#72271540

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13005 ⋊ C₂
Sylow	C₃ ⋊ #500001

Subgroups

	subgroup	G/H
center	C₂	#10827342
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13005	C₂
socle	C₂×C₃	#23960
hypercenter	#13005	D₃
hypocenter	C₃	#500001
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13005

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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=eg,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bg,c^a=ce,c^b=cegi,d^a=defi,d^b=defi,d^c=dh,e^b=ei,e^c=eh,e^d=ehi,f^b=fi,f^c=fi,f^d=fi,g^a=gi,g^b=gi,h^c=hi,h^d=hi,j^b=j^2,j^c=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271549
maximal normal subgroup of	#72271549

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