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Cayley diagram

C7×AGL(1,𝔽7) #64419

order294
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C2×C3×AGL(1,𝔽7) (order 252) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C7)

Splittings

Fitting, socle, SylowC72 ⋊ C2×C3
derivedC7 ⋊ C2×C3×C7
HallC7×ΓL(1,𝔽8) ⋊ C2
HallC7×D7 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C3×C7
FrattiniC1G
Fitting, socleC72C2×C3
LCSG ▷ C7
UCSC1 ◁ C7

17 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,𝔽7), C2×C7×AGL(1,𝔽7), ΓL(1,𝔽8)×AGL(1,𝔽7), …
maximal normal subgroup ofD7×AGL(1,𝔽7), C2×C7×AGL(1,𝔽7), ΓL(1,𝔽8)×AGL(1,𝔽7), …
maximal quotient ofC7×ΓL(1,𝔽8)⋊C4, C2×C7×AGL(1,𝔽7), C7×D7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 8);