Finite Group Databasebeta

← Prev (#6273) Next (#6275) →

Cayley diagram

C22×AGL(1,ℤ/9) #6274

order216
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = S4×AGL(1,ℤ/9) (order 1296) · Inn(G) = AGL(1,ℤ/9) · Out(G) = S4
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C22×C3)
unambiguous semidirect product3−1+2⋊C23
unambiguous semidirect product(C2×3−1+2)⋊C22

Splittings

FittingC22×3−1+2 ⋊ C2
derivedC9 ⋊ C23×C3
Sylow3−1+2 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C23×C3
FrattiniC3C22×C3×D3
FittingC22×3−1+2C2
socleC22×C3C3×D3
LCSG ▷ C9
UCSC1 ◁ C22

101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

(6,10,13,8,9,12)(7,11);(3,4)(5,13,8,11,9,10,7,6,12);(1,2)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,ℤ/9))⋊C4, C2×C4×AGL(1,ℤ/9), C2×C4⋊AGL(1,ℤ/9), …
maximal normal subgroup of(C2×AGL(1,ℤ/9))⋊C4, C2×C4×AGL(1,ℤ/9), C2×C4⋊AGL(1,ℤ/9), …
maximal quotient ofC2×3−1+2⋊Q8, C2×C4×AGL(1,ℤ/9), C2×C4⋊AGL(1,ℤ/9), …
automorphism group ofC9⋊C8, C4×D9, #611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 111);