#72271503

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12953 ⋊ C₂
Sylow	C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831705
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12953	C₂
socle	C₂×C₃	#24338
prime radical	C₂×#5147	C₂²
hypercenter	#12953	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #12953

1426 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)(7,21)(8,22)(9,20)(11,25)(12,13)(15,28)(16,31)(18,33)(19,32)(23,29)(24,27)(26,35)(30,34);(2,3)(4,7,20,21,10,22,9,8)(5,12,28,13,14,29,15,23)(6,16,19,33,18,32,31,17)(11,24,27,25,26,30,34,35);(2,3)(4,6,8,16,20,31,7,17,10,18,21,32,9,19,22,33)(5,11,13,24,28,34,29,35,14,26,23,30,15,27,12,25);(4,5)(6,11,18,26)(7,12)(8,23)(9,15)(10,14)(13,21)(16,30,32,24)(17,25,33,35)(19,27,31,34)(20,28)(22,29)

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271549, D₃×#500940
maximal normal subgroup of	#72271549, D₃×#500940

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