metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C157⋊C3, SmallGroup(471,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C157 — C157⋊C3 |
Generators and relations for C157⋊C3
G = < a,b | a157=b3=1, bab-1=a144 >
Smallest permutation representation of C157⋊C3
►On 157 points: primitive
Generators in S157
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157)
(2 13 145)(3 25 132)(4 37 119)(5 49 106)(6 61 93)(7 73 80)(8 85 67)(9 97 54)(10 109 41)(11 121 28)(12 133 15)(14 157 146)(16 24 120)(17 36 107)(18 48 94)(19 60 81)(20 72 68)(21 84 55)(22 96 42)(23 108 29)(26 144 147)(27 156 134)(30 35 95)(31 47 82)(32 59 69)(33 71 56)(34 83 43)(38 131 148)(39 143 135)(40 155 122)(44 46 70)(45 58 57)(50 118 149)(51 130 136)(52 142 123)(53 154 110)(62 105 150)(63 117 137)(64 129 124)(65 141 111)(66 153 98)(74 92 151)(75 104 138)(76 116 125)(77 128 112)(78 140 99)(79 152 86)(87 91 139)(88 103 126)(89 115 113)(90 127 100)(101 102 114)
G:=sub<Sym(157)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157), (2,13,145)(3,25,132)(4,37,119)(5,49,106)(6,61,93)(7,73,80)(8,85,67)(9,97,54)(10,109,41)(11,121,28)(12,133,15)(14,157,146)(16,24,120)(17,36,107)(18,48,94)(19,60,81)(20,72,68)(21,84,55)(22,96,42)(23,108,29)(26,144,147)(27,156,134)(30,35,95)(31,47,82)(32,59,69)(33,71,56)(34,83,43)(38,131,148)(39,143,135)(40,155,122)(44,46,70)(45,58,57)(50,118,149)(51,130,136)(52,142,123)(53,154,110)(62,105,150)(63,117,137)(64,129,124)(65,141,111)(66,153,98)(74,92,151)(75,104,138)(76,116,125)(77,128,112)(78,140,99)(79,152,86)(87,91,139)(88,103,126)(89,115,113)(90,127,100)(101,102,114)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157), (2,13,145)(3,25,132)(4,37,119)(5,49,106)(6,61,93)(7,73,80)(8,85,67)(9,97,54)(10,109,41)(11,121,28)(12,133,15)(14,157,146)(16,24,120)(17,36,107)(18,48,94)(19,60,81)(20,72,68)(21,84,55)(22,96,42)(23,108,29)(26,144,147)(27,156,134)(30,35,95)(31,47,82)(32,59,69)(33,71,56)(34,83,43)(38,131,148)(39,143,135)(40,155,122)(44,46,70)(45,58,57)(50,118,149)(51,130,136)(52,142,123)(53,154,110)(62,105,150)(63,117,137)(64,129,124)(65,141,111)(66,153,98)(74,92,151)(75,104,138)(76,116,125)(77,128,112)(78,140,99)(79,152,86)(87,91,139)(88,103,126)(89,115,113)(90,127,100)(101,102,114) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157)], [(2,13,145),(3,25,132),(4,37,119),(5,49,106),(6,61,93),(7,73,80),(8,85,67),(9,97,54),(10,109,41),(11,121,28),(12,133,15),(14,157,146),(16,24,120),(17,36,107),(18,48,94),(19,60,81),(20,72,68),(21,84,55),(22,96,42),(23,108,29),(26,144,147),(27,156,134),(30,35,95),(31,47,82),(32,59,69),(33,71,56),(34,83,43),(38,131,148),(39,143,135),(40,155,122),(44,46,70),(45,58,57),(50,118,149),(51,130,136),(52,142,123),(53,154,110),(62,105,150),(63,117,137),(64,129,124),(65,141,111),(66,153,98),(74,92,151),(75,104,138),(76,116,125),(77,128,112),(78,140,99),(79,152,86),(87,91,139),(88,103,126),(89,115,113),(90,127,100),(101,102,114)]])
55 conjugacy classes
| class | 1 | 3A | 3B | 157A | ··· | 157AZ |
| order | 1 | 3 | 3 | 157 | ··· | 157 |
| size | 1 | 157 | 157 | 3 | ··· | 3 |
55 irreducible representations
| dim | 1 | 1 | 3 |
| type | + | ||
| image | C1 | C3 | C157⋊C3 |
| kernel | C157⋊C3 | C157 | C1 |
| # reps | 1 | 2 | 52 |
Matrix representation of C157⋊C3 ►in GL3(𝔽3769) generated by
| 2212 | 1 | 0 |
| 1463 | 0 | 1 |
| 3232 | 3455 | 3419 |
| 1215 | 2917 | 913 |
| 1018 | 2898 | 3064 |
| 1408 | 2925 | 3425 |
G:=sub<GL(3,GF(3769))| [2212,1463,3232,1,0,3455,0,1,3419],[1215,1018,1408,2917,2898,2925,913,3064,3425] >;
C157⋊C3 in GAP, Magma, Sage, TeX
C_{157}\rtimes C_3 % in TeX
G:=Group("C157:C3"); // GroupNames label
G:=SmallGroup(471,1);
// by ID
G=gap.SmallGroup(471,1);
# by ID
G:=PCGroup([2,-3,-157,145]);
// Polycyclic
G:=Group<a,b|a^157=b^3=1,b*a*b^-1=a^144>;
// generators/relations
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