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Cayley diagram

C8×SL(2,𝔽3) #4641

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C23×S4 (order 192) · Inn(G) = A4 · Out(G) = C24
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productC8×SL(2,𝔽3)
unambiguous semidirect product(C8×Q8)⋊C3

Splittings

Fitting, SylowC8×Q8 ⋊ C3
derivedQ8 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC2×C8A4
commutator, hypocenterQ8C3×C8
FrattiniC2×C4C2×A4
FittingC8×Q8C3
socleC22C4×A4
prime radicalC2×SL(2,𝔽3)C4
cosocleC4×Q8C2×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C8

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711453, #11711454, #11711455, …
maximal subgroup ofSL(2,𝔽3)⋊C16, C16×SL(2,𝔽3), C8×#226, …
maximal normal subgroup ofSL(2,𝔽3)⋊C16, C16×SL(2,𝔽3), C8×#226, …
maximal quotient ofC16×SL(2,𝔽3), C2×C8×SL(2,𝔽3), C8×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 200);