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Cayley diagram

C22×SL(2,𝔽3) #991

order96
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = S42 (order 576) · Inn(G) = A4 · Out(G) = C2×S4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC22×SL(2,𝔽3)

Splittings

Fitting, SylowC22×Q8 ⋊ C3
derivedQ8 ⋊ C22×C3

Subgroups

subgroupG/H
center, socle, hypercenterC23A4
commutator, cosocle, hypocenterQ8C22×C3
FrattiniC2C22×A4
FittingC22×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C23

61 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×#226, A4×GL(2,𝔽3), #11711027, …
maximal subgroup ofC2×#859, #5419, #5420, …
maximal normal subgroup ofC2×#859, #5419, #5420, …
maximal quotient ofC2×C4×SL(2,𝔽3), D4×SL(2,𝔽3), Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 198);