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Cayley diagram

C22×PSU(3,𝔽2) #64384

order288
isoclinismisoclinic class of PSU(3,𝔽2)
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C3

Aliases

direct productC22×PSU(3,𝔽2)

Splittings

Fitting, socleC22×C32 ⋊ Q8
SylowC32 ⋊ C22×Q8

Subgroups

subgroupG/H
center, hypercenterC22PSU(3,𝔽2)
commutator, cosocleC3⋊D3C24
FrattiniC1G
Fitting, socleC22×C32Q8
prime radicalC22×C3⋊D3C22
hypocenterC32C22×Q8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 13 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC22⋊(PSU(3,𝔽2)⋊C3), #10593570⋊C3, C2×C22⋊(PSU(3,𝔽2)⋊C3), …
maximal subgroup of#10593555, #10593556, #10593570, …
maximal normal subgroup of#10593555, #10593556, #10593570, …
maximal quotient ofC2×#64243, C2×C4×PSU(3,𝔽2), C2×C4⋊PSU(3,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1032);