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Cayley diagram

C9×SL(2,𝔽3) #6201

order216
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C3×D3×S4 (order 432) · Inn(G) = A4 · Out(G) = C2×C3×D3
homologyAbelianization (H1) = C3×C9 · Schur multiplier (H2) = C3

Aliases

direct productC9×SL(2,𝔽3)

Splittings

FittingQ8×C9 ⋊ C3
derived, SylowQ8 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC2×C9A4
commutator, hypocenterQ8C3×C9
Frattini, socleC2×C3C3×A4
FittingQ8×C9C3
prime radicalC3×SL(2,𝔽3)C3
cosocleC3×Q8C32
LCSG ▷ Q8
UCSC1 ◁ C2×C9

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC9⋊#226, C9⋊GL(2,𝔽3), C9⋊#859, …
maximal subgroup ofC9×#226, C9×GL(2,𝔽3), C9⋊#226, …
maximal normal subgroup ofC9×#226, C9×GL(2,𝔽3), C9⋊#226, …
maximal quotient ofC2×C9×SL(2,𝔽3), SL(2,𝔽3)×C27, C27⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 38);