#72258824

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#7179 ⋊ C₂
Sylow	C₃ ⋊ #136361

Subgroups

	subgroup	G/H
center	C₂²	#68195
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×#7179	C₂
socle	C₂²×C₃	#1629
prime radical	C₂×#904	D₄
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#7179	D₃
hypocenter	C₃	#136361
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #7179

742 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)(4,6,7,5)(9,13,16,22)(10,18)(11,20)(14,15,17,21)(19,23);(2,3)(4,5,7,6)(10,18)(11,20)(13,21)(14,17)(15,22)(19,23);(2,3)(8,9)(10,13)(11,14)(12,16)(15,23)(17,20)(18,21)(19,22)

Polycyclic presentation:

2^9,3|b^2=d,c^2=dg,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=fh,g^a=gi,h^a=hi,h^b=hi,h^c=hi,j^a=j^2,j^b=j^2,j^c=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#136361
maximal normal subgroup of	D₃×#136361

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, 3, F.4*F.7);;
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);;
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> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);