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Cayley diagram

C3×GU(2,𝔽3) #63752

order288
isoclinismisoclinic class of #226
automorphismAut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×GU(2,𝔽3)

Splittings

derivedSL(2,𝔽3) ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC3×C4S4
commutator, hypocenterSL(2,𝔽3)C3×C4
FrattiniC4C3×S4
FittingC3×#41D3
socleC2×C3SL(2,ℤ/4)
prime radicalC3×#41⋊C3C2
cosocle#41⋊C3C2×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C3×C4

53 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 27 points:

(1,6)(2,16,8,5)(3,10)(4,17,15,20)(7,11,18,12)(9,23,19,21)(13,24,14,22)(25,26,27);(1,4,12)(2,7,11)(3,9,17)(5,15,20)(6,14,23)(8,19,21)(10,18,22)(13,24,16)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593220, #10593222, D3×GU(2,𝔽3), …
maximal normal subgroup of#10593220, #10593222, D3×GU(2,𝔽3), …
maximal quotient ofC3×SL(2,𝔽3)⋊C8, C2×C3×GU(2,𝔽3), C9×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 400);