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## G = C31⋊C12order 372 = 22·3·31

### The semidirect product of C31 and C12 acting via C12/C2=C6

Aliases: C31⋊C12, C62.C6, Dic31⋊C3, C31⋊C3⋊C4, C2.(C31⋊C6), (C2×C31⋊C3).C2, SmallGroup(372,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C31⋊C12
 Chief series C1 — C31 — C62 — C2×C31⋊C3 — C31⋊C12
 Lower central C31 — C31⋊C12
 Upper central C1 — C2

Generators and relations for C31⋊C12
G = < a,b | a31=b12=1, bab-1=a26 >

Character table of C31⋊C12

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 31A 31B 31C 31D 31E 62A 62B 62C 62D 62E size 1 1 31 31 31 31 31 31 31 31 31 31 6 6 6 6 6 6 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 ζ3 ζ32 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 1 -i i -1 -1 i i -i -i 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 1 i -i -1 -1 -i -i i i 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ9 1 -1 ζ32 ζ3 i -i ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 12 ρ11 1 -1 ζ32 ζ3 -i i ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 12 ρ13 6 6 0 0 0 0 0 0 0 0 0 0 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 orthogonal lifted from C31⋊C6 ρ14 6 6 0 0 0 0 0 0 0 0 0 0 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 orthogonal lifted from C31⋊C6 ρ15 6 6 0 0 0 0 0 0 0 0 0 0 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 orthogonal lifted from C31⋊C6 ρ16 6 6 0 0 0 0 0 0 0 0 0 0 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 orthogonal lifted from C31⋊C6 ρ17 6 6 0 0 0 0 0 0 0 0 0 0 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 orthogonal lifted from C31⋊C6 ρ18 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 symplectic faithful, Schur index 2 ρ19 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 symplectic faithful, Schur index 2 ρ20 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 symplectic faithful, Schur index 2 ρ21 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 symplectic faithful, Schur index 2 ρ22 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 -ζ3127-ζ3124-ζ3120-ζ3111-ζ317-ζ314 -ζ3130-ζ3126-ζ3125-ζ316-ζ315-ζ31 -ζ3123-ζ3122-ζ3117-ζ3114-ζ319-ζ318 -ζ3129-ζ3121-ζ3119-ζ3112-ζ3110-ζ312 -ζ3128-ζ3118-ζ3116-ζ3115-ζ3113-ζ313 symplectic faithful, Schur index 2

Smallest permutation representation of C31⋊C12
On 124 points
Generators in S124
```(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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62)(63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93)(94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 94 60 63)(2 100 34 93 26 120 61 69 6 124 54 89)(3 106 39 92 20 115 62 75 11 123 48 84)(4 112 44 91 14 110 32 81 16 122 42 79)(5 118 49 90 8 105 33 87 21 121 36 74)(7 99 59 88 27 95 35 68 31 119 55 64)(9 111 38 86 15 116 37 80 10 117 43 85)(12 98 53 83 28 101 40 67 25 114 56 70)(13 104 58 82 22 96 41 73 30 113 50 65)(17 97 47 78 29 107 45 66 19 109 57 76)(18 103 52 77 23 102 46 72 24 108 51 71)```

`G:=sub<Sym(124)| (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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,94,60,63)(2,100,34,93,26,120,61,69,6,124,54,89)(3,106,39,92,20,115,62,75,11,123,48,84)(4,112,44,91,14,110,32,81,16,122,42,79)(5,118,49,90,8,105,33,87,21,121,36,74)(7,99,59,88,27,95,35,68,31,119,55,64)(9,111,38,86,15,116,37,80,10,117,43,85)(12,98,53,83,28,101,40,67,25,114,56,70)(13,104,58,82,22,96,41,73,30,113,50,65)(17,97,47,78,29,107,45,66,19,109,57,76)(18,103,52,77,23,102,46,72,24,108,51,71)>;`

`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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,94,60,63)(2,100,34,93,26,120,61,69,6,124,54,89)(3,106,39,92,20,115,62,75,11,123,48,84)(4,112,44,91,14,110,32,81,16,122,42,79)(5,118,49,90,8,105,33,87,21,121,36,74)(7,99,59,88,27,95,35,68,31,119,55,64)(9,111,38,86,15,116,37,80,10,117,43,85)(12,98,53,83,28,101,40,67,25,114,56,70)(13,104,58,82,22,96,41,73,30,113,50,65)(17,97,47,78,29,107,45,66,19,109,57,76)(18,103,52,77,23,102,46,72,24,108,51,71) );`

`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),(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,94,60,63),(2,100,34,93,26,120,61,69,6,124,54,89),(3,106,39,92,20,115,62,75,11,123,48,84),(4,112,44,91,14,110,32,81,16,122,42,79),(5,118,49,90,8,105,33,87,21,121,36,74),(7,99,59,88,27,95,35,68,31,119,55,64),(9,111,38,86,15,116,37,80,10,117,43,85),(12,98,53,83,28,101,40,67,25,114,56,70),(13,104,58,82,22,96,41,73,30,113,50,65),(17,97,47,78,29,107,45,66,19,109,57,76),(18,103,52,77,23,102,46,72,24,108,51,71)]])`

Matrix representation of C31⋊C12 in GL7(𝔽373)

 1 0 0 0 0 0 0 0 372 1 0 0 0 0 0 372 0 1 0 0 0 0 372 0 0 1 0 0 0 372 0 0 0 1 0 0 372 0 0 0 0 1 0 314 139 74 299 234 58
,
 69 0 0 0 0 0 0 0 105 272 125 26 43 42 0 68 231 371 276 323 150 0 86 116 68 55 63 358 0 249 255 347 103 366 273 0 315 295 293 295 315 279 0 226 37 136 266 368 297

`G:=sub<GL(7,GF(373))| [1,0,0,0,0,0,0,0,372,372,372,372,372,314,0,1,0,0,0,0,139,0,0,1,0,0,0,74,0,0,0,1,0,0,299,0,0,0,0,1,0,234,0,0,0,0,0,1,58],[69,0,0,0,0,0,0,0,105,68,86,249,315,226,0,272,231,116,255,295,37,0,125,371,68,347,293,136,0,26,276,55,103,295,266,0,43,323,63,366,315,368,0,42,150,358,273,279,297] >;`

C31⋊C12 in GAP, Magma, Sage, TeX

`C_{31}\rtimes C_{12}`
`% in TeX`

`G:=Group("C31:C12");`
`// GroupNames label`

`G:=SmallGroup(372,1);`
`// by ID`

`G=gap.SmallGroup(372,1);`
`# by ID`

`G:=PCGroup([4,-2,-3,-2,-31,24,5763,2407]);`
`// Polycyclic`

`G:=Group<a,b|a^31=b^12=1,b*a*b^-1=a^26>;`
`// generators/relations`

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