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Cayley diagram

C8×AGL(1,𝔽7) #66538

order336
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C23×AGL(1,𝔽7) (order 336) · Inn(G) = AGL(1,𝔽7) · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C8)
unambiguous semidirect product(C8×D7)⋊C3

Splittings

FittingC7×C8 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C8
HallC8×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C8
FrattiniC4C2×AGL(1,𝔽7)
FittingC7×C8C2×C3
socleC2×C7C2×C3×C4
prime radicalC2×AGL(1,𝔽7)C4
cosocleC4×C7C22×C3
LCSG ▷ C7
UCSC1 ◁ C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×AGL(1,𝔽7), #6372⋊C3, C2×C8×AGL(1,𝔽7), …
maximal normal subgroup ofC16×AGL(1,𝔽7), #6372⋊C3, C2×C8×AGL(1,𝔽7), …
maximal quotient ofC16×AGL(1,𝔽7), #6372⋊C3, C8×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 7);