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Cayley diagram

C3×#997 #64339

order288
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC3×#997

Splittings

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

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3C22⋊A4
commutator, cosocle, hypocenterH(5,𝔽2)C32
FrattiniC2C3×C22⋊A4
FittingC3×H(5,𝔽2)C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C2×C3

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊#5932, C3⋊#5933, C3⋊#5934, …
maximal subgroup ofC3⋊(#458⋊C3), C3⋊(B(4,𝔽2)⋊C3), C3×B(4,𝔽2)⋊C3, …
maximal normal subgroup ofC3⋊(#458⋊C3), C3⋊(B(4,𝔽2)⋊C3), C3×B(4,𝔽2)⋊C3, …
maximal quotient ofC3×#543⋊C3, C3×Q8⋊SL(2,𝔽3), C3×#5465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 987);