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Cayley diagram

C3×#1007 #64349

order288
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#1007

Splittings

FittingC2×C32×Q8 ⋊ C2
SylowC32 ⋊ 2−1+4

Subgroups

subgroupG/H
centerC2×C3C23×D3
commutator, cosocleC2×C3C24×C3
FrattiniC2C23×C3×D3
FittingC2×C32×Q8C2
socleC2×C32C24
hypercenterC2×C3×Q8D3
hypocenterC3C3×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×Q8

311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10591645, #10591646, #10591648, …
maximal normal subgroup of#10591645, #10591646, #10591648, …
maximal quotient ofC3×#5500, C3×#5504, C3×#5507, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 997);