Finite Group Databasebeta

← Prev (#747) Next (#749) →

Cayley diagram

C2×AGL(1,𝔽7) #748

order84
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C2×AGL(1,𝔽7) (order 84) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2
holomorphG ⋊ Aut(G) = #72222066
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3)
unambiguous semidirect productΓL(1,𝔽8)⋊C22
unambiguous semidirect product(C2×D7)⋊C3
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊C2

Splittings

Fitting, socleC2×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C3
HallC2×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C3
FrattiniC1G
Fitting, socleC2×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 14

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AGL(1,𝔽7), C4⋊AGL(1,𝔽7), (C22⋊D7)⋊C3, …
maximal normal subgroup ofC4×AGL(1,𝔽7), C4⋊AGL(1,𝔽7), (C22⋊D7)⋊C3, …
maximal quotient ofΓL(1,𝔽8)⋊Q8, C4×AGL(1,𝔽7), C4⋊AGL(1,𝔽7), …
automorphism group ofDic7, C2×D7, C3×D7, …
holomorph ofC2×C7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 7);