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Cayley diagram

C2×AΓL(1,𝔽9) #64379

order288
isoclinismisoclinic class of AΓL(1,𝔽9)
automorphismAut(G) = C22×AΓL(1,𝔽9) (order 576) · Inn(G) = AΓL(1,𝔽9) · Out(G) = C22
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×AΓL(1,𝔽9)
unambiguous semidirect productAGL(1,𝔽9)⋊C22
unambiguous semidirect productPSU(3,𝔽2)⋊C22

Splittings

Fitting, socleC2×C32 ⋊ SD16
SylowC32 ⋊ C2×SD16

Subgroups

subgroupG/H
center, hypercenterC2AΓL(1,𝔽9)
commutator, cosocle#157C23
FrattiniC1G
Fitting, socleC2×C32SD16
prime radicalC2×GO+(4,𝔽2)C2
hypocenterC32C2×SD16
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AΓL(1,𝔽9), #10596408, #10596409, …
maximal normal subgroup ofC4×AΓL(1,𝔽9), #10596408, #10596409, …
maximal quotient of#10596402, GO+(4,𝔽2)⋊Q8, #10596404, …
automorphism group of#622, C2×#157, C3×#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1027);