Finite Group Databasebeta

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Cayley diagram

C3×#834 #63620

order288
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#834

Splittings

SylowC32 ⋊ #99

Subgroups

subgroupG/H
centerC2×C3#217
commutatorC22×C3C2×C3×C4
FrattiniC23C2×C3×D3
Fitting, prime radicalC2×C32×D4C2
socleC2×C32C22⋊C4
cosocleC23×C3C22×C3
hypercenterC2×C3×D4D3
hypocenterC3C3×#99
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C2×C3×D4

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 24

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

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589172, #10589174, #10589175, …
maximal normal subgroup of#10589172, #10589174, #10589175, …
maximal quotient ofC3×#4525, C3×#4526, C3×#4528, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 268);