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Cayley diagram

C9×AGL(1,𝔽7) #67354

order378
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C3×D3×AGL(1,𝔽7) (order 756) · Inn(G) = AGL(1,𝔽7) · Out(G) = C3×D3
homologyAbelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3

Aliases

direct productC9×AGL(1,𝔽7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C9)

Splittings

FittingC7×C9 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C9
HallC9×ΓL(1,𝔽8) ⋊ C2
HallD7 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC9AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C9
FrattiniC3C3×AGL(1,𝔽7)
FittingC7×C9C2×C3
socle, cosocleC3×C7C2×C32
prime radicalC3×AGL(1,𝔽7)C3
LCSG ▷ C7
UCSC1 ◁ C9

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(2,5,6,3,4,7);(1,2,4,7,6,5,3)(8,9,11,10,12,14,13,15,16)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×AGL(1,𝔽7), C2×C9×AGL(1,𝔽7), C27×AGL(1,𝔽7), …
maximal normal subgroup ofD9×AGL(1,𝔽7), C2×C9×AGL(1,𝔽7), C27×AGL(1,𝔽7), …
maximal quotient ofC9×ΓL(1,𝔽8)⋊C4, C2×C9×AGL(1,𝔽7), C27×AGL(1,𝔽7), …
automorphism group ofC7⋊C27, D7⋊C27, C2×C7⋊C27

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 7);