#72271760

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13310 ⋊ C₂
Sylow	C₃ ⋊ #500001

Subgroups

	subgroup	G/H
center	C₂	#10827333
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13310	C₂
socle	C₂×C₃	#23960
hypercenter	#13310	D₃
hypocenter	C₃	#500001
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13310

1828 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)(6,17,19,34)(7,22)(8,25)(10,28,29,32)(11,21)(13,24)(14,30,27,35)(15,23)(16,20,18,33)(26,31);(6,16)(7,15)(8,25)(10,27)(11,26)(13,24)(14,29)(17,20)(18,19)(21,31)(22,23)(28,35)(30,32)(33,34);(4,6,15,17,25,33,23,18,9,19,22,34,8,20,7,16)(5,10,26,32,24,35,31,14,12,29,21,28,13,30,11,27);(2,3)(4,5)(6,14,19,27)(7,21)(8,24)(9,12)(10,18,29,16)(11,22)(13,25)(15,31)(17,30,34,35)(20,28,33,32)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271782
maximal normal subgroup of	#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, 1, F.7);;
gap> SetPower(coll, 2, F.5*F.7*F.8*F.9);;
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*F.8*F.9);;
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*F.9);;
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, 1, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);