#72271687

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#61414 ⋊ C₂
Sylow	C₃ ⋊ #500650

Subgroups

	subgroup	G/H
center	C₂	#10830066
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#61414	C₂
socle	C₂×C₃	#24196
hypercenter	#61414	D₃
hypocenter	C₃	#500650
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ #61414

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=cefgh,d^b=deg,e^c=eh,e^d=ehi,f^a=fi,g^b=gi,h^c=hi,h^d=hi,j^b=j^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271714
maximal normal subgroup of	#72271714

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