Finite Group Databasebeta

← Prev (#89050) Next (#89052) →

Cayley diagram

C2×C32×SL(2,𝔽3) #89051

order432
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C33

Aliases

direct productC2×C32×SL(2,𝔽3)
unambiguous semidirect product(C2×Q8)⋊C33
unambiguous semidirect product(C2×C3×Q8)⋊C32

Splittings

FittingC2×C32×Q8 ⋊ C3
derivedQ8 ⋊ C2×C33
SylowC2×Q8 ⋊ C33

Subgroups

subgroupG/H
center, socle, hypercenterC22×C32A4
commutator, cosocle, hypocenterQ8C2×C33
FrattiniC2C2×C32×A4
FittingC2×C32×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C22×C32

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13165944, #13165946, #13165947, …
maximal subgroup ofC32×#859, C3×C3⋊#859, C3⋊(C3⋊#859), …
maximal normal subgroup ofC32×#859, C3×C3⋊#859, C3⋊(C3⋊#859), …
maximal quotient ofC32×C4×SL(2,𝔽3), C22×C32×SL(2,𝔽3), C2×C3×C9×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 698);