#72258795

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#9837 ⋊ C₂
Sylow	C₃ ⋊ #136868

Subgroups

	subgroup	G/H
center	C₂²	#71281
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×#9837	C₂
socle	C₂²×C₃	#1619
prime radical	C₂×#71295	C₂
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#9837	D₃
hypocenter	C₃	#136868
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #9837

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(9,13)(10,19)(11,21)(14,15)(16,23)(17,22)(18,20);(4,5,6,7)(10,18)(11,21)(13,22)(14,17)(15,23)(19,20);(2,3)(8,9)(10,13)(11,14)(12,16)(15,19)(17,21)(18,22)(20,23)

Polycyclic presentation:

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

Conjugacy classes & character table

108 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.4);;
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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, 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, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);