#72270260

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13291 ⋊ C₂
Sylow	C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831704
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13291	C₂
socle	C₂×C₃	#24338
prime radical	C₂×#4908	C₂²
hypercenter	#13291	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13291

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(2,3)(6,18)(7,16)(8,24)(10,26)(11,30)(12,13)(15,28)(17,34)(19,33)(20,35)(21,31)(22,23)(25,29)(27,32);(1,2,3)(6,17,20,33)(7,22)(8,24)(10,25,28,32)(11,21)(12,13)(15,27,26,29)(16,23)(18,34,35,19)(30,31);(4,6,16,33,24,34,23,35,9,20,22,17,8,19,7,18)(5,10,21,25,12,27,11,15,14,28,30,32,13,29,31,26);(4,5)(6,15,20,26)(7,21)(8,12)(9,14)(10,18,28,35)(11,22)(13,24)(16,31)(17,29,33,27)(19,32,34,25)(23,30)

Polycyclic presentation:

2^9,3|a^2=g,b^2=efi,c^2=g,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cg,c^b=cei,d^b=defi,e^a=eh,e^c=eh,e^d=ehi,f^c=fi,f^d=fi,g^b=gi,h^a=hi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270277, D₃×#500940
maximal normal subgroup of	#72270277, D₃×#500940

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