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Cayley diagram

C2×C3⋊GL(2,𝔽3) #64263

order288
isoclinismisoclinic class of C3⋊#226
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊GL(2,𝔽3)
unambiguous semidirect product(C2×C3)⋊GL(2,𝔽3)

Splittings

FittingC2×C3×Q8 ⋊ D3
derivedC3×SL(2,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC22C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22
FrattiniC2C2×C3⋊S4
FittingC2×C3×Q8D3
socleC22×C3S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 48

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

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593206, #10593207, #10593208, …
maximal normal subgroup of#10593206, #10593207, #10593208, …
maximal quotient of#10593620, C4×C3⋊GL(2,𝔽3), C4⋊(C3⋊GL(2,𝔽3)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 911);