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Cayley diagram

C2×C4⋊AGL(1,𝔽7) #66654

order336
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×D28)⋊C3

Splittings

FittingC2×C4×C7 ⋊ C2×C3
HallC2×D28 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×D4
SylowC7 ⋊ C2×C3×D4

Subgroups

subgroupG/H
centerC22C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3
FrattiniC2C22×AGL(1,𝔽7)
FittingC2×C4×C7C2×C3
socleC22×C7C22×C3
hypercenterC2×C4AGL(1,𝔽7)
hypocenterC7C2×C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6383⋊C3, #6395⋊C3, #6397⋊C3, …
maximal normal subgroup of#6383⋊C3, #6395⋊C3, #6397⋊C3, …
maximal quotient of#6432⋊C3, C4×C4⋊AGL(1,𝔽7), #10620710, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 123);