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Cayley diagram

C4×AGL(1,𝔽9) #64215

order288
isoclinismisoclinic class of AGL(1,𝔽9)
automorphismAut(G) = D4×AΓL(1,𝔽9) (order 1152) · Inn(G) = AGL(1,𝔽9) · Out(G) = C2×D4
homologyAbelianization (H1) = C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC4×AGL(1,𝔽9)

Splittings

FittingC32×C4 ⋊ C8
derived, SylowC32 ⋊ C4×C8

Subgroups

subgroupG/H
center, hypercenterC4AGL(1,𝔽9)
commutator, hypocenterC32C4×C8
FrattiniC2C2×AGL(1,𝔽9)
FittingC32×C4C8
socleC2×C32C2×C8
prime radicalC2×C3⋊D3C2×C4
cosocleC2×#157C22
LCSG ▷ C32
UCSC1 ◁ C4

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGO+(4,𝔽2)⋊C8, PSU(3,𝔽2)⋊C8, AGL(1,𝔽9)⋊C8, …
maximal normal subgroup ofGO+(4,𝔽2)⋊C8, PSU(3,𝔽2)⋊C8, AGL(1,𝔽9)⋊C8, …
maximal quotient ofC8×AGL(1,𝔽9), #10593251, #10593254, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 863);