#72226246

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33443 ⋊ C₂
Sylow	C₃ ⋊ #483512

Subgroups

	subgroup	G/H
center	C₈	C₂²×#213
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#33443	C₂
socle	C₂×C₃	C₂×C₈×D₈
prime radical	#83894	C₄
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#33443	D₃
hypocenter	C₃	#483512
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ #33443

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

Polycyclic presentation:

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

Conjugacy classes & character table

216 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, 1, F.9);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 4, 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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);