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Cayley diagram

C7×ΓL(1,𝔽16) #88297

order420
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphismAut(G) = C2×C3×D3×AGL(1,𝔽5) (order 720) · Inn(G) = ΓL(1,𝔽16) · Out(G) = C22×C3
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C7×AGL(1,𝔽5))

Splittings

Fitting, socle, HallC3×C5×C7 ⋊ C4
derived, HallC3×C5 ⋊ C4×C7
HallC5×C7 ⋊ Dic3
HallC3×C7 ⋊ AGL(1,𝔽5)
SylowC5 ⋊ C7×Dic3
SylowC3 ⋊ C7×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C7
FrattiniC1G
Fitting, socleC3×C5×C7C4
prime radicalC3×C7×D5C2
cosocleC3×D5C2×C7
LCSG ▷ C3×C5
UCSC1 ◁ C7

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×ΓL(1,𝔽16), D3×C7×AGL(1,𝔽5), D21⋊AGL(1,𝔽5), …
maximal normal subgroup ofD7×ΓL(1,𝔽16), D3×C7×AGL(1,𝔽5), D21⋊AGL(1,𝔽5), …
maximal quotient ofC7×C3⋊#170, C2×C7×ΓL(1,𝔽16), C7×C9⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 22);