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Cayley diagram

C2×C9×SL(2,𝔽3) #88680

order432
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = A4 · Out(G) = C22×C3×D3
homologyAbelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3

Aliases

direct productC2×C9×SL(2,𝔽3)

Splittings

FittingC2×Q8×C9 ⋊ C3
derivedQ8 ⋊ C2×C3×C9
SylowC2×Q8 ⋊ C3×C9

Subgroups

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

82 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)(4,5,7)(8,10,9);(3,7,6,9)(4,10,8,5)(11,15,19,13,18,14,16,12,17)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146209, #13146210, #13146211, …
maximal subgroup ofC9×#859, C9⋊#859, SL(2,𝔽3)×Dic9, …
maximal normal subgroup ofC9×#859, C9⋊#859, SL(2,𝔽3)×Dic9, …
maximal quotient ofC4×C9×SL(2,𝔽3), C22×C9×SL(2,𝔽3), C2×SL(2,𝔽3)×C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 327);