#72225834

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #129159

Subgroups

	subgroup	G/H
center	C₈	#5106
commutator	C₂×C₃×C₈	C₂×C₄²
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×C₈×M₃₂	C₂
socle	C₂×C₃	#8283
prime radical	C₂³×C₃	#443
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	C₈×M₃₂	D₃
hypocenter	C₃	#129159
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ C₈×M₃₂

458 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)(4,6,18,34,9,19,23,35,10,21,8,20,7,22,24,28)(5,12,15,17,13,31,27,33,16,30,26,32,14,29,11,25);(5,11,14,26,16,27,13,15)(12,25,29,32,30,33,31,17);(2,3)(4,5,7,13,10,16,9,14)(6,17,19,25,21,32,22,33)(8,11,23,15,18,27,24,26)(12,28,29,34,30,35,31,20)

Polycyclic presentation:

2^9,3|a^2=d,b^2=e,c^2=fgi,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bf,c^a=cfgi,d^c=di,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,j^a=j^2

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226305, #72228279
maximal normal subgroup of	#72226305, #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, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.7*F.9);;
gap> SetPower(coll, 4, F.9);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, 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);