Finite Group Databasebeta

← Prev (#92700) Next (#92702) →

Cayley diagram

C3×C7⋊SL(2,𝔽3) #92701

order504
isoclinismisoclinic class of C7⋊SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×C7⋊SL(2,𝔽3)
unambiguous semidirect product(C3×C7)⋊SL(2,𝔽3)
unambiguous semidirect product(C3×Q8)⋊ΓL(1,𝔽8)

Splittings

FittingC3×C7×Q8 ⋊ C3
derived, HallC7×Q8 ⋊ C32
SylowQ8 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×SL(2,𝔽3)

Subgroups

subgroupG/H
center, hypercenterC2×C3C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7×Q8C32
FrattiniC2C3×C22⋊ΓL(1,𝔽8)
FittingC3×C7×Q8C3
socleC2×C3×C7A4
LCSG ▷ C7×Q8
UCSC1 ◁ C2×C3

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(5,6,7)(8,10,9)(12,13,15)(16,18,17);(1,2,3)(4,5,6,8,7,9,10)(11,15,14,17)(12,18,16,13)

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#66665, C3×D7⋊SL(2,𝔽3), C3⋊(#41⋊ΓL(1,𝔽8)), …
maximal normal subgroup ofC3×#66665, C3×D7⋊SL(2,𝔽3), C3⋊(#41⋊ΓL(1,𝔽8)), …
maximal quotient ofC2×C3×C7⋊SL(2,𝔽3), C9×C7⋊SL(2,𝔽3), #13094437, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 127);