Finite Group Databasebeta

← Prev (#72243179) Next (#72243181) →

C3×3−1+2×AGL(1,ℤ/9) #72243180

order4374
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C35

Aliases

direct productC3×3−1+2×AGL(1,ℤ/9)

Splittings

Fitting, SylowC3×3−1+22 ⋊ C2

Subgroups

subgroupG/H
centerC32C32×AGL(1,ℤ/9)
commutator, cosocleC3×C9C2×C34
FrattiniC32C34×D3
FittingC3×3−1+22C2
socleC33C33×D3
prime radicalC33×AGL(1,ℤ/9)C3
hypercenterC3×3−1+2AGL(1,ℤ/9)
hypocenterC9C2×C32×3−1+2
LCSG ▷ C3×C9 ▷ C9
UCSC1 ◁ C32 ◁ C3×3−1+2

3832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,ℤ/9)×#6900, 3−1+2×#92265, AGL(1,ℤ/9)×C3⋊AGL(1,ℤ/9), …
maximal normal subgroup ofAGL(1,ℤ/9)×#6900, 3−1+2×#92265, AGL(1,ℤ/9)×C3⋊AGL(1,ℤ/9), …
maximal quotient ofAGL(1,ℤ/9)×#6900, 3−1+2×#92265, #72243574, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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