#72271578

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#29271 ⋊ C₂
Sylow	C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831708
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#29271	C₂
socle	C₂×C₃	#24338
prime radical	C₂×#5193	C₂²
hypercenter	#29271	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ #29271

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 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, 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);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
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, 2, F.8*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);