#72270488

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33659 ⋊ C₂
Sylow	C₃ ⋊ #500168

Subgroups

	subgroup	G/H
center	C₂	#10828134
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33659	C₂
socle	C₂×C₃	#24033
prime radical	#11671075	C₂
hypercenter	#33659	D₃
hypocenter	C₃	#500168
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #33659

1956 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)(6,15)(7,21)(9,23)(10,25)(11,30)(13,31)(14,32)(16,18)(17,33)(19,29)(20,22)(24,27)(26,34)(28,35);(1,2,3)(6,14,17,16)(7,20)(9,23)(10,24,28,26)(11,29)(13,31)(15,32,33,18)(19,30)(21,22)(25,27,35,34);(4,6,7,16,9,18,20,15,8,17,22,14,23,32,21,33)(5,10,11,26,13,27,29,35,12,28,19,24,31,34,30,25);(4,5)(6,10)(7,19)(8,12)(9,13)(11,22)(14,26)(15,25)(16,24)(17,28)(18,27)(20,30)(21,29)(23,31)(32,34)(33,35)

Polycyclic presentation:

2^9,3|b^2=e,c^2=gi,e^2=h,h^2=i,b^a=bg,c^a=cgi,c^b=ceh,d^b=def,e^a=ei,e^c=eh,e^d=ehi,f^a=fi,g^b=gi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168
maximal normal subgroup of	D₃×#500168

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