direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C23, C7⋊C46, C161⋊3C2, SmallGroup(322,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C23 |
Generators and relations for D7×C23
G = < a,b,c | a23=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C23
►On 161 points
Generators in S161
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161)
(1 90 39 94 60 152 120)(2 91 40 95 61 153 121)(3 92 41 96 62 154 122)(4 70 42 97 63 155 123)(5 71 43 98 64 156 124)(6 72 44 99 65 157 125)(7 73 45 100 66 158 126)(8 74 46 101 67 159 127)(9 75 24 102 68 160 128)(10 76 25 103 69 161 129)(11 77 26 104 47 139 130)(12 78 27 105 48 140 131)(13 79 28 106 49 141 132)(14 80 29 107 50 142 133)(15 81 30 108 51 143 134)(16 82 31 109 52 144 135)(17 83 32 110 53 145 136)(18 84 33 111 54 146 137)(19 85 34 112 55 147 138)(20 86 35 113 56 148 116)(21 87 36 114 57 149 117)(22 88 37 115 58 150 118)(23 89 38 93 59 151 119)
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 116)(21 117)(22 118)(23 119)(24 68)(25 69)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)
G:=sub<Sym(161)| (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161), (1,90,39,94,60,152,120)(2,91,40,95,61,153,121)(3,92,41,96,62,154,122)(4,70,42,97,63,155,123)(5,71,43,98,64,156,124)(6,72,44,99,65,157,125)(7,73,45,100,66,158,126)(8,74,46,101,67,159,127)(9,75,24,102,68,160,128)(10,76,25,103,69,161,129)(11,77,26,104,47,139,130)(12,78,27,105,48,140,131)(13,79,28,106,49,141,132)(14,80,29,107,50,142,133)(15,81,30,108,51,143,134)(16,82,31,109,52,144,135)(17,83,32,110,53,145,136)(18,84,33,111,54,146,137)(19,85,34,112,55,147,138)(20,86,35,113,56,148,116)(21,87,36,114,57,149,117)(22,88,37,115,58,150,118)(23,89,38,93,59,151,119), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,116)(21,117)(22,118)(23,119)(24,68)(25,69)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)>;
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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161), (1,90,39,94,60,152,120)(2,91,40,95,61,153,121)(3,92,41,96,62,154,122)(4,70,42,97,63,155,123)(5,71,43,98,64,156,124)(6,72,44,99,65,157,125)(7,73,45,100,66,158,126)(8,74,46,101,67,159,127)(9,75,24,102,68,160,128)(10,76,25,103,69,161,129)(11,77,26,104,47,139,130)(12,78,27,105,48,140,131)(13,79,28,106,49,141,132)(14,80,29,107,50,142,133)(15,81,30,108,51,143,134)(16,82,31,109,52,144,135)(17,83,32,110,53,145,136)(18,84,33,111,54,146,137)(19,85,34,112,55,147,138)(20,86,35,113,56,148,116)(21,87,36,114,57,149,117)(22,88,37,115,58,150,118)(23,89,38,93,59,151,119), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,116)(21,117)(22,118)(23,119)(24,68)(25,69)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154) );
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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)], [(1,90,39,94,60,152,120),(2,91,40,95,61,153,121),(3,92,41,96,62,154,122),(4,70,42,97,63,155,123),(5,71,43,98,64,156,124),(6,72,44,99,65,157,125),(7,73,45,100,66,158,126),(8,74,46,101,67,159,127),(9,75,24,102,68,160,128),(10,76,25,103,69,161,129),(11,77,26,104,47,139,130),(12,78,27,105,48,140,131),(13,79,28,106,49,141,132),(14,80,29,107,50,142,133),(15,81,30,108,51,143,134),(16,82,31,109,52,144,135),(17,83,32,110,53,145,136),(18,84,33,111,54,146,137),(19,85,34,112,55,147,138),(20,86,35,113,56,148,116),(21,87,36,114,57,149,117),(22,88,37,115,58,150,118),(23,89,38,93,59,151,119)], [(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,116),(21,117),(22,118),(23,119),(24,68),(25,69),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154)]])
115 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 23A | ··· | 23V | 46A | ··· | 46V | 161A | ··· | 161BN |
| order | 1 | 2 | 7 | 7 | 7 | 23 | ··· | 23 | 46 | ··· | 46 | 161 | ··· | 161 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C23 | C46 | D7 | D7×C23 |
| kernel | D7×C23 | C161 | D7 | C7 | C23 | C1 |
| # reps | 1 | 1 | 22 | 22 | 3 | 66 |
Matrix representation of D7×C23 ►in GL2(𝔽967) generated by
| 641 | 0 |
| 0 | 641 |
| 256 | 1 |
| 966 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(967))| [641,0,0,641],[256,966,1,0],[0,1,1,0] >;
D7×C23 in GAP, Magma, Sage, TeX
D_7\times C_{23} % in TeX
G:=Group("D7xC23"); // GroupNames label
G:=SmallGroup(322,1);
// by ID
G=gap.SmallGroup(322,1);
# by ID
G:=PCGroup([3,-2,-23,-7,2486]);
// Polycyclic
G:=Group<a,b,c|a^23=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export