#72270152

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #500940

Subgroups

	subgroup	G/H
center	C₂	#10831703
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13037	C₂
socle	C₂×C₃	#24338
prime radical	C₂×C₃×#509	C₂²
hypercenter	#13037	D₃
hypocenter	C₃	#500940
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13037

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,c^2=g,e^2=hi,f^2=i,h^2=i,b^a=bhi,c^a=cg,c^b=cf,d^b=de,e^a=ei,e^c=ei,e^d=eh,f^a=fi,f^d=fi,g^b=gi,h^d=hi,j^a=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270183, D₃×#500940
maximal normal subgroup of	#72270183, 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);;
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.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
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, 4, 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, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);