#72270350

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12934 ⋊ C₂
Sylow	C₃ ⋊ #500349

Subgroups

	subgroup	G/H
center	C₂	#10828765
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12934	C₂
socle	C₂×C₃	#24089
prime radical	#11674709	C₂
hypercenter	#12934	D₃
hypocenter	C₃	#500349
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12934

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500349, #72271549
maximal normal subgroup of	D₃×#500349, #72271549

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, 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.5*F.8*F.9);;
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.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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, 3, 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);