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Cayley diagram

D5⋊(C3⋊C16) #91218

order480
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C2×C16 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD5⋊(C3⋊C16)

Splittings

derived, HallC3×C5 ⋊ C2×C16
SylowC5 ⋊ C2×C3⋊C16
SylowC3 ⋊ D5⋊C16

Subgroups

subgroupG/H
center, hypercenterC8ΓL(1,𝔽16)
commutator, hypocenterC3×C5C2×C16
FrattiniC4C2×ΓL(1,𝔽16)
FittingC3×C5×C8C4
socleC2×C3×C5C2×C8
prime radicalC2×C3×D5C8
cosocleC3×#168C22
LCSG ▷ C3×C5
UCSC1 ◁ 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:

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

Polycyclic presentation:

2^5,3,5|a^2=c,b^2=d,c^2=d,d^2=e,f^a=f^2,g^a=g^3,g^c=g^4

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊C16, #11746450, #11746456, …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊C16, #11746450, #11746456, …
maximal quotient ofD5⋊(C3⋊C32), #11746713, #11746756, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 294);