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Cayley diagram

C22×C3×C4×D3 #64341

order288
isoclinismisoclinic class of D3
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×C3×C4×D3

Splittings

FittingC22×C32×C4 ⋊ C2
derivedC3 ⋊ C23×C3×C4
SylowC32 ⋊ C23×C4

Subgroups

subgroupG/H
center, hypercenterC22×C3×C4D3
commutator, hypocenterC3C23×C3×C4
FrattiniC2C23×C3×D3
FittingC22×C32×C4C2
socleC23×C32C22
prime radicalC23×C3×D3C2
cosocleC2×C3C24×C3
LCSG ▷ C3
UCSC1 ◁ C22×C3×C4

499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×(C2×D3)⋊(C3⋊C8), #10591069, D3×#830, …
maximal normal subgroup ofC2×(C2×D3)⋊(C3⋊C8), #10591069, D3×#830, …
maximal quotient ofC2×C3×C4×Dic6, C2×C3×C42×D3, C2×C3×#872, …
automorphism group ofC13×C3⋊C8, C4×D3×C13, C13×#67, …
outer automorphism group ofC13×#131, C22×C3×C5×C7, C7×#504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 989);