#72270646

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12759 ⋊ C₂
Sylow	C₃ ⋊ #501097

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12759	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	#12759	D₃
hypocenter	C₃	#501097
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #12759

3084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^9,3|a^2=h,c^2=g,f^2=i,c^a=cf,c^b=ce,d^a=di,d^b=dg,d^c=dg,e^a=eh,f^b=fhi,f^c=fi,f^e=fi,h^c=hi,j^d=j^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270741, #72270758, D₃×#501097
maximal normal subgroup of	#72270741, #72270758, D₃×#501097

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