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Cayley diagram

C2×C3×#239 #64348

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

Aliases

direct productC2×C3×#239

Splittings

FittingC2×C32×Q8 ⋊ C2
SylowC32 ⋊ C2×#41

Subgroups

subgroupG/H
centerC22×C3C22×D3
commutator, cosocleC2×C3C24×C3
FrattiniC2C23×C3×D3
FittingC2×C32×Q8C2
socleC22×C32C23
hypercenterC2×C3×Q8D3
hypocenterC3C2×C3×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 16 points:

(1,2)(3,4,5)(6,7,8)(9,11,12,15)(10,13,14,16);(9,10,12,14)(11,16,15,13);(4,5)(11,15)(13,16);(1,2)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10591550, #10591556, #10591586, …
maximal normal subgroup of#10591550, #10591556, #10591586, …
maximal quotient ofC2×C3×#890, C2×C3×#892, C2×C3×#893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 996);