#72270518

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13400 ⋊ C₂
Sylow	C₃ ⋊ #500168

Subgroups

	subgroup	G/H
center	C₂	#10828138
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13400	C₂
socle	C₂×C₃	#24033
prime radical	#11671075	C₂
hypercenter	#13400	D₃
hypocenter	C₃	#500168
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13400

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

Polycyclic presentation:

2^9,3|b^2=e,c^2=g,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ceh,d^b=defghi,e^c=eh,e^d=ehi,f^a=fi,f^b=fi,f^c=fi,f^d=fi,g^b=gi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168, #72271782
maximal normal subgroup of	D₃×#500168, #72271782

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