#72271722

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13373 ⋊ C₂
Sylow	C₃ ⋊ #500001

Subgroups

	subgroup	G/H
center	C₂	#10827339
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13373	C₂
socle	C₂×C₃	#23960
hypercenter	#13373	D₃
hypocenter	C₃	#500001
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13373

1828 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)(6,18,21,35)(7,25)(8,26)(10,29,31,34)(11,22)(12,13)(15,28,32,30)(16,24)(17,23,20,19)(27,33);(1,2,3)(6,17,21,20)(7,24)(10,28,31,30)(11,33)(15,29,32,34)(16,25)(18,23,35,19)(22,27);(4,6,16,18,26,17,24,19,9,21,25,35,8,20,7,23)(5,10,27,34,12,30,33,15,14,31,22,29,13,28,11,32);(4,5)(6,15,21,32)(7,22)(8,12)(9,14)(10,19,31,23)(11,25)(13,26)(16,33)(17,34,20,29)(18,28,35,30)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271743
maximal normal subgroup of	#72271743

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