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Cayley diagram

C22×AGL(1,𝔽7) #4277

order168
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = AGL(1,𝔽7) · Out(G) = S4
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C22×C3)
unambiguous semidirect productΓL(1,𝔽8)⋊C23
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊C22

Splittings

Fitting, socleC22×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C23×C3
HallC22×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C23×C3
FrattiniC1G
Fitting, socleC22×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 28

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽7))⋊C4, C2×C4×AGL(1,𝔽7), C2×C4⋊AGL(1,𝔽7), …
maximal normal subgroup of(C2×AGL(1,𝔽7))⋊C4, C2×C4×AGL(1,𝔽7), C2×C4⋊AGL(1,𝔽7), …
maximal quotient ofC2×ΓL(1,𝔽8)⋊Q8, C2×C4×AGL(1,𝔽7), C2×C4⋊AGL(1,𝔽7), …
automorphism group ofC7⋊C8, C4×D7, C22⋊D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 47);