direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D23, C23⋊C14, C161⋊2C2, SmallGroup(322,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C7×D23 |
Generators and relations for C7×D23
G = < a,b,c | a7=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D23
►On 161 points
Generators in S161
(1 141 119 111 81 50 46)(2 142 120 112 82 51 24)(3 143 121 113 83 52 25)(4 144 122 114 84 53 26)(5 145 123 115 85 54 27)(6 146 124 93 86 55 28)(7 147 125 94 87 56 29)(8 148 126 95 88 57 30)(9 149 127 96 89 58 31)(10 150 128 97 90 59 32)(11 151 129 98 91 60 33)(12 152 130 99 92 61 34)(13 153 131 100 70 62 35)(14 154 132 101 71 63 36)(15 155 133 102 72 64 37)(16 156 134 103 73 65 38)(17 157 135 104 74 66 39)(18 158 136 105 75 67 40)(19 159 137 106 76 68 41)(20 160 138 107 77 69 42)(21 161 116 108 78 47 43)(22 139 117 109 79 48 44)(23 140 118 110 80 49 45)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(45 46)(47 52)(48 51)(49 50)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(106 115)(107 114)(108 113)(109 112)(110 111)(116 121)(117 120)(118 119)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(139 142)(140 141)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)
G:=sub<Sym(161)| (1,141,119,111,81,50,46)(2,142,120,112,82,51,24)(3,143,121,113,83,52,25)(4,144,122,114,84,53,26)(5,145,123,115,85,54,27)(6,146,124,93,86,55,28)(7,147,125,94,87,56,29)(8,148,126,95,88,57,30)(9,149,127,96,89,58,31)(10,150,128,97,90,59,32)(11,151,129,98,91,60,33)(12,152,130,99,92,61,34)(13,153,131,100,70,62,35)(14,154,132,101,71,63,36)(15,155,133,102,72,64,37)(16,156,134,103,73,65,38)(17,157,135,104,74,66,39)(18,158,136,105,75,67,40)(19,159,137,106,76,68,41)(20,160,138,107,77,69,42)(21,161,116,108,78,47,43)(22,139,117,109,79,48,44)(23,140,118,110,80,49,45), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(45,46)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,115)(107,114)(108,113)(109,112)(110,111)(116,121)(117,120)(118,119)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(139,142)(140,141)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153)>;
G:=Group( (1,141,119,111,81,50,46)(2,142,120,112,82,51,24)(3,143,121,113,83,52,25)(4,144,122,114,84,53,26)(5,145,123,115,85,54,27)(6,146,124,93,86,55,28)(7,147,125,94,87,56,29)(8,148,126,95,88,57,30)(9,149,127,96,89,58,31)(10,150,128,97,90,59,32)(11,151,129,98,91,60,33)(12,152,130,99,92,61,34)(13,153,131,100,70,62,35)(14,154,132,101,71,63,36)(15,155,133,102,72,64,37)(16,156,134,103,73,65,38)(17,157,135,104,74,66,39)(18,158,136,105,75,67,40)(19,159,137,106,76,68,41)(20,160,138,107,77,69,42)(21,161,116,108,78,47,43)(22,139,117,109,79,48,44)(23,140,118,110,80,49,45), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(45,46)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,115)(107,114)(108,113)(109,112)(110,111)(116,121)(117,120)(118,119)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(139,142)(140,141)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153) );
G=PermutationGroup([[(1,141,119,111,81,50,46),(2,142,120,112,82,51,24),(3,143,121,113,83,52,25),(4,144,122,114,84,53,26),(5,145,123,115,85,54,27),(6,146,124,93,86,55,28),(7,147,125,94,87,56,29),(8,148,126,95,88,57,30),(9,149,127,96,89,58,31),(10,150,128,97,90,59,32),(11,151,129,98,91,60,33),(12,152,130,99,92,61,34),(13,153,131,100,70,62,35),(14,154,132,101,71,63,36),(15,155,133,102,72,64,37),(16,156,134,103,73,65,38),(17,157,135,104,74,66,39),(18,158,136,105,75,67,40),(19,159,137,106,76,68,41),(20,160,138,107,77,69,42),(21,161,116,108,78,47,43),(22,139,117,109,79,48,44),(23,140,118,110,80,49,45)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(45,46),(47,52),(48,51),(49,50),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(106,115),(107,114),(108,113),(109,112),(110,111),(116,121),(117,120),(118,119),(122,138),(123,137),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(139,142),(140,141),(143,161),(144,160),(145,159),(146,158),(147,157),(148,156),(149,155),(150,154),(151,153)]])
91 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 14A | ··· | 14F | 23A | ··· | 23K | 161A | ··· | 161BN |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 23 | ··· | 23 | 161 | ··· | 161 |
| size | 1 | 23 | 1 | ··· | 1 | 23 | ··· | 23 | 2 | ··· | 2 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D23 | C7×D23 |
| kernel | C7×D23 | C161 | D23 | C23 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 11 | 66 |
Matrix representation of C7×D23 ►in GL2(𝔽967) generated by
| 97 | 0 |
| 0 | 97 |
| 535 | 313 |
| 966 | 914 |
| 660 | 226 |
| 182 | 307 |
G:=sub<GL(2,GF(967))| [97,0,0,97],[535,966,313,914],[660,182,226,307] >;
C7×D23 in GAP, Magma, Sage, TeX
C_7\times D_{23} % in TeX
G:=Group("C7xD23"); // GroupNames label
G:=SmallGroup(322,2);
// by ID
G=gap.SmallGroup(322,2);
# by ID
G:=PCGroup([3,-2,-7,-23,2774]);
// Polycyclic
G:=Group<a,b,c|a^7=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export