Finite Group Databasebeta

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Cayley diagram

C3×#836 #63622

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

Aliases

direct productC3×#836

Splittings

SylowC32 ⋊ #101

Subgroups

subgroupG/H
centerC2×C3#217
commutatorC22×C3C2×C3×C4
FrattiniC2×C4C2×C3×D3
FittingC2×C32×Q8C2
socleC2×C32C22⋊C4
prime radicalC22×C32C2×C4
cosocleC2×C3×C4C22×C3
hypercenterC2×C3×Q8D3
hypocenterC3C3×#101
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

95 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589182, #10589183, #10589184, …
maximal normal subgroup of#10589182, #10589183, #10589184, …
maximal quotient ofC3×#4528, C3×#4530, C3×#4543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 270);