#72271409

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13310 ⋊ C₂
Sylow	C₃ ⋊ #499269

Subgroups

	subgroup	G/H
center	C₂	#10823405
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13310	C₂
socle	C₂×C₃	#23573
hypercenter	#13310	D₃
hypocenter	C₃	#499269
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13310

1638 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)(6,17,19,35)(7,15)(10,27,28,33)(11,31)(14,32,29,26)(16,21,34,18)(20,30)(22,23);(1,2,3)(6,16)(7,22)(9,24)(10,26)(11,30)(12,25)(14,27)(15,23)(17,21)(18,35)(19,34)(20,31)(28,32)(29,33);(4,6,15,18,9,17,22,34,8,19,7,21,24,35,23,16)(5,10,11,29,12,27,20,32,13,28,31,14,25,33,30,26);(4,5)(6,14,19,29)(7,20)(8,13)(9,25)(10,18,28,21)(11,23)(12,24)(15,30)(16,33,34,27)(17,32,35,26)(22,31)

Polycyclic presentation:

2^9,3|a^2=g,b^2=ef,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cg,c^b=cef,d^b=dhi,e^a=eh,e^b=ei,e^c=ehi,e^d=ei,f^a=fi,f^b=fi,g^b=gi,h^a=hi,h^c=hi,j^d=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271460, D₃×#499269
maximal normal subgroup of	#72271460, D₃×#499269

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