#72258635

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#7702719 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	C₂×#560⋊C₃
commutator, cosocle	C₂×C₄²	C₂⁴×C₃
Frattini, socle	C₂³	C₂⁴×A₄
Fitting	#7702719	C₃
hypercenter	C₂×C₄	#560⋊C₃
hypocenter	C₄²	C₂×C₃×#41
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

2437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 48

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

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

Polycyclic presentation:

2^4,3,2^5|c^2=f,d^2=f,g^2=i,h^2=j,d^b=df,e^c=ei,g^a=gij,g^b=gj,g^e=hi,h^a=hi,h^b=hij,h^e=gh,i^e=j,j^e=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72258655

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