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Cayley diagram

C2×C7⋊SL(2,𝔽3) #66703

order336
isoclinismisoclinic class of C7⋊SL(2,𝔽3)
automorphismAut(G) = C2×A4×AGL(1,𝔽7) (order 1008) · Inn(G) = C22⋊ΓL(1,𝔽8) · Out(G) = C22×C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×C7⋊SL(2,𝔽3)
unambiguous semidirect product(C2×C7)⋊SL(2,𝔽3)
unambiguous semidirect product(C2×Q8)⋊ΓL(1,𝔽8)

Splittings

Fitting, HallC2×C7×Q8 ⋊ C3
derivedC7×Q8 ⋊ C2×C3
SylowC2×Q8 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C2×SL(2,𝔽3)

Subgroups

subgroupG/H
center, hypercenterC22C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7×Q8C2×C3
FrattiniC2C2×C22⋊ΓL(1,𝔽8)
FittingC2×C7×Q8C3
socleC22×C7A4
LCSG ▷ C7×Q8
UCSC1 ◁ C22

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic7⋊SL(2,𝔽3), C4×C7⋊SL(2,𝔽3), C2×#66665, …
maximal normal subgroup ofDic7⋊SL(2,𝔽3), C4×C7⋊SL(2,𝔽3), C2×#66665, …
maximal quotient ofC4×C7⋊SL(2,𝔽3), C22×C7⋊SL(2,𝔽3), C2×C7⋊(Q8⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 172);