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Cayley diagram

C4×AGL(1,𝔽7) #4238

order168
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C22×AGL(1,𝔽7) (order 168) · Inn(G) = AGL(1,𝔽7) · Out(G) = C22
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4)
unambiguous semidirect product(C4×D7)⋊C3

Splittings

FittingC4×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C4
HallC4×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C4
FrattiniC2C2×AGL(1,𝔽7)
FittingC4×C7C2×C3
socle, cosocleC2×C7C22×C3
prime radicalC2×AGL(1,𝔽7)C2
LCSG ▷ C7
UCSC1 ◁ C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×AGL(1,𝔽7), #1131⋊C3, C2×C4×AGL(1,𝔽7), …
maximal normal subgroup ofC8×AGL(1,𝔽7), #1131⋊C3, C2×C4×AGL(1,𝔽7), …
maximal quotient ofC8×AGL(1,𝔽7), #1131⋊C3, C4×ΓL(1,𝔽8)⋊C4, …
automorphism group ofC5×D7, C5×ΓL(1,𝔽8), C5×AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 8);