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Cayley diagram

C2×SL(2,ℤ/4) #987

order96
isoclinismisoclinic class of S4
automorphismAut(G) = D4×S4 (order 192) · Inn(G) = S4 · Out(G) = D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×SL(2,ℤ/4)
unambiguous semidirect product(C2×A4)⋊C4

Splittings

derivedA4 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22S4
commutator, hypocenterA4C2×C4
FrattiniC2C2×S4
Fitting, socleC24D3
prime radicalC22×A4C2
cosocleC2×A4C22
LCSG ▷ A4
UCSC1 ◁ C22

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

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

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×SL(2,ℤ/4), SL(2,ℤ/4)⋊C4, C4⋊SL(2,ℤ/4), …
maximal normal subgroup ofC4×SL(2,ℤ/4), SL(2,ℤ/4)⋊C4, C4⋊SL(2,ℤ/4), …
maximal quotient ofC2×A4⋊C8, #5409, C4×SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 194);