#72271426

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13005 ⋊ C₂
Sylow	C₃ ⋊ #500473

Subgroups

	subgroup	G/H
center	C₂	#10829225
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13005	C₂
socle	C₂×C₃	#24128
hypercenter	#13005	D₃
hypocenter	C₃	#500473
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13005

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=egh,c^2=g,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cg,c^b=ceg,d^b=def,e^a=eh,e^b=ei,e^c=eh,e^d=eh,f^a=fi,f^b=fi,g^b=gi,h^a=hi,h^c=hi,h^d=hi,j^c=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271460, #72271549
maximal normal subgroup of	#72271460, #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.7*F.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, 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, 2, 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);