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Cayley diagram

C2×C5×ΓL(1,𝔽8) #6145

order210
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C4×AGL(1,𝔽7) (order 168) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C4
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×C5×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7)⋊C3

Splittings

Fitting, socle, HallC2×C5×C7 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C5
HallC5×C7 ⋊ C2×C3
HallC2×C7 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7C3
LCSG ▷ C7
UCSC1 ◁ C2×C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 70

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×ΓL(1,𝔽8)⋊C4, Dic5×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊Dic5, …
maximal normal subgroup ofC5×ΓL(1,𝔽8)⋊C4, Dic5×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊Dic5, …
maximal quotient ofC4×C5×ΓL(1,𝔽8), C22×C5×ΓL(1,𝔽8), C2×C5×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(210, 4);