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## G = C4×S4order 96 = 25·3

### Direct product of C4 and S4

Aliases: C4×S4, C23.2D6, (C2×S4).C2, C22⋊(C4×S3), C4(A4⋊C4), A4⋊C42C2, (C4×A4)⋊2C2, A41(C2×C4), C2.1(C2×S4), (C22×C4)⋊1S3, (C2×A4).2C22, SmallGroup(96,186)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C4×S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4
 Lower central A4 — C4×S4
 Upper central C1 — C4

Generators and relations for C4×S4
G = < a,b,c,d,e | a4=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 172 in 56 conjugacy classes, 14 normal (12 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×5], C22, C22 [×6], S3 [×2], C6, C2×C4 [×7], D4 [×4], C23, C23, Dic3, C12, A4, D6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C4×S3, S4 [×2], C2×A4, C4×D4, A4⋊C4, C4×A4, C2×S4, C4×S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4

Character table of C4×S4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6 12A 12B size 1 1 3 3 6 6 8 1 1 3 3 6 6 6 6 6 6 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 -i i -i i i -1 1 -i -i i -1 i -i linear of order 4 ρ6 1 -1 1 -1 1 -1 1 i -i i -i -i -1 1 i i -i -1 -i i linear of order 4 ρ7 1 -1 1 -1 -1 1 1 -i i -i i -i 1 -1 i i -i -1 i -i linear of order 4 ρ8 1 -1 1 -1 -1 1 1 i -i i -i i 1 -1 -i -i i -1 -i i linear of order 4 ρ9 2 2 2 2 0 0 -1 -2 -2 -2 -2 0 0 0 0 0 0 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 2 -2 0 0 -1 -2i 2i -2i 2i 0 0 0 0 0 0 1 -i i complex lifted from C4×S3 ρ12 2 -2 2 -2 0 0 -1 2i -2i 2i -2i 0 0 0 0 0 0 1 i -i complex lifted from C4×S3 ρ13 3 3 -1 -1 1 1 0 3 3 -1 -1 -1 -1 -1 1 -1 1 0 0 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 -1 -1 0 3 3 -1 -1 1 1 1 -1 1 -1 0 0 0 orthogonal lifted from S4 ρ15 3 3 -1 -1 -1 -1 0 -3 -3 1 1 -1 1 1 1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 -1 -1 1 1 0 -3 -3 1 1 1 -1 -1 -1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ17 3 -3 -1 1 1 -1 0 3i -3i -i i i 1 -1 i -i -i 0 0 0 complex faithful ρ18 3 -3 -1 1 -1 1 0 -3i 3i i -i i -1 1 i -i -i 0 0 0 complex faithful ρ19 3 -3 -1 1 -1 1 0 3i -3i -i i -i -1 1 -i i i 0 0 0 complex faithful ρ20 3 -3 -1 1 1 -1 0 -3i 3i i -i -i 1 -1 -i i i 0 0 0 complex faithful

Permutation representations of C4×S4
On 12 points - transitive group 12T53
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(9 11)(10 12)
(5 7)(6 8)(9 11)(10 12)
(1 9 7)(2 10 8)(3 11 5)(4 12 6)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)

G:=sub<Sym(12)| (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(9,11)(10,12), (5,7)(6,8)(9,11)(10,12), (1,9,7)(2,10,8)(3,11,5)(4,12,6), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)>;

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

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(9,11),(10,12)], [(5,7),(6,8),(9,11),(10,12)], [(1,9,7),(2,10,8),(3,11,5),(4,12,6)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)])

G:=TransitiveGroup(12,53);

On 16 points - transitive group 16T181
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 16)(2 13)(3 14)(4 15)(5 9)(6 10)(7 11)(8 12)
(5 9 15)(6 10 16)(7 11 13)(8 12 14)
(5 9)(6 10)(7 11)(8 12)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,16)(2,13)(3,14)(4,15)(5,9)(6,10)(7,11)(8,12), (5,9,15)(6,10,16)(7,11,13)(8,12,14), (5,9)(6,10)(7,11)(8,12)>;

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

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(1,16),(2,13),(3,14),(4,15),(5,9),(6,10),(7,11),(8,12)], [(5,9,15),(6,10,16),(7,11,13),(8,12,14)], [(5,9),(6,10),(7,11),(8,12)])

G:=TransitiveGroup(16,181);

On 24 points - transitive group 24T129
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 17 9)(6 18 10)(7 19 11)(8 20 12)
(1 3)(2 4)(9 17)(10 18)(11 19)(12 20)(13 23)(14 24)(15 21)(16 22)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22), (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,17,9)(6,18,10)(7,19,11)(8,20,12), (1,3)(2,4)(9,17)(10,18)(11,19)(12,20)(13,23)(14,24)(15,21)(16,22)>;

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

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

G:=TransitiveGroup(24,129);

On 24 points - transitive group 24T130
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 17 9)(6 18 10)(7 19 11)(8 20 12)
(1 8)(2 5)(3 6)(4 7)(9 24)(10 21)(11 22)(12 23)(13 18)(14 19)(15 20)(16 17)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22), (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,17,9)(6,18,10)(7,19,11)(8,20,12), (1,8)(2,5)(3,6)(4,7)(9,24)(10,21)(11,22)(12,23)(13,18)(14,19)(15,20)(16,17)>;

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

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

G:=TransitiveGroup(24,130);

On 24 points - transitive group 24T167
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 11 13)(2 12 14)(3 9 15)(4 10 16)(5 18 21)(6 19 22)(7 20 23)(8 17 24)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 20)(10 17)(11 18)(12 19)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,13)(2,12,14)(3,9,15)(4,10,16)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,20)(10,17)(11,18)(12,19)>;

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

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

G:=TransitiveGroup(24,167);

On 24 points - transitive group 24T168
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 16)(2 13)(3 14)(4 15)(5 10)(6 11)(7 12)(8 9)
(5 10)(6 11)(7 12)(8 9)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 19 13)(6 20 14)(7 17 15)(8 18 16)
(5 19)(6 20)(7 17)(8 18)(9 23)(10 24)(11 21)(12 22)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,13)(3,14)(4,15)(5,10)(6,11)(7,12)(8,9), (5,10)(6,11)(7,12)(8,9)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,19,13)(6,20,14)(7,17,15)(8,18,16), (5,19)(6,20)(7,17)(8,18)(9,23)(10,24)(11,21)(12,22)>;

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

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

G:=TransitiveGroup(24,168);

On 24 points - transitive group 24T169
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)
(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(1 8)(2 5)(3 6)(4 7)(9 20)(10 17)(11 18)(12 19)(13 24)(14 21)(15 22)(16 23)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13), (9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (1,8)(2,5)(3,6)(4,7)(9,20)(10,17)(11,18)(12,19)(13,24)(14,21)(15,22)(16,23)>;

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

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

G:=TransitiveGroup(24,169);

C4×S4 is a maximal subgroup of
C8⋊S4  C24.10D6  D42S4  Q84S4  Dic32S4  Dic52S4
C4×S4 is a maximal quotient of
CSU2(𝔽3)⋊C4  GL2(𝔽3)⋊C4  C8⋊S4  CU2(𝔽3)  C8.5S4  C24.3D6  C24.5D6  Dic32S4  Dic52S4

Polynomial with Galois group C4×S4 over ℚ
actionf(x)Disc(f)
12T53x12-11x10-x9+46x8+12x7-90x6-48x5+78x4+73x3-19x2-36x-4220·59·2294

Matrix representation of C4×S4 in GL3(𝔽5) generated by

 3 0 0 0 3 0 0 0 3
,
 4 0 0 0 4 0 0 0 1
,
 1 0 0 0 4 0 0 0 4
,
 0 0 4 2 0 0 0 2 0
,
 1 0 0 0 0 3 0 2 0
G:=sub<GL(3,GF(5))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,4,0,0,0,1],[1,0,0,0,4,0,0,0,4],[0,2,0,0,0,2,4,0,0],[1,0,0,0,0,2,0,3,0] >;

C4×S4 in GAP, Magma, Sage, TeX

C_4\times S_4
% in TeX

G:=Group("C4xS4");
// GroupNames label

G:=SmallGroup(96,186);
// by ID

G=gap.SmallGroup(96,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,31,387,1444,202,869,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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