#72270355

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13459 ⋊ C₂
Sylow	C₃ ⋊ #500349

Subgroups

	subgroup	G/H
center	C₂	#10828762
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13459	C₂
socle	C₂×C₃	#24089
prime radical	#11674711	C₂
hypercenter	#13459	D₃
hypocenter	C₃	#500349
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13459

1464 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)(6,18,21,34)(7,24)(8,25)(10,27,29,33)(11,22)(12,13)(15,26,32,28)(16,23)(17,35,20,19)(30,31);(1,2,3)(6,17,21,20)(7,23)(10,26,29,28)(11,30)(15,27,32,33)(16,24)(18,35,34,19)(22,31);(4,6,16,34,25,17,23,35,9,21,24,18,8,20,7,19)(5,10,22,33,12,28,11,15,14,29,31,27,13,26,30,32);(4,5)(6,15,21,32)(7,22)(8,12)(9,14)(10,19,29,35)(11,24)(13,25)(16,30)(17,33,20,27)(18,26,34,28)(23,31)

Polycyclic presentation:

2^9,3|a^2=g,b^2=efi,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cg,c^b=cf,d^b=defh,e^a=eh,e^c=ei,e^d=eh,f^d=fi,g^b=gi,h^a=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×#500349, #72271743
maximal normal subgroup of	D₃×#500349, #72271743

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