#72268950

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×3₋¹⁺²×H(5,𝔽₂) ⋊ C₂
Sylow	3₋¹⁺² ⋊ #3498

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂⁵×C₃
Frattini, socle	C₂²×C₃	C₂⁴×C₃×D₃
Fitting	C₂×3₋¹⁺²×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#3498
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

5693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 5 generators on 21 points:

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

Polycyclic presentation:

2^5,3,2^2,3^2|d^2=h,i^3=j,b^a=bg,c^a=cg,d^a=dh,d^b=dh,e^c=eh,i^a=i^2j^2,i^f=ij,j^a=j^2

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#3498
maximal normal subgroup of	AGL(1,ℤ/9)×#3498

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