#72226258

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12011 ⋊ C₂
Sylow	C₃ ⋊ #483512

Subgroups

	subgroup	G/H
center	C₈	C₂×D₃×D₈
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#12011	C₂
socle	C₂×C₃	C₂×C₈×D₈
prime radical	#82009	C₄
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#12011	D₃
hypocenter	C₃	#483512
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #12011

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

Polycyclic presentation:

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

Conjugacy classes & character table

192 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.5);;
gap> SetPower(coll, 2, F.5*F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.6);;
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, 4, 1, F.4*F.6*F.8);;
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);