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Cayley diagram

C7×SL(2,𝔽3) #4252

order168
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C3×S4 (order 144) · Inn(G) = A4 · Out(G) = C22×C3
holomorphG ⋊ Aut(G) = AGL(1,𝔽7)×SO+(4,𝔽3)
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×SL(2,𝔽3)
unambiguous semidirect productQ8⋊(C3×C7)

Splittings

Fitting, HallC7×Q8 ⋊ C3
derived, SylowQ8 ⋊ C3×C7

Subgroups

subgroupG/H
center, socle, hypercenterC2×C7A4
commutator, cosocle, hypocenterQ8C3×C7
FrattiniC2C7×A4
FittingC7×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C2×C7

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7⋊#226, C7⋊GL(2,𝔽3), C7⋊#859, …
maximal subgroup ofC7×#226, C7×GL(2,𝔽3), C7⋊#226, …
maximal normal subgroup ofC7×#226, C7×GL(2,𝔽3), C7⋊#226, …
maximal quotient ofC2×C7×SL(2,𝔽3), C7×Q8⋊C9, C3×C7×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 22);