#72271634

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13417 ⋊ C₂
Sylow	C₃ ⋊ #499269

Subgroups

	subgroup	G/H
center	C₂	#10823411
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13417	C₂
socle	C₂×C₃	#23573
hypercenter	#13417	D₃
hypocenter	C₃	#499269
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13417

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,d^2=ghi,e^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=cefgh,d^b=dh,d^c=dhi,e^c=eh,e^d=ei,f^a=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, #72271714
maximal normal subgroup of	D₃×#499269, #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, 4, F.7*F.8*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.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*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, 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);