#72270428

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12828 ⋊ C₂
Sylow	C₃ ⋊ #500473

Subgroups

	subgroup	G/H
center	C₂	#10829229
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12828	C₂
socle	C₂×C₃	#24128
hypercenter	#12828	D₃
hypocenter	C₃	#500473
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #12828

1534 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)(6,17,20,35)(7,16)(8,24)(10,28,27,34)(11,31)(12,13)(15,29,30,33)(18,21,23,19)(22,32)(25,26);(2,3)(10,27)(15,30)(28,34)(29,33);(4,6,16,35,8,21,7,23,9,20,25,17,24,19,26,18)(5,10,22,28,13,29,32,15,14,27,31,34,12,33,11,30);(4,5)(6,15)(7,22)(8,12)(9,14)(10,18)(11,25)(13,24)(16,32)(17,29)(19,34)(20,30)(21,28)(23,27)(26,31)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270453, #72271714
maximal normal subgroup of	#72270453, #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, 2, F.5*F.6*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.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, 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, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);