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Cayley diagram

C2×PSU(3,𝔽2)⋊C3 #89088

order432
isoclinismisoclinic class of PSU(3,𝔽2)⋊C3
automorphismAut(G) = AGL(2,𝔽3) (order 432) · Inn(G) = PSU(3,𝔽2)⋊C3 · Out(G) = C2
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

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

Splittings

Fitting, socleC2×C32 ⋊ SL(2,𝔽3)
derivedPSU(3,𝔽2) ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2PSU(3,𝔽2)⋊C3
commutator, cosocle, hypocenterPSU(3,𝔽2)C2×C3
FrattiniC1G
Fitting, socleC2×C32SL(2,𝔽3)
LCSG ▷ PSU(3,𝔽2)
UCSC1 ◁ C2

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup of(PSU(3,𝔽2)⋊C3)⋊Q8, C4⋊AGL(2,𝔽3), #13180131, …
maximal subgroup of(PSU(3,𝔽2)⋊C3)⋊C4, C4×PSU(3,𝔽2)⋊C3, C2×AGL(2,𝔽3), …
maximal normal subgroup of(PSU(3,𝔽2)⋊C3)⋊C4, C4×PSU(3,𝔽2)⋊C3, C2×AGL(2,𝔽3), …
maximal quotient ofC4×PSU(3,𝔽2)⋊C3, C22×PSU(3,𝔽2)⋊C3, C2×#10620013, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 735);