#72271717

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13460 ⋊ C₂
Sylow	C₃ ⋊ #500351

Subgroups

	subgroup	G/H
center	C₂	#10828762
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13460	C₂
socle	C₂×C₃	#24089
prime radical	#11674097	C₂
hypercenter	#13460	D₃
hypocenter	C₃	#500351
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13460

1460 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)(6,15,16,14)(7,19)(8,21)(10,22,24,23)(11,26)(13,27)(17,29,28,30)(18,31)(20,32)(25,35,33,34);(1,2,3)(6,14,16,15)(7,19)(8,21)(10,22,24,23)(11,26)(13,27)(17,30,28,29)(18,31)(20,32)(25,35,33,34);(4,6,7,15,8,17,19,30,9,16,20,14,21,28,32,29)(5,10,11,23,13,25,26,35,12,24,18,22,27,33,31,34);(4,5)(6,10,16,24)(7,18)(8,13)(9,12)(11,20)(14,23,15,22)(17,25,28,33)(19,31)(21,27)(26,32)(29,35,30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271743, #72271782
maximal normal subgroup of	#72271743, #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, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
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, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
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, 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);