Finite Group Databasebeta

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Cayley diagram

C3×#904 #64027

order288
isoclinismisoclinic class of #204
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#904

Splittings

FittingC2×C32×C8 ⋊ C2
SylowC32 ⋊ #135

Subgroups

subgroupG/H
centerC3×C4D12
commutator, cosocleC3×C4C23×C3
FrattiniC4C22×C3×D3
FittingC2×C32×C8C2
socleC2×C32C2×D4
hypercenterC2×C3×C8D3
hypocenterC3C3×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589120, #10589121, #10589123, …
maximal normal subgroup of#10589120, #10589121, #10589123, …
maximal quotient ofC3×#4683, C3×C4×#204, C3×C4×D24, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 675);