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Cayley diagram

C11×AGL(1,𝔽7) #90756

order462
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C2×C5×AGL(1,𝔽7) (order 420) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C5
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C11)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C11)
unambiguous semidirect product(C11×D7)⋊C3

Splittings

Fitting, socle, HallC7×C11 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C11
HallC11×ΓL(1,𝔽8) ⋊ C2
HallC11×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C11
HallD7 ⋊ C3×C11

Subgroups

subgroupG/H
center, hypercenterC11AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C3×C11
FrattiniC1G
Fitting, socleC7×C11C2×C3
LCSG ▷ C7
UCSC1 ◁ C11

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 77

A minimal faithful permutation representation — 2 generators on 18 points:

(2,5,6,3,4,7)(8,9,10,11,12,13,14,15,16,17,18);(1,2,4,7,6,5,3)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×AGL(1,𝔽7), C2×C11×AGL(1,𝔽7), C3×C11×AGL(1,𝔽7), …
maximal normal subgroup ofD11×AGL(1,𝔽7), C2×C11×AGL(1,𝔽7), C3×C11×AGL(1,𝔽7), …
maximal quotient ofC11×ΓL(1,𝔽8)⋊C4, C2×C11×AGL(1,𝔽7), C11×D7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(462, 2);