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Cayley diagram

C2×ΓL(1,𝔽8)⋊Q8 #66652

order336
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×ΓL(1,𝔽8)⋊Q8
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊Q8
unambiguous semidirect product(C2×Dic14)⋊C3

Splittings

HallC2×Dic14 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×Q8
SylowC7 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
centerC22C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3
FrattiniC2C22×AGL(1,𝔽7)
FittingC2×C4×C7C2×C3
socleC22×C7C22×C3
prime radicalC22×ΓL(1,𝔽8)C22
hypercenterC2×C4AGL(1,𝔽7)
hypocenterC7C2×C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2)(3,4,9)(5,6,8)(10,12,13,17)(11,14,15,16);(4,5)(6,8)(7,9)(10,11,13,15)(12,16,17,14);(1,2)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6384⋊C3, #6390⋊C3, #6398⋊C3, …
maximal normal subgroup of#6384⋊C3, #6390⋊C3, #6398⋊C3, …
maximal quotient ofC4×ΓL(1,𝔽8)⋊Q8, #6432⋊C3, #6433⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 121);