metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C127⋊C3, SmallGroup(381,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C127 — C127⋊C3 |
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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