#72269809

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#11404 ⋊ C₂
Sylow	C₃ ⋊ #479908

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄²×D₃
commutator	C₃×C₄	C₂²×C₄×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#11404	C₂
socle	C₂²×C₃	C₂²×#97
prime radical	C₂×#235	C₄²
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#11404	D₃
hypocenter	C₃	#479908
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #11404

1268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 43 points:

(1,2)(12,14,15,24,16,22,26,35,17,23,25,36,27,34,38,37)(13,19,20,32,21,31,33,42,18,28,29,40,30,39,41,43);(2,3)(13,18)(19,28)(20,29)(21,30)(31,39)(32,40)(33,41)(42,43);(4,5,6,8,7,9,10,11)(14,22,23,34)(15,25)(19,31,28,39)(20,29)(24,37,36,35)(26,38)(32,43,40,42)(33,41);(2,3)(12,13)(14,19)(15,20)(16,21)(17,18)(22,31)(23,28)(24,32)(25,29)(26,33)(27,30)(34,39)(35,42)(36,40)(37,43)(38,41)

Polycyclic presentation:

2^9,3|b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,c^a=ci,d^b=dh,e^d=ei,f^b=fi,j^a=j^2,j^c=j^2,j^d=j^2

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

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