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Cayley diagram

C2×C3×#272 #66453

order324
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×#272

Splittings

FittingC2×C3×H(3,𝔽3) ⋊ C2
derivedH(3,𝔽3) ⋊ C22×C3
SylowC3×H(3,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC2×C32C3⋊D3
commutator, cosocle, hypocenterH(3,𝔽3)C22×C3
FrattiniC3C2×C3×C3⋊D3
FittingC2×C3×H(3,𝔽3)C2
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C32

184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72260000
maximal subgroup ofC3×#6199, C3×#6200, Dic3×#272, …
maximal normal subgroup ofC3×#6199, C3×#6200, Dic3×#272, …
maximal quotient ofC3×#6229, C3×C4×#272, C3×C4⋊#272, …
automorphism group ofC7×3−1+2, C2×C7×3−1+2
outer automorphism group of#6866, #6867, #6868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 145);