#72228552

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10586097 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂²	C₂⁵×A₄
commutator, cosocle	C₂×Q₈	C₂⁵×C₃
Fitting	#10586097	C₃
hypercenter	#3499	A₄
hypocenter	Q₈	C₂³×C₃×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂² ◁ #3499

21857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^5,3,2^4|c^2=j,d^2=j,e^2=j,h^2=j,i^2=j,b^a=bg,c^a=cj,c^b=cj,e^d=ej,h^f=hi,i^f=h,i^h=ij

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228568
maximal normal subgroup of	#72228568

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