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Cayley diagram

ΓL(1,𝔽8)×SL(2,𝔽3) #92648

order504
isoclinismisoclinic class of ΓL(1,𝔽8)×SL(2,𝔽3)
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = A4×ΓL(1,𝔽8) · Out(G) = C22
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productΓL(1,𝔽8)×SL(2,𝔽3)
unambiguous semidirect product(Q8×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C7×SL(2,𝔽3))⋊C3
unambiguous semidirect product(C7⋊SL(2,𝔽3))⋊C3

Splittings

Fitting, derived, HallC7×Q8 ⋊ C32
SylowQ8 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×ΓL(1,𝔽8)
commutator, Fitting, cosocle, hypocenterC7×Q8C32
socleC2×C7C3×A4
LCSG ▷ C7×Q8
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,5,4,6,7)(9,10,12)(13,15,14);(2,3,4)(5,7,6)(8,14,11,12)(9,10,13,15)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#226, ΓL(1,𝔽8)×GL(2,𝔽3), ΓL(1,𝔽8)⋊#226, …
maximal normal subgroup ofΓL(1,𝔽8)×#226, ΓL(1,𝔽8)×GL(2,𝔽3), ΓL(1,𝔽8)⋊#226, …
maximal quotient ofC2×ΓL(1,𝔽8)×SL(2,𝔽3), ΓL(1,𝔽8)×Q8⋊C9, SL(2,𝔽3)×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 74);