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Cayley diagram

C2×C3⋊PSU(3,𝔽2) #89111

order432
isoclinismisoclinic class of C3⋊PSU(3,𝔽2)
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C3

Aliases

direct productC2×C3⋊PSU(3,𝔽2)
unambiguous semidirect product(C2×C3)⋊PSU(3,𝔽2)

Splittings

Fitting, socleC2×C33 ⋊ Q8
SylowC33 ⋊ C2×Q8

Subgroups

subgroupG/H
center, hypercenterC2C3⋊PSU(3,𝔽2)
commutator, cosocleC3×C3⋊D3C23
FrattiniC1G
Fitting, socleC2×C33Q8
prime radicalC2×C3×C3⋊D3C22
hypocenterC33C2×Q8
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2)(3,4,5)(6,12,11,13)(8,9,14,10);(4,5)(7,9,11,13)(8,12,10,14);(1,2)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11724099, #11724105, #11724148, …
maximal normal subgroup of#11724099, #11724105, #11724148, …
maximal quotient of#11724361, #11724362, C4×C3⋊PSU(3,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 758);