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Cayley diagram

C3×#994 #64336

order288
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×#994

Splittings

FittingC3×H(5,𝔽2) ⋊ C3
SylowH(5,𝔽2) ⋊ C32

Subgroups

subgroupG/H
center, socleC2×C3C22×A4
commutator, cosocle, hypocenterQ8C22×C32
FrattiniC2C22×C3×A4
FittingC3×H(5,𝔽2)C3
hypercenterC3×Q8A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3 ◁ C3×Q8

114 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:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13164043, #13164047, C3⋊(#226⋊A4), …
maximal subgroup ofC3×#5428, C3×#5429, #10593662, …
maximal normal subgroup ofC3×#5428, C3×#5429, #10593662, …
maximal quotient ofC3×#520⋊C3, C3×C4⋊(#41⋊C3), C3×Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 984);