#72228272

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #134947

Subgroups

	subgroup	G/H
center	C₂	#10688231
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×#33443	C₂
socle	C₂×C₃	#8642
prime radical	C₂×C₃×#135	D₄
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#33443	D₃
hypocenter	C₃	#134947
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #33443

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228279
maximal normal subgroup of	#72228279

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