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Cayley diagram

C2×C9⋊SL(2,𝔽3) #88681

order432
isoclinismisoclinic class of C9⋊SL(2,𝔽3)
automorphismAut(G) = C2×C3×D3×A4 (order 432) · Inn(G) = C3×A4 · Out(G) = C22×C3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

direct productC2×C9⋊SL(2,𝔽3)
unambiguous semidirect product(C2×C9)⋊SL(2,𝔽3)

Splittings

FittingC2×Q8×C9 ⋊ C3
SylowC2×Q8 ⋊ 3−1+2

Subgroups

subgroupG/H
center, socleC22×C3C3×A4
commutator, cosocleC3×Q8C2×C32
FrattiniC2×C3C2×C3×A4
FittingC2×Q8×C9C3
prime radicalC2×C3×SL(2,𝔽3)C3
hypercenterC22×C9A4
hypocenterQ8C2×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C22×C9

68 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)(11,14,19,13,17,15,16,12,18);(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

maximal subgroup ofDic9⋊SL(2,𝔽3), C4×C9⋊SL(2,𝔽3), C2×#88614, …
maximal normal subgroup ofDic9⋊SL(2,𝔽3), C4×C9⋊SL(2,𝔽3), C2×#88614, …
maximal quotient ofC4×C9⋊SL(2,𝔽3), C22×C9⋊SL(2,𝔽3), C2×(Q8⋊C9)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 328);