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Cayley diagram

C3×D5⋊C16 #91193

order480
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×#41×AGL(1,𝔽5) (order 640) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×#41
homologyAbelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2

Aliases

direct productC3×D5⋊C16
unambiguous semidirect productD5⋊(C3×C16)

Splittings

derived, SylowC5 ⋊ C2×C3×C16
HallC3×C5 ⋊ C2×C16

Subgroups

subgroupG/H
center, hypercenterC3×C8AGL(1,𝔽5)
commutator, hypocenterC5C2×C3×C16
FrattiniC4C2×C3×AGL(1,𝔽5)
FittingC3×C5×C8C4
socleC2×C3×C5C2×C8
prime radicalC2×C3×D5C8
cosocle#168C22×C3
LCSG ▷ C5
UCSC1 ◁ C3×C8

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽16)⋊C16, #11746451, #11746457, …
maximal normal subgroup ofΓL(1,𝔽16)⋊C16, #11746451, #11746457, …
maximal quotient ofC3×D5⋊C32, C3×#64842, C3×#64885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 269);