Finite Group Databasebeta

← Prev (#6987) Next (#6989) →

Cayley diagram

A4×ΓL(1,𝔽8) #6988

order252
isoclinismisoclinic class of A4×ΓL(1,𝔽8)
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = G · Out(G) = C22
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productA4×ΓL(1,𝔽8)
unambiguous semidirect product(C22×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C7×A4)⋊C3
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊C3

Splittings

Fitting, socle, derived, HallC22×C7 ⋊ C32
SylowC7 ⋊ C3×A4
SylowC22 ⋊ C3×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C7C32
LCSG ▷ C22×C7
UCSC1

28 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:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×S4, A4⋊AGL(1,𝔽7), A4×AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)×S4, A4⋊AGL(1,𝔽7), A4×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×SL(2,𝔽3), C2×A4×ΓL(1,𝔽8), ΓL(1,𝔽8)×C22⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 27);