#72271404

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13037 ⋊ C₂
Sylow	C₃ ⋊ #500650

Subgroups

	subgroup	G/H
center	C₂	#10830069
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13037	C₂
socle	C₂×C₃	#24196
hypercenter	#13037	D₃
hypocenter	C₃	#500650
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13037

1558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271460
maximal normal subgroup of	#72271460

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