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Cayley diagram

C2×Q8×ΓL(1,𝔽8) #66697

order336
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×Q8×ΓL(1,𝔽8)

Splittings

Fitting, HallC2×C7×Q8 ⋊ C3
SylowC7 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
centerC22C22×ΓL(1,𝔽8)
commutator, cosocleC2×C7C23×C3
FrattiniC2C23×ΓL(1,𝔽8)
FittingC2×C7×Q8C3
socleC22×C7C22×C3
prime radicalC22×ΓL(1,𝔽8)C22
hypercenterC2×Q8ΓL(1,𝔽8)
hypocenterC7C2×C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×Q8

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6409⋊C3, #6410⋊C3, ΓL(1,𝔽8)×#101, …
maximal normal subgroup of#6409⋊C3, #6410⋊C3, ΓL(1,𝔽8)×#101, …
maximal quotient ofC2×C4⋊C4×ΓL(1,𝔽8), C4×Q8×ΓL(1,𝔽8), ΓL(1,𝔽8)×C22⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 166);