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Cayley diagram

C5×#131⋊C3 #91584

order480
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = A4 · Out(G) = C23×C4
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#131⋊C3
unambiguous semidirect product#131⋊(C3×C5)
unambiguous semidirect product(C5×#131)⋊C3

Splittings

Fitting, HallC5×#131 ⋊ C3
Sylow#131 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5×C8A4
commutator, hypocenterQ8C3×C4×C5
FrattiniC4C2×C5×A4
FittingC5×#131C3
socleC2×C5C4×A4
prime radicalC5×#41⋊C3C2
cosocle#41C2×C3×C5
LCSG ▷ Q8
UCSC1 ◁ C5×C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,15,8,25,11,7,18,12)(9,29,30,35,22,26,31,24,19,27,17,36,20,14,32,23,10,28,16,37,21,13,33,34);(6,19,11,21)(7,26,15,28)(8,30,18,32)(9,22,20,10)(12,34,25,36)(13,29,27,14)(16,33,31,17)(23,37,35,24)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14511738, #14511739, #14511740, …
maximal subgroup ofC5×#4628, #11746600, #65429⋊C3, …
maximal normal subgroup ofC5×#4628, #11746600, #65429⋊C3, …
maximal quotient ofC5×C8×SL(2,𝔽3), C2×C5×#131⋊C3, C5×#131⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 660);