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Cayley diagram

C2×D4×ΓL(1,𝔽8) #66696

order336
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×D4×ΓL(1,𝔽8)
unambiguous semidirect productC4⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D4)⋊C3

Splittings

Fitting, HallC2×C7×D4 ⋊ C3
SylowC7 ⋊ C2×C3×D4

Subgroups

subgroupG/H
centerC22C22×ΓL(1,𝔽8)
commutator, cosocleC2×C7C23×C3
FrattiniC2C23×ΓL(1,𝔽8)
FittingC2×C7×D4C3
socleC22×C7C22×C3
hypercenterC2×D4ΓL(1,𝔽8)
hypocenterC7C2×C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×D4

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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6406⋊C3, #6407⋊C3, #6408⋊C3, …
maximal normal subgroup of#6406⋊C3, #6407⋊C3, #6408⋊C3, …
maximal quotient ofC2×C22⋊C4×ΓL(1,𝔽8), C2×C4⋊C4×ΓL(1,𝔽8), C4×D4×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 165);