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Cayley diagram

C4×PSU(3,𝔽2) #64244

order288
isoclinismisoclinic class of PSU(3,𝔽2)
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productC4×PSU(3,𝔽2)

Splittings

FittingC32×C4 ⋊ Q8
SylowC32 ⋊ C4×Q8

Subgroups

subgroupG/H
center, hypercenterC4PSU(3,𝔽2)
commutatorC3⋊D3C22×C4
FrattiniC2C2×PSU(3,𝔽2)
FittingC32×C4Q8
socleC2×C32C2×Q8
prime radical, cosocleC2×C3⋊D3C23
hypocenterC32C4×Q8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C4

78 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:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup of#106×PSU(3,𝔽2)⋊C3, C2×D8×PSU(3,𝔽2)⋊C3, #108×PSU(3,𝔽2)⋊C3, …
maximal subgroup ofPSU(3,𝔽2)⋊C8, C8×PSU(3,𝔽2), #10593532, …
maximal normal subgroup ofPSU(3,𝔽2)⋊C8, C8×PSU(3,𝔽2), #10593532, …
maximal quotient ofC8×PSU(3,𝔽2), #10593532, #10593548, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 892);