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Cayley diagram

ΓL(1,𝔽8)×#67 #92658

order504
isoclinismisoclinic class of ΓL(1,𝔽8)×Dic6
automorphismAut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×D3×ΓL(1,𝔽8) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#67
unambiguous semidirect product(C7×#67)⋊C3

Splittings

FittingC22×C3×C7 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ D4
SylowC7 ⋊ C3×#67

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C3×C7C22×C3
FittingC22×C3×C7C2×C3
hypercenterC22D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

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

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66684⋊C3, #66686⋊C3, #67×AGL(1,𝔽7), …
maximal normal subgroup of#66684⋊C3, #66686⋊C3, #67×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×Dic3⋊C4, ΓL(1,𝔽8)×(C2×D3)⋊C4, ΓL(1,𝔽8)×#213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 84);