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Cayley diagram

C22×AGL(1,𝔽9) #64382

order288
isoclinismisoclinic class of AGL(1,𝔽9)
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,𝔽9)

Splittings

Fitting, socleC22×C32 ⋊ C8
derived, SylowC32 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽9)
commutator, hypocenterC32C22×C8
FrattiniC1G
Fitting, socleC22×C32C8
prime radicalC22×C3⋊D3C4
cosocle#157C23
LCSG ▷ C32
UCSC1 ◁ C22

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593166, #10593267, C2×GO+(4,𝔽2)⋊C4, …
maximal normal subgroup of#10593166, #10593267, C2×GO+(4,𝔽2)⋊C4, …
maximal quotient ofC2×#64213, C2×#64214, #10596481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1030);