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Cayley diagram

C2×Q8⋊H(3,𝔽3) #88689

order432
isoclinismisoclinic class of C9⋊SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productC2×Q8⋊H(3,𝔽3)
unambiguous semidirect product(C2×Q8)⋊H(3,𝔽3)
unambiguous semidirect product(C2×C3×SL(2,𝔽3))⋊C3

Splittings

FittingC2×C32×Q8 ⋊ C3
SylowC2×Q8 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
center, socleC22×C3C3×A4
commutator, cosocleC3×Q8C2×C32
FrattiniC2×C3C2×C3×A4
FittingC2×C32×Q8C3
hypercenterC22×C32A4
hypocenterQ8C2×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C22×C32

101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2)(3,9,6,7)(4,5,8,10)(11,12,15)(13,14,18)(16,17,19);(4,5,7)(8,10,9)(12,14,17)(15,19,18)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146331, #13146332, #13146334, …
maximal subgroup of(C3⋊#859)⋊C3, #11720738, #11720765, …
maximal normal subgroup of(C3⋊#859)⋊C3, #11720738, #11720765, …
maximal quotient ofC4×Q8⋊H(3,𝔽3), C22×Q8⋊H(3,𝔽3), C2×#10619574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 336);