#72270385

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12741 ⋊ C₂
Sylow	C₃ ⋊ #499742

Subgroups

	subgroup	G/H
center	C₂	#10826186
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12741	C₂
socle	C₂×C₃	#23848
hypercenter	#12741	D₃
hypocenter	C₃	#499742
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12741

2716 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)(6,14)(7,19)(9,20)(10,23)(11,26)(13,27)(15,16)(17,29)(18,32)(21,33)(22,34)(24,35)(25,30)(28,31);(2,3)(10,22)(23,34)(24,30)(25,35);(4,6,7,15,20,31,33,29,8,17,21,28,9,16,19,14)(5,10,11,24,27,35,32,23,12,22,18,30,13,25,26,34);(4,5)(6,10)(7,18)(8,12)(9,13)(11,21)(14,23)(15,30)(16,25)(17,22)(19,32)(20,27)(24,28)(26,33)(29,34)(31,35)

Polycyclic presentation:

2^9,3|b^2=e,e^2=hi,h^2=i,b^a=bg,c^a=cgi,c^b=cf,d^b=def,e^a=ei,e^d=eh,f^a=fi,g^b=gi,h^d=hi,j^c=j^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270453
maximal normal subgroup of	#72270453

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