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Cayley diagram

C5×C3⋊C32 #90925

order480
isoclinismisoclinic class of D3
automorphismAut(G) = C2×C4×D3×C8 (order 384) · Inn(G) = D3 · Out(G) = C2×C4×C8
homologyAbelianization (H1) = C5×C32 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊C32
unambiguous semidirect productC3⋊(C5×C32)

Splittings

derived, SylowC3 ⋊ C5×C32
HallC3×C5 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC5×C16D3
commutator, hypocenterC3C5×C32
FrattiniC16C5×D3
FittingC3×C5×C16C2
socle, prime radicalC2×C3×C5C16
cosocleC3×C16C2×C5
LCSG ▷ C3
UCSC1 ◁ C5×C16

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 2 generators on 40 points:

(2,3)(4,11,15,30,18,25,34,10,8,23,16,31,29,5,14,21,9,24,28,35,7,12,26,22,19,33,6,20,17,13,27,32)(36,37,38,39,40);(1,2,3)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×C3⋊C32, #11745814, #11745815, …
maximal normal subgroup ofD5×C3⋊C32, #11745814, #11745815, …
maximal quotient ofC5×C3⋊C64, C2×C5×C3⋊C32, C5×C9⋊C32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1);