#72225945

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12506 ⋊ C₂
Sylow	C₃ ⋊ #491500

Subgroups

	subgroup	G/H
center	C₂²	D₃×#493
commutator, cosocle	C₃×C₄×C₈	C₂⁴
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#12506	C₂
socle	C₂²×C₃	C₂×#493
hypercenter	#12506	D₃
hypocenter	C₃	#491500
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #12506

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 35 points:

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

Polycyclic presentation:

2^9,3|b^2=fh,c^2=fhi,d^2=eg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cfghi,c^b=ci,d^a=de,d^c=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,j^b=j^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#491500
maximal normal subgroup of	D₃×#491500

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