socle	C₂⁵	C₂³×D₃
hypercenter	#386	S₄
hypocenter	A₄	#2410
LCS	G ▷ C₂³×A₄ ▷ A₄
UCS	C₁ ◁ C₂³ ◁ #386

8678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^7,3,2^2|b^2=e,b^a=bf,c^a=cf,c^b=ce,d^a=dg,d^c=dg,h^c=h^2,i^c=j,i^h=j,j^c=i,j^h=ij

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259267
maximal normal subgroup of	#72259267

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