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Cayley diagram

C2×#504⋊C3 #73275

order384
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×#504⋊C3
unambiguous semidirect product(C2×#504)⋊C3

Splittings

Fitting, SylowC2×#504 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C16A4
commutator, hypocenterQ8C2×C3×C8
FrattiniC8C22×A4
FittingC2×#504C3
socleC22C8×A4
prime radicalC2×#41⋊C3C4
cosocle#131C22×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C16

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 66 points:

(3,4,5,10,6,11,15,26,9,14,18,29,21,32,39,50)(7,13,40,27,20,55,37,49,24,34,43,53,45,58,61,64,8,33,16,28,44,30,38,63,23,36,42,52,47,57,60,66,22,12,17,51,19,31,59,48,25,35,41,54,46,56,62,65);(1,2)(3,22,9,24)(4,33,14,35)(5,40,18,42)(6,44,21,46)(7,25,23,8)(10,51,29,53)(11,55,32,57)(12,36,34,13)(15,59,39,61)(16,43,41,17)(19,47,45,20)(26,63,50,65)(27,54,52,28)(30,58,56,31)(37,62,60,38)(48,66,64,49)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#504⋊C3, #33326⋊C3, #33334⋊C3, …
maximal normal subgroup ofC4×#504⋊C3, #33326⋊C3, #33334⋊C3, …
maximal quotient ofC4×#504⋊C3, #33334⋊C3, C4⋊(#504⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5851);