#72225831

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈×M₃₂ ⋊ C₂
Sylow	C₃ ⋊ #483512

Subgroups

	subgroup	G/H
center	C₈	C₂²×D₂₄
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×C₈×M₃₂	C₂
socle	C₂×C₃	C₂×C₈×D₈
prime radical	#82009	C₄
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	C₈×M₃₂	D₃
hypocenter	C₃	#483512
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ C₈×M₃₂

962 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)(4,6,7,18,8,19,20,35,9,17,21,33,22,34,30,26)(5,12,14,28,15,29,23,32,10,24,25,31,13,27,11,16);(5,11,13,25,10,23,15,14)(12,16,27,31,24,32,29,28);(5,10)(6,17)(11,23)(13,15)(14,25)(18,33)(19,34)(26,35);(2,3)(4,5)(6,16)(7,11)(8,13)(9,10)(12,26)(14,30)(15,22)(17,32)(18,27)(19,31)(20,25)(21,23)(24,35)(28,34)(29,33)

Polycyclic presentation:

2^9,3|c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bi,c^a=cf,d^a=dfhi,d^b=di,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,j^a=j^2

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#483512
maximal normal subgroup of	D₃×#483512

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