#72253102

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×S₄
commutator	C₂²×A₄	C₂²×C₈
Fitting	C₂⁵×C₈	D₃
socle	C₂⁴	C₂²×C₃⋊C₈
prime radical	C₂⁴×A₄	C₈
cosocle	C₂×C₈×A₄	C₂³
hypercenter	C₂³×C₈	S₄
hypocenter	A₄	#2112
LCS	G ▷ C₂²×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

2115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72253105

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.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
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);