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Cayley diagram

C2×SL(2,𝔽3) #230

order48
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×S4 (order 48) · Inn(G) = A4 · Out(G) = C22
holomorphG ⋊ Aut(G) = #72224794
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×SL(2,𝔽3)
unambiguous semidirect product(C2×Q8)⋊C3

Splittings

Fitting, SylowC2×Q8 ⋊ C3
derivedQ8 ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC22A4
commutator, cosocle, hypocenterQ8C2×C3
FrattiniC2C2×A4
FittingC2×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C22

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

derived subgroup of#5386, #5389, C4⋊#226, …
maximal subgroup of#859, C4×SL(2,𝔽3), C2×#226, …
maximal normal subgroup of#859, C4×SL(2,𝔽3), C2×#226, …
maximal quotient ofC4×SL(2,𝔽3), C22×SL(2,𝔽3), C2×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 32);