#72270320

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33657 ⋊ C₂
Sylow	C₃ ⋊ #500349

Subgroups

	subgroup	G/H
center	C₂	#10828758
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33657	C₂
socle	C₂×C₃	#24089
prime radical	#11674708	C₂
hypercenter	#33657	D₃
hypocenter	C₃	#500349
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #33657

1464 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)(4,7,8,24)(5,12,16,13)(6,18)(9,25,10,23)(11,27)(14,32,15,31)(17,22)(20,21)(26,33)(29,30);(1,2,3)(6,17)(7,23)(9,10)(11,26)(12,31)(13,32)(14,15)(18,19)(20,22)(21,35)(24,25)(27,28)(29,34)(30,33);(4,6,9,21,8,20,10,18)(5,11,14,29,16,30,15,27)(7,22,23,35,24,17,25,19)(12,26,31,28,13,33,32,34);(4,5)(6,11)(7,12)(8,16)(9,14)(10,15)(13,24)(17,33)(18,27)(19,34)(20,30)(21,29)(22,26)(23,31)(25,32)(28,35)

Polycyclic presentation:

2^9,3|b^2=g,d^2=ghi,e^2=h,g^2=i,h^2=i,c^a=cgh,c^b=ce,d^b=def,d^c=dhi,e^a=ei,e^b=eh,e^c=ehi,e^d=eh,f^a=fi,g^c=gi,g^d=gi,h^b=hi,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×#500349
maximal normal subgroup of	D₃×#500349

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