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Cayley diagram

C2×#4208 #66456

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

Aliases

direct productC2×#4208

Splittings

FittingC2×#737 ⋊ C2
Sylow#737 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2×C9C3⋊D3
commutator, cosocle, hypocenterH(3,𝔽3)C22×C3
FrattiniC3C2×C3×C3⋊D3
FittingC2×#737C2
socleC2×C3C3×C3⋊D3
prime radicalC2×#272C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C9

101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 29 points:

(3,9,23,6,17,7,12,4,14)(5,25,29,19,26,21,20,15,22,10,16,27,11,24,13,18,8,28);(1,2)(3,5,10)(4,8,15)(6,11,19)(7,13,21)(9,16,25)(12,20,18)(14,22,28)(17,26,24)(23,29,27);(5,11,20)(8,16,26)(10,18,19)(13,22,29)(15,24,25)(21,27,28)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10619908, #10619909, C4×#4208, …
maximal normal subgroup of#10619908, #10619909, C4×#4208, …
maximal quotient of#10619966, C4×#4208, C4⋊#4208, …
automorphism group ofC3×C27, C2×C3×C27, C4×C27⋊C3
outer automorphism group ofC3×C27, C2×C3×C27, C32⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 148);