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Cayley diagram

C8×ΓL(1,𝔽16) #91220

order480
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC8×ΓL(1,𝔽16)

Splittings

FittingC3×C5×C8 ⋊ C4
derived, HallC3×C5 ⋊ C4×C8
SylowC5 ⋊ C8×Dic3
SylowC3 ⋊ C8×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC8ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C8
FrattiniC4C2×ΓL(1,𝔽16)
FittingC3×C5×C8C4
socleC2×C3×C5C42
prime radicalC2×C3×D5C2×C4
cosocleC3×C4×D5C22
LCSG ▷ C3×C5
UCSC1 ◁ C8

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3,4,5)(7,8);(1,2,5,3,4)(6,7,8)(9,10,11,13,12,14,15,16)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽16)⋊C16, #11746451, C16×ΓL(1,𝔽16), …
maximal normal subgroup ofΓL(1,𝔽16)⋊C16, #11746451, C16×ΓL(1,𝔽16), …
maximal quotient ofC16×ΓL(1,𝔽16), #11746715, C8×C3⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 296);