#72259271

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂⁵×S₄
commutator, cosocle	C₂×A₄	C₂⁶
Fitting	C₂³×H(5,𝔽₂)	D₃
socle	C₂³	C₂⁵×D₃
hypercenter	C₂×H(5,𝔽₂)	S₄
hypocenter	A₄	H(7,𝔽₂)
LCS	G ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

54775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3,2^2|a^2=g,c^a=cg,d^a=dg,d^b=dg,f^e=fg,h^b=h^2,h^e=h^2,i^b=j,i^e=j,i^h=j,j^b=i,j^e=i,j^h=ij

Conjugacy classes & character table

164 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 5, F.9);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);