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Cayley diagram

C16×SL(2,𝔽3) #68040

order384
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = A4 · Out(G) = C23×C4
homologyAbelianization (H1) = C3×C16 · Schur multiplier (H2) = C1

Aliases

direct productC16×SL(2,𝔽3)
unambiguous semidirect product(Q8×C16)⋊C3

Splittings

Fitting, SylowQ8×C16 ⋊ C3
derivedQ8 ⋊ C3×C16

Subgroups

subgroupG/H
center, hypercenterC2×C16A4
commutator, hypocenterQ8C3×C16
FrattiniC2×C8C2×A4
FittingQ8×C16C3
socleC22C8×A4
prime radicalC2×SL(2,𝔽3)C8
cosocleC8×Q8C2×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C16

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 24 points:

(2,3,5)(6,8,7)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,5,4,7)(2,8,6,3)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSL(2,𝔽3)×C32, C2×C16×SL(2,𝔽3), SL(2,𝔽3)×M32, …
maximal normal subgroup ofSL(2,𝔽3)×C32, C2×C16×SL(2,𝔽3), SL(2,𝔽3)×M32, …
maximal quotient ofSL(2,𝔽3)×C32, C2×C16×SL(2,𝔽3), C3×C16×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 616);