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Cayley diagram

C7×AGL(1,𝔽9) #92738

order504
isoclinismisoclinic class of AGL(1,𝔽9)
automorphismAut(G) = C2×C3×AΓL(1,𝔽9) (order 864) · Inn(G) = AGL(1,𝔽9) · Out(G) = C22×C3
homologyAbelianization (H1) = C7×C8 · Schur multiplier (H2) = C1

Aliases

direct productC7×AGL(1,𝔽9)

Splittings

Fitting, socle, HallC32×C7 ⋊ C8
derived, SylowC32 ⋊ C7×C8

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽9)
commutator, hypocenterC32C7×C8
FrattiniC1G
Fitting, socleC32×C7C8
prime radicalC7×C3⋊D3C4
cosocle#157C2×C7
LCSG ▷ C32
UCSC1 ◁ C7

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,𝔽9), C7×AΓL(1,𝔽9), AGL(1,𝔽9)⋊D7, …
maximal normal subgroup ofD7×AGL(1,𝔽9), C7×AΓL(1,𝔽9), AGL(1,𝔽9)⋊D7, …
maximal quotient ofC7×#3771, C2×C7×AGL(1,𝔽9), C7×#6249, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 164);