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Cayley diagram

C4×SL(2,ℤ/4) #5410

order192
isoclinismisoclinic class of S4
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = S4 · Out(G) = #120
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productC4×SL(2,ℤ/4)
unambiguous semidirect productA4⋊C42

Splittings

derivedA4 ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC2×C4S4
commutator, hypocenterA4C42
FrattiniC22C2×S4
FittingC23×C4D3
socleC24C2×D3
prime radical, cosocleC22×A4C22
LCSG ▷ A4
UCSC1 ◁ C2×C4

125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

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

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×SL(2,ℤ/4), SL(2,ℤ/4)⋊C8, #73108, …
maximal normal subgroup ofC8×SL(2,ℤ/4), SL(2,ℤ/4)⋊C8, #73108, …
maximal quotient ofC4×A4⋊C8, #73044, C4×#859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 969);