#72271562

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13269 ⋊ C₂
Sylow	C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831709
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13269	C₂
socle	C₂×C₃	#24338
prime radical	C₂×#4908	C₂²
hypercenter	#13269	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ #13269

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,c^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=cfhi,d^b=de,e^c=ei,e^d=eh,f^a=fi,g^b=gi,h^d=hi,j^d=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500940, #72271714
maximal normal subgroup of	D₃×#500940, #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, 3, 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.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 3, 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, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);