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Cayley diagram

C9×AGL(1,𝔽8) #92627

order504
isoclinismisoclinic class of AGL(1,𝔽8)
automorphismAut(G) = C2×C3×AΓL(1,𝔽8) (order 1008) · Inn(G) = AGL(1,𝔽8) · Out(G) = C2×C32
homologyAbelianization (H1) = C7×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×AGL(1,𝔽8)
unambiguous semidirect product(C23×C9)⋊C7

Splittings

Fitting, HallC23×C9 ⋊ C7
derived, SylowC23 ⋊ C7×C9

Subgroups

subgroupG/H
center, hypercenterC9AGL(1,𝔽8)
commutator, hypocenterC23C7×C9
FrattiniC3C3×AGL(1,𝔽8)
FittingC23×C9C7
socle, cosocleC23×C3C3×C7
prime radicalC3×AGL(1,𝔽8)C3
LCSG ▷ C23
UCSC1 ◁ C9

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×AGL(1,𝔽8), C2×C9×AGL(1,𝔽8), AGL(1,𝔽8)⋊C27, …
maximal normal subgroup ofD9×AGL(1,𝔽8), C2×C9×AGL(1,𝔽8), AGL(1,𝔽8)⋊C27, …
maximal quotient ofC2×C9×AGL(1,𝔽8), C27×AGL(1,𝔽8), C3×C9×AGL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 53);