#72271500

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13030 ⋊ C₂
Sylow	C₃ ⋊ #500003

Subgroups

	subgroup	G/H
center	C₂	#10827336
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13030	C₂
socle	C₂×C₃	#23960
hypercenter	#13030	D₃
hypocenter	C₃	#500003
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13030

1840 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)(6,16,15,34)(7,19)(9,20)(10,24,14,32)(12,29)(13,28)(17,23,35,18)(21,22)(25,31,33,26)(27,30);(2,3)(5,11)(6,15)(10,14)(12,27)(13,28)(16,34)(17,35)(18,23)(24,32)(25,33)(26,31)(29,30);(4,6)(5,10)(7,16)(8,15)(9,23)(11,14)(12,24)(13,31)(17,22)(18,20)(19,35)(21,34)(25,30)(26,28)(27,32)(29,33);(4,5,8,11)(6,14)(7,12,21,27)(9,13,20,28)(10,15)(16,32)(17,33)(18,31)(19,29,22,30)(23,26)(24,34)(25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271549, #72271714
maximal normal subgroup of	#72271549, #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, 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.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, 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, 2, F.8*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);