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Cayley diagram

C5×AGL(1,𝔽7) #6142

order210
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C4×AGL(1,𝔽7) (order 168) · Inn(G) = AGL(1,𝔽7) · Out(G) = C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽7)2
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C5)
unambiguous semidirect product(C5×D7)⋊C3

Splittings

Fitting, socle, HallC5×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C2
HallC5×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C5
HallD7 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C3×C5
FrattiniC1G
Fitting, socleC5×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×AGL(1,𝔽7), C2×C5×AGL(1,𝔽7), C3×C5×AGL(1,𝔽7), …
maximal normal subgroup ofD5×AGL(1,𝔽7), C2×C5×AGL(1,𝔽7), C3×C5×AGL(1,𝔽7), …
maximal quotient ofC5×ΓL(1,𝔽8)⋊C4, C2×C5×AGL(1,𝔽7), C5×D7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(210, 1);