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Cayley diagram

C2×C4×SL(2,𝔽3) #5437

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = A4 · Out(G) = C2×#120
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×C4×SL(2,𝔽3)
unambiguous semidirect product(C2×C4×Q8)⋊C3

Splittings

Fitting, SylowC2×C4×Q8 ⋊ C3
derivedQ8 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC22×C4A4
commutator, hypocenterQ8C2×C3×C4
FrattiniC22C22×A4
FittingC2×C4×Q8C3
socleC23C2×A4
prime radicalC22×SL(2,𝔽3)C2
cosocleC2×Q8C22×C3
LCSG ▷ Q8
UCSC1 ◁ C22×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72258519, #72268859
maximal subgroup ofC2×SL(2,𝔽3)⋊C8, #73055, #73056, …
maximal normal subgroup ofC2×SL(2,𝔽3)⋊C8, #73055, #73056, …
maximal quotient ofC42×SL(2,𝔽3), C22⋊C4×SL(2,𝔽3), C4⋊C4×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 996);