#72271642

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13396 ⋊ C₂
Sylow	C₃ ⋊ #499269

Subgroups

	subgroup	G/H
center	C₂	#10823407
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13396	C₂
socle	C₂×C₃	#23573
hypercenter	#13396	D₃
hypocenter	C₃	#499269
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13396

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 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);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, 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, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);