#72271643

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33468 ⋊ C₂
Sylow	C₃ ⋊ #499269

Subgroups

	subgroup	G/H
center	C₂	#10823408
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33468	C₂
socle	C₂×C₃	#23573
hypercenter	#33468	D₃
hypocenter	C₃	#499269
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #33468

1638 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)(6,18,20,19)(7,23)(11,27,28,31)(12,29)(15,30,26,32)(16,22)(17,33,34,35)(21,25);(2,3)(6,17)(7,22)(8,9)(11,26)(12,21)(13,24)(15,28)(16,23)(18,33)(19,35)(20,34)(25,29)(27,32)(30,31);(4,6,16,33,9,18,7,17,10,20,22,35,8,19,23,34)(5,11,25,30,13,27,12,15,14,28,21,32,24,31,29,26);(4,5)(6,15)(7,21)(8,13)(9,24)(10,14)(11,17)(12,22)(16,29)(18,30)(19,32)(20,26)(23,25)(27,33)(28,34)(31,35)

Polycyclic presentation:

2^9,3|b^2=ehi,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cg,c^b=cefi,d^b=dhi,e^a=eh,e^c=eh,e^d=ei,f^c=fi,g^b=gi,h^a=hi,h^c=hi,j^c=j^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#499269
maximal normal subgroup of	D₃×#499269

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