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Cayley diagram

C2×D5×ΓL(1,𝔽8) #88293

order420
isoclinismisoclinic class of D5×ΓL(1,𝔽8)
automorphismAut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽7) (order 1680) · Inn(G) = D5×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×D5×ΓL(1,𝔽8)
unambiguous semidirect productC5⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C5)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D5)⋊C3

Splittings

Fitting, socleC2×C5×C7 ⋊ C2×C3
derived, HallC5×C7 ⋊ C22×C3
HallC2×C7×D5 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C22
SylowC7 ⋊ C2×C3×D5
SylowC5 ⋊ C22×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2D5×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC5×C7C22×C3
FrattiniC1G
Fitting, socleC2×C5×C7C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2

40 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)(4,5)(6,7)(9,10,11)(12,14,13);(1,2)(3,4,6,7,5)(8,9,10,12,11,13,14)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)⋊C4, #63305⋊C3, #63307⋊C3, …
maximal normal subgroup ofD5×ΓL(1,𝔽8)⋊C4, #63305⋊C3, #63307⋊C3, …
maximal quotient ofΓL(1,𝔽8)×Dic10, C4×D5×ΓL(1,𝔽8), ΓL(1,𝔽8)×D20, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 18);