#72271699

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13037 ⋊ C₂
Sylow	C₃ ⋊ #500001

Subgroups

	subgroup	G/H
center	C₂	#10827336
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13037	C₂
socle	C₂×C₃	#23960
hypercenter	#13037	D₃
hypocenter	C₃	#500001
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13037

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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=efi,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bf,c^a=ce,c^b=ceg,d^a=defgi,d^b=defgi,e^c=eh,e^d=ehi,f^a=fi,f^c=fi,f^d=fi,g^a=gi,g^b=gi,h^c=hi,h^d=hi,j^b=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

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