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## G = C127⋊C3order 381 = 3·127

### The semidirect product of C127 and C3 acting faithfully

Aliases: C127⋊C3, SmallGroup(381,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C127 — C127⋊C3
 Chief series C1 — C127 — C127⋊C3
 Lower central C127 — C127⋊C3
 Upper central C1

Generators and relations for C127⋊C3
G = < a,b | a127=b3=1, bab-1=a19 >

Smallest permutation representation of C127⋊C3
On 127 points: primitive
Generators in S127
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127)
(2 108 20)(3 88 39)(4 68 58)(5 48 77)(6 28 96)(7 8 115)(9 95 26)(10 75 45)(11 55 64)(12 35 83)(13 15 102)(14 122 121)(16 82 32)(17 62 51)(18 42 70)(19 22 89)(21 109 127)(23 69 38)(24 49 57)(25 29 76)(27 116 114)(30 56 44)(31 36 63)(33 123 101)(34 103 120)(37 43 50)(40 110 107)(41 90 126)(46 117 94)(47 97 113)(52 124 81)(53 104 100)(54 84 119)(59 111 87)(60 91 106)(61 71 125)(65 118 74)(66 98 93)(67 78 112)(72 105 80)(73 85 99)(79 92 86)```

`G:=sub<Sym(127)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127), (2,108,20)(3,88,39)(4,68,58)(5,48,77)(6,28,96)(7,8,115)(9,95,26)(10,75,45)(11,55,64)(12,35,83)(13,15,102)(14,122,121)(16,82,32)(17,62,51)(18,42,70)(19,22,89)(21,109,127)(23,69,38)(24,49,57)(25,29,76)(27,116,114)(30,56,44)(31,36,63)(33,123,101)(34,103,120)(37,43,50)(40,110,107)(41,90,126)(46,117,94)(47,97,113)(52,124,81)(53,104,100)(54,84,119)(59,111,87)(60,91,106)(61,71,125)(65,118,74)(66,98,93)(67,78,112)(72,105,80)(73,85,99)(79,92,86)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127), (2,108,20)(3,88,39)(4,68,58)(5,48,77)(6,28,96)(7,8,115)(9,95,26)(10,75,45)(11,55,64)(12,35,83)(13,15,102)(14,122,121)(16,82,32)(17,62,51)(18,42,70)(19,22,89)(21,109,127)(23,69,38)(24,49,57)(25,29,76)(27,116,114)(30,56,44)(31,36,63)(33,123,101)(34,103,120)(37,43,50)(40,110,107)(41,90,126)(46,117,94)(47,97,113)(52,124,81)(53,104,100)(54,84,119)(59,111,87)(60,91,106)(61,71,125)(65,118,74)(66,98,93)(67,78,112)(72,105,80)(73,85,99)(79,92,86) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127)], [(2,108,20),(3,88,39),(4,68,58),(5,48,77),(6,28,96),(7,8,115),(9,95,26),(10,75,45),(11,55,64),(12,35,83),(13,15,102),(14,122,121),(16,82,32),(17,62,51),(18,42,70),(19,22,89),(21,109,127),(23,69,38),(24,49,57),(25,29,76),(27,116,114),(30,56,44),(31,36,63),(33,123,101),(34,103,120),(37,43,50),(40,110,107),(41,90,126),(46,117,94),(47,97,113),(52,124,81),(53,104,100),(54,84,119),(59,111,87),(60,91,106),(61,71,125),(65,118,74),(66,98,93),(67,78,112),(72,105,80),(73,85,99),(79,92,86)]])`

45 conjugacy classes

 class 1 3A 3B 127A ··· 127AP order 1 3 3 127 ··· 127 size 1 127 127 3 ··· 3

45 irreducible representations

 dim 1 1 3 type + image C1 C3 C127⋊C3 kernel C127⋊C3 C127 C1 # reps 1 2 42

Matrix representation of C127⋊C3 in GL3(𝔽2287) generated by

 1160 1 0 1396 0 1 1 0 0
,
 1 126 1822 0 313 1664 0 1795 1973
`G:=sub<GL(3,GF(2287))| [1160,1396,1,1,0,0,0,1,0],[1,0,0,126,313,1795,1822,1664,1973] >;`

C127⋊C3 in GAP, Magma, Sage, TeX

`C_{127}\rtimes C_3`
`% in TeX`

`G:=Group("C127:C3");`
`// GroupNames label`

`G:=SmallGroup(381,1);`
`// by ID`

`G=gap.SmallGroup(381,1);`
`# by ID`

`G:=PCGroup([2,-3,-127,1285]);`
`// Polycyclic`

`G:=Group<a,b|a^127=b^3=1,b*a*b^-1=a^19>;`
`// generators/relations`

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