Finite Group Databasebeta

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Cayley diagram

C3×#823 #63609

order288
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#823

Splittings

FittingC32×M16 ⋊ C2
SylowC32 ⋊ #100

Subgroups

subgroupG/H
centerC2×C3(C2×D3)⋊C4
commutatorC22×C3C2×C3×C4
FrattiniC2×C4C2×C3×D3
FittingC32×M16C2
socleC2×C32C22⋊C4
prime radicalC2×C3×D12C2
cosocleC2×C3×C4C22×C3
hypercenterC3×M16D3
hypocenterC3C3×#100
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,3)(4,5,6)(7,12,13,8,10,14,9,11);(5,6)(8,11)(9,13)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10590979, #10590982, #10590987, …
maximal normal subgroup of#10590979, #10590982, #10590987, …
maximal quotient ofC3×#4464, C3×#4468, C3×#4481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 257);