#72271757

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13417 ⋊ C₂
Sylow	C₃ ⋊ #500650

Subgroups

	subgroup	G/H
center	C₂	#10830067
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13417	C₂
socle	C₂×C₃	#24196
hypercenter	#13417	D₃
hypocenter	C₃	#500650
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13417

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=fg,d^2=gi,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=ce,d^a=di,d^b=defgi,d^c=dhi,e^a=ei,e^b=eh,e^c=eh,e^d=ehi,f^a=fi,f^b=fi,g^b=gi,h^b=hi,h^c=hi,h^d=hi,j^b=j^2,j^d=j^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271782
maximal normal subgroup of	#72271782

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