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## G = C11⋊S4order 264 = 23·3·11

### The semidirect product of C11 and S4 acting via S4/A4=C2

Aliases: C11⋊S4, A4⋊D11, C22⋊D33, (C2×C22)⋊2S3, (C11×A4)⋊1C2, SmallGroup(264,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×A4 — C11⋊S4
 Chief series C1 — C22 — C2×C22 — C11×A4 — C11⋊S4
 Lower central C11×A4 — C11⋊S4
 Upper central C1

Generators and relations for C11⋊S4
G = < a,b,c,d,e | a11=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
66C2
4C3
33C22
33C4
44S3
3C22
6D11
4C33
33D4
3D22
4D33
11S4

Character table of C11⋊S4

 class 1 2A 2B 3 4 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 33A 33B 33C 33D 33E 33F 33G 33H 33I 33J size 1 3 66 8 66 2 2 2 2 2 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 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 1 1 1 1 1 linear of order 2 ρ3 2 2 0 -1 0 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D11 ρ5 2 2 0 2 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D11 ρ6 2 2 0 2 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D11 ρ7 2 2 0 2 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D11 ρ8 2 2 0 2 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D11 ρ9 2 2 0 -1 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ3ζ117+ζ3ζ114-ζ117 ζ3ζ119-ζ3ζ112-ζ112 ζ32ζ118-ζ32ζ113-ζ113 ζ3ζ118-ζ3ζ113-ζ113 ζ32ζ119-ζ32ζ112-ζ112 ζ3ζ117-ζ3ζ114-ζ114 ζ32ζ1110-ζ32ζ11-ζ11 ζ3ζ116-ζ3ζ115-ζ115 -ζ3ζ116+ζ3ζ115-ζ116 -ζ32ζ1110+ζ32ζ11-ζ1110 orthogonal lifted from D33 ρ10 2 2 0 -1 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ3ζ119-ζ3ζ112-ζ112 ζ32ζ1110-ζ32ζ11-ζ11 -ζ3ζ117+ζ3ζ114-ζ117 ζ3ζ117-ζ3ζ114-ζ114 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ32ζ119-ζ32ζ112-ζ112 -ζ3ζ116+ζ3ζ115-ζ116 ζ3ζ118-ζ3ζ113-ζ113 ζ32ζ118-ζ32ζ113-ζ113 ζ3ζ116-ζ3ζ115-ζ115 orthogonal lifted from D33 ρ11 2 2 0 -1 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ3ζ117-ζ3ζ114-ζ114 ζ32ζ119-ζ32ζ112-ζ112 ζ3ζ118-ζ3ζ113-ζ113 ζ32ζ118-ζ32ζ113-ζ113 ζ3ζ119-ζ3ζ112-ζ112 -ζ3ζ117+ζ3ζ114-ζ117 -ζ32ζ1110+ζ32ζ11-ζ1110 -ζ3ζ116+ζ3ζ115-ζ116 ζ3ζ116-ζ3ζ115-ζ115 ζ32ζ1110-ζ32ζ11-ζ11 orthogonal lifted from D33 ρ12 2 2 0 -1 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ3ζ116-ζ3ζ115-ζ115 ζ32ζ119-ζ32ζ112-ζ112 ζ3ζ119-ζ3ζ112-ζ112 -ζ3ζ116+ζ3ζ115-ζ116 ζ32ζ1110-ζ32ζ11-ζ11 ζ3ζ118-ζ3ζ113-ζ113 -ζ3ζ117+ζ3ζ114-ζ117 ζ3ζ117-ζ3ζ114-ζ114 ζ32ζ118-ζ32ζ113-ζ113 orthogonal lifted from D33 ρ13 2 2 0 -1 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ3ζ116+ζ3ζ115-ζ116 ζ32ζ118-ζ32ζ113-ζ113 ζ32ζ1110-ζ32ζ11-ζ11 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ3ζ118-ζ3ζ113-ζ113 ζ3ζ116-ζ3ζ115-ζ115 -ζ3ζ117+ζ3ζ114-ζ117 ζ32ζ119-ζ32ζ112-ζ112 ζ3ζ119-ζ3ζ112-ζ112 ζ3ζ117-ζ3ζ114-ζ114 orthogonal lifted from D33 ρ14 2 2 0 -1 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ32ζ118-ζ32ζ113-ζ113 -ζ3ζ117+ζ3ζ114-ζ117 -ζ3ζ116+ζ3ζ115-ζ116 ζ3ζ116-ζ3ζ115-ζ115 ζ3ζ117-ζ3ζ114-ζ114 ζ3ζ118-ζ3ζ113-ζ113 ζ3ζ119-ζ3ζ112-ζ112 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ32ζ1110-ζ32ζ11-ζ11 ζ32ζ119-ζ32ζ112-ζ112 orthogonal lifted from D33 ρ15 2 2 0 -1 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ32ζ1110-ζ32ζ11-ζ11 -ζ3ζ116+ζ3ζ115-ζ116 ζ3ζ119-ζ3ζ112-ζ112 ζ32ζ119-ζ32ζ112-ζ112 ζ3ζ116-ζ3ζ115-ζ115 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ32ζ118-ζ32ζ113-ζ113 ζ3ζ117-ζ3ζ114-ζ114 -ζ3ζ117+ζ3ζ114-ζ117 ζ3ζ118-ζ3ζ113-ζ113 orthogonal lifted from D33 ρ16 2 2 0 -1 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ3ζ116-ζ3ζ115-ζ115 ζ3ζ118-ζ3ζ113-ζ113 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ32ζ1110-ζ32ζ11-ζ11 ζ32ζ118-ζ32ζ113-ζ113 -ζ3ζ116+ζ3ζ115-ζ116 ζ3ζ117-ζ3ζ114-ζ114 ζ3ζ119-ζ3ζ112-ζ112 ζ32ζ119-ζ32ζ112-ζ112 -ζ3ζ117+ζ3ζ114-ζ117 orthogonal lifted from D33 ρ17 2 2 0 -1 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ3ζ118-ζ3ζ113-ζ113 ζ3ζ117-ζ3ζ114-ζ114 ζ3ζ116-ζ3ζ115-ζ115 -ζ3ζ116+ζ3ζ115-ζ116 -ζ3ζ117+ζ3ζ114-ζ117 ζ32ζ118-ζ32ζ113-ζ113 ζ32ζ119-ζ32ζ112-ζ112 ζ32ζ1110-ζ32ζ11-ζ11 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ3ζ119-ζ3ζ112-ζ112 orthogonal lifted from D33 ρ18 2 2 0 -1 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ32ζ119-ζ32ζ112-ζ112 -ζ32ζ1110+ζ32ζ11-ζ1110 ζ3ζ117-ζ3ζ114-ζ114 -ζ3ζ117+ζ3ζ114-ζ117 ζ32ζ1110-ζ32ζ11-ζ11 ζ3ζ119-ζ3ζ112-ζ112 ζ3ζ116-ζ3ζ115-ζ115 ζ32ζ118-ζ32ζ113-ζ113 ζ3ζ118-ζ3ζ113-ζ113 -ζ3ζ116+ζ3ζ115-ζ116 orthogonal lifted from D33 ρ19 3 -1 -1 0 1 3 3 3 3 3 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ20 3 -1 1 0 -1 3 3 3 3 3 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ21 6 -2 0 0 0 3ζ118+3ζ113 3ζ116+3ζ115 3ζ119+3ζ112 3ζ117+3ζ114 3ζ1110+3ζ11 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ22 6 -2 0 0 0 3ζ1110+3ζ11 3ζ119+3ζ112 3ζ118+3ζ113 3ζ116+3ζ115 3ζ117+3ζ114 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ23 6 -2 0 0 0 3ζ117+3ζ114 3ζ118+3ζ113 3ζ1110+3ζ11 3ζ119+3ζ112 3ζ116+3ζ115 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 6 -2 0 0 0 3ζ116+3ζ115 3ζ1110+3ζ11 3ζ117+3ζ114 3ζ118+3ζ113 3ζ119+3ζ112 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 -2 0 0 0 3ζ119+3ζ112 3ζ117+3ζ114 3ζ116+3ζ115 3ζ1110+3ζ11 3ζ118+3ζ113 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C11⋊S4
On 44 points
Generators in S44
```(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)(23 24 25 26 27 28 29 30 31 32 33)(34 35 36 37 38 39 40 41 42 43 44)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 12)(11 13)(23 41)(24 42)(25 43)(26 44)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(33 40)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 31)(13 32)(14 33)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)
(12 38 31)(13 39 32)(14 40 33)(15 41 23)(16 42 24)(17 43 25)(18 44 26)(19 34 27)(20 35 28)(21 36 29)(22 37 30)
(2 11)(3 10)(4 9)(5 8)(6 7)(12 24)(13 23)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(34 35)(36 44)(37 43)(38 42)(39 41)```

`G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,12)(11,13)(23,41)(24,42)(25,43)(26,44)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,31)(13,32)(14,33)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30), (12,38,31)(13,39,32)(14,40,33)(15,41,23)(16,42,24)(17,43,25)(18,44,26)(19,34,27)(20,35,28)(21,36,29)(22,37,30), (2,11)(3,10)(4,9)(5,8)(6,7)(12,24)(13,23)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(34,35)(36,44)(37,43)(38,42)(39,41)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,12)(11,13)(23,41)(24,42)(25,43)(26,44)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,31)(13,32)(14,33)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30), (12,38,31)(13,39,32)(14,40,33)(15,41,23)(16,42,24)(17,43,25)(18,44,26)(19,34,27)(20,35,28)(21,36,29)(22,37,30), (2,11)(3,10)(4,9)(5,8)(6,7)(12,24)(13,23)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(34,35)(36,44)(37,43)(38,42)(39,41) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33),(34,35,36,37,38,39,40,41,42,43,44)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,12),(11,13),(23,41),(24,42),(25,43),(26,44),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(33,40)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,31),(13,32),(14,33),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30)], [(12,38,31),(13,39,32),(14,40,33),(15,41,23),(16,42,24),(17,43,25),(18,44,26),(19,34,27),(20,35,28),(21,36,29),(22,37,30)], [(2,11),(3,10),(4,9),(5,8),(6,7),(12,24),(13,23),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(34,35),(36,44),(37,43),(38,42),(39,41)]])`

Matrix representation of C11⋊S4 in GL5(𝔽397)

 267 385 0 0 0 12 279 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 396 0 0 0 1 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 396 396 0 0 396 0 1 0 0 0 0 396
,
 396 396 0 0 0 1 0 0 0 0 0 0 396 396 0 0 0 1 0 0 0 0 396 0 1
,
 12 130 0 0 0 118 385 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 396

`G:=sub<GL(5,GF(397))| [267,12,0,0,0,385,279,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,396,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,396,0,0,0,396,0,0,0,0,396,1,396],[396,1,0,0,0,396,0,0,0,0,0,0,396,1,396,0,0,396,0,0,0,0,0,0,1],[12,118,0,0,0,130,385,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,396] >;`

C11⋊S4 in GAP, Magma, Sage, TeX

`C_{11}\rtimes S_4`
`% in TeX`

`G:=Group("C11:S4");`
`// GroupNames label`

`G:=SmallGroup(264,32);`
`// by ID`

`G=gap.SmallGroup(264,32);`
`# by ID`

`G:=PCGroup([5,-2,-3,-11,-2,2,41,902,2643,1328,1654,2484]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^11=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,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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