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Cayley diagram

C3×AGL(1,𝔽7) #1258

order126
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = D3×AGL(1,𝔽7) (order 252) · Inn(G) = AGL(1,𝔽7) · Out(G) = D3
holomorphG ⋊ Aut(G) = #72230337
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×AGL(1,𝔽7)
unambiguous semidirect productD7⋊C32
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C3)
unambiguous semidirect product(C3×D7)⋊C3

Splittings

Fitting, socleC3×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C32
HallC3×ΓL(1,𝔽8) ⋊ C2
HallD7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C32
FrattiniC1G
Fitting, socleC3×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C3

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 21

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽7), C2×C3×AGL(1,𝔽7), C9×AGL(1,𝔽7), …
maximal normal subgroup ofD3×AGL(1,𝔽7), C2×C3×AGL(1,𝔽7), C9×AGL(1,𝔽7), …
maximal quotient ofC3×ΓL(1,𝔽8)⋊C4, C2×C3×AGL(1,𝔽7), C9×AGL(1,𝔽7), …
automorphism group ofC7⋊C9, D7⋊C9, C2×C7⋊C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 7);