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Cayley diagram

C4×ΓL(1,𝔽16) #6759

order240
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphismAut(G) = D3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = ΓL(1,𝔽16) · Out(G) = C2×D4
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

direct productC4×ΓL(1,𝔽16)

Splittings

FittingC3×C4×C5 ⋊ C4
derived, HallC3×C5 ⋊ C42
SylowC5 ⋊ C4×Dic3
SylowC3 ⋊ C4×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽16)
commutator, hypocenterC3×C5C42
FrattiniC2C2×ΓL(1,𝔽16)
FittingC3×C4×C5C4
socleC2×C3×C5C2×C4
prime radical, cosocleC2×C3×D5C22
LCSG ▷ C3×C5
UCSC1 ◁ C4

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽16)⋊C8, #91150, #91155, …
maximal normal subgroup ofΓL(1,𝔽16)⋊C8, #91150, #91155, …
maximal quotient ofC8×ΓL(1,𝔽16), #91221, C4×C3⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 120);