#72270490

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#24404 ⋊ C₂
Sylow	C₃ ⋊ #501097

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5032
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#24404	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	#24404	D₃
hypocenter	C₃	#501097
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #24404

3084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=hi,f^2=i,c^a=ce,c^b=cefi,d^a=dg,d^b=dgh,d^c=dgi,e^a=ei,e^b=eh,f^a=fh,f^b=fh,f^d=fi,f^e=fi,h^c=hi,j^b=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270508, D₃×#501097, #72270771
maximal normal subgroup of	#72270508, D₃×#501097, #72270771

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