#72270170

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13318 ⋊ C₂
Sylow	C₃ ⋊ #500473

Subgroups

	subgroup	G/H
center	C₂	#10829221
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13318	C₂
socle	C₂×C₃	#24128
hypercenter	#13318	D₃
hypocenter	C₃	#500473
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13318

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270183, #72271782
maximal normal subgroup of	#72270183, #72271782

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, 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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*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.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
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, 3, 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);