#72269837

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁵×C₈ ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊S₄
commutator, cosocle	C₂×C₄×A₄	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂⁵×C₈	D₃
socle	C₂⁴	C₂²×D₁₂
hypercenter	C₂³×C₈	S₄
hypocenter	A₄	#3050
LCS	G ▷ C₂×C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4)(6,8);(9,11,18,24,13,20,12,19)(10,15,21,23,14,22,16,17);(10,14)(15,22)(16,21)(17,23);(2,3)(5,6)(7,8)(9,10)(11,17)(12,21)(13,14)(15,19)(16,18)(20,23)(22,24)

Polycyclic presentation:

2^7,3,2^2|c^2=eg,e^2=g,b^a=bg,c^a=ce,d^a=df,e^a=eg,h^a=h^2,i^a=j,i^h=j,j^a=i,j^h=ij

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72269839

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