#72259072

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×#120 ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×S₄
commutator, cosocle	C₂²×A₄	C₂⁵
Fitting	C₂³×#120	D₃
socle	C₂⁴	C₂⁴×D₃
hypercenter	C₂×#120	S₄
hypocenter	A₄	#3484
LCS	G ▷ C₂²×A₄ ▷ A₄
UCS	C₁ ◁ C₂² ◁ C₂×#120

15893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3,2^2|c^a=cf,c^b=cg,e^d=eg,h^d=h^2,i^d=j,i^h=j,j^d=i,j^h=ij

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259102
maximal quotient of	#120×GL(2,ℤ/4)

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