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Cayley diagram

ΓL(1,𝔽8)×S4 #92733

order504
isoclinismisoclinic class of ΓL(1,𝔽8)×S4
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×S4
unambiguous semidirect productA4⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C7×S4)⋊C3

Splittings

Fitting, socleC22×C7 ⋊ C3×D3
derivedC7×A4 ⋊ C2×C3
SylowC7 ⋊ C3×S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC7×A4C2×C3
Fitting, socleC22×C7C3×D3
LCSG ▷ C7×A4
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×S4, C3×ΓL(1,𝔽8)×S4, …
maximal normal subgroup ofS4×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×S4, C3×ΓL(1,𝔽8)×S4, …
maximal quotient ofΓL(1,𝔽8)×#226, ΓL(1,𝔽8)×GL(2,𝔽3), ΓL(1,𝔽8)×SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 159);