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Cayley diagram

C22×C3×SL(2,𝔽3) #64333

order288
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×C3×SL(2,𝔽3)

Splittings

FittingC22×C3×Q8 ⋊ C3
derivedQ8 ⋊ C22×C32
SylowC22×Q8 ⋊ C32

Subgroups

subgroupG/H
center, socle, hypercenterC23×C3A4
commutator, cosocle, hypocenterQ8C22×C32
FrattiniC2C22×C3×A4
FittingC22×C3×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C23×C3

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13164030, #13164038, A4×C3⋊#226, …
maximal subgroup ofC2×C3×#859, C3×#5419, C3×#5420, …
maximal normal subgroup ofC2×C3×#859, C3×#5419, C3×#5420, …
maximal quotient ofC2×C3×C4×SL(2,𝔽3), C3×D4×SL(2,𝔽3), C3×Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 981);