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Cayley diagram

C23×AGL(1,ℤ/9) #88912

order432
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC23×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C23×C3)
unambiguous semidirect product3−1+2⋊C24
unambiguous semidirect product(C2×3−1+2)⋊C23

Splittings

FittingC23×3−1+2 ⋊ C2
derivedC9 ⋊ C24×C3
Sylow3−1+2 ⋊ C24

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C24×C3
FrattiniC3C23×C3×D3
FittingC23×3−1+2C2
socleC23×C3C3×D3
LCSG ▷ C9
UCSC1 ◁ C23

418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 4 generators on 15 points:

(8,12,15,10,11,14)(9,13);(5,6)(7,15,10,13,11,12,9,8,14);(3,4);(1,2)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22⋊C4×AGL(1,ℤ/9), #63444⋊C3, C2×(C2×AGL(1,ℤ/9))⋊C4, …
maximal normal subgroup ofC22⋊C4×AGL(1,ℤ/9), #63444⋊C3, C2×(C2×AGL(1,ℤ/9))⋊C4, …
maximal quotient ofC22×3−1+2⋊Q8, C22×C4×AGL(1,ℤ/9), C22×C4⋊AGL(1,ℤ/9), …
automorphism group ofC8×D9, #3663, C3×C9⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 559);