direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D29, C58⋊C6, C174⋊2C2, C87⋊3C22, C29⋊(C2×C6), SmallGroup(348,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C6×D29 |
Generators and relations for C6×D29
G = < a,b,c | a6=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D29
►On 174 points
Generators in S174
(1 144 84 109 36 151)(2 145 85 110 37 152)(3 117 86 111 38 153)(4 118 87 112 39 154)(5 119 59 113 40 155)(6 120 60 114 41 156)(7 121 61 115 42 157)(8 122 62 116 43 158)(9 123 63 88 44 159)(10 124 64 89 45 160)(11 125 65 90 46 161)(12 126 66 91 47 162)(13 127 67 92 48 163)(14 128 68 93 49 164)(15 129 69 94 50 165)(16 130 70 95 51 166)(17 131 71 96 52 167)(18 132 72 97 53 168)(19 133 73 98 54 169)(20 134 74 99 55 170)(21 135 75 100 56 171)(22 136 76 101 57 172)(23 137 77 102 58 173)(24 138 78 103 30 174)(25 139 79 104 31 146)(26 140 80 105 32 147)(27 141 81 106 33 148)(28 142 82 107 34 149)(29 143 83 108 35 150)
(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)(146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 120)(31 119)(32 118)(33 117)(34 145)(35 144)(36 143)(37 142)(38 141)(39 140)(40 139)(41 138)(42 137)(43 136)(44 135)(45 134)(46 133)(47 132)(48 131)(49 130)(50 129)(51 128)(52 127)(53 126)(54 125)(55 124)(56 123)(57 122)(58 121)(59 146)(60 174)(61 173)(62 172)(63 171)(64 170)(65 169)(66 168)(67 167)(68 166)(69 165)(70 164)(71 163)(72 162)(73 161)(74 160)(75 159)(76 158)(77 157)(78 156)(79 155)(80 154)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)
G:=sub<Sym(174)| (1,144,84,109,36,151)(2,145,85,110,37,152)(3,117,86,111,38,153)(4,118,87,112,39,154)(5,119,59,113,40,155)(6,120,60,114,41,156)(7,121,61,115,42,157)(8,122,62,116,43,158)(9,123,63,88,44,159)(10,124,64,89,45,160)(11,125,65,90,46,161)(12,126,66,91,47,162)(13,127,67,92,48,163)(14,128,68,93,49,164)(15,129,69,94,50,165)(16,130,70,95,51,166)(17,131,71,96,52,167)(18,132,72,97,53,168)(19,133,73,98,54,169)(20,134,74,99,55,170)(21,135,75,100,56,171)(22,136,76,101,57,172)(23,137,77,102,58,173)(24,138,78,103,30,174)(25,139,79,104,31,146)(26,140,80,105,32,147)(27,141,81,106,33,148)(28,142,82,107,34,149)(29,143,83,108,35,150), (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)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,120)(31,119)(32,118)(33,117)(34,145)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,146)(60,174)(61,173)(62,172)(63,171)(64,170)(65,169)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)>;
G:=Group( (1,144,84,109,36,151)(2,145,85,110,37,152)(3,117,86,111,38,153)(4,118,87,112,39,154)(5,119,59,113,40,155)(6,120,60,114,41,156)(7,121,61,115,42,157)(8,122,62,116,43,158)(9,123,63,88,44,159)(10,124,64,89,45,160)(11,125,65,90,46,161)(12,126,66,91,47,162)(13,127,67,92,48,163)(14,128,68,93,49,164)(15,129,69,94,50,165)(16,130,70,95,51,166)(17,131,71,96,52,167)(18,132,72,97,53,168)(19,133,73,98,54,169)(20,134,74,99,55,170)(21,135,75,100,56,171)(22,136,76,101,57,172)(23,137,77,102,58,173)(24,138,78,103,30,174)(25,139,79,104,31,146)(26,140,80,105,32,147)(27,141,81,106,33,148)(28,142,82,107,34,149)(29,143,83,108,35,150), (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)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,120)(31,119)(32,118)(33,117)(34,145)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,146)(60,174)(61,173)(62,172)(63,171)(64,170)(65,169)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147) );
G=PermutationGroup([[(1,144,84,109,36,151),(2,145,85,110,37,152),(3,117,86,111,38,153),(4,118,87,112,39,154),(5,119,59,113,40,155),(6,120,60,114,41,156),(7,121,61,115,42,157),(8,122,62,116,43,158),(9,123,63,88,44,159),(10,124,64,89,45,160),(11,125,65,90,46,161),(12,126,66,91,47,162),(13,127,67,92,48,163),(14,128,68,93,49,164),(15,129,69,94,50,165),(16,130,70,95,51,166),(17,131,71,96,52,167),(18,132,72,97,53,168),(19,133,73,98,54,169),(20,134,74,99,55,170),(21,135,75,100,56,171),(22,136,76,101,57,172),(23,137,77,102,58,173),(24,138,78,103,30,174),(25,139,79,104,31,146),(26,140,80,105,32,147),(27,141,81,106,33,148),(28,142,82,107,34,149),(29,143,83,108,35,150)], [(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),(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)], [(1,108),(2,107),(3,106),(4,105),(5,104),(6,103),(7,102),(8,101),(9,100),(10,99),(11,98),(12,97),(13,96),(14,95),(15,94),(16,93),(17,92),(18,91),(19,90),(20,89),(21,88),(22,116),(23,115),(24,114),(25,113),(26,112),(27,111),(28,110),(29,109),(30,120),(31,119),(32,118),(33,117),(34,145),(35,144),(36,143),(37,142),(38,141),(39,140),(40,139),(41,138),(42,137),(43,136),(44,135),(45,134),(46,133),(47,132),(48,131),(49,130),(50,129),(51,128),(52,127),(53,126),(54,125),(55,124),(56,123),(57,122),(58,121),(59,146),(60,174),(61,173),(62,172),(63,171),(64,170),(65,169),(66,168),(67,167),(68,166),(69,165),(70,164),(71,163),(72,162),(73,161),(74,160),(75,159),(76,158),(77,157),(78,156),(79,155),(80,154),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 29A | ··· | 29N | 58A | ··· | 58N | 87A | ··· | 87AB | 174A | ··· | 174AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 | 174 | ··· | 174 |
| size | 1 | 1 | 29 | 29 | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D29 | D58 | C3×D29 | C6×D29 |
| kernel | C6×D29 | C3×D29 | C174 | D58 | D29 | C58 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 14 | 14 | 28 | 28 |
Matrix representation of C6×D29 ►in GL3(𝔽349) generated by
| 226 | 0 | 0 |
| 0 | 348 | 0 |
| 0 | 0 | 348 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 348 | 322 |
| 348 | 0 | 0 |
| 0 | 0 | 348 |
| 0 | 348 | 0 |
G:=sub<GL(3,GF(349))| [226,0,0,0,348,0,0,0,348],[1,0,0,0,0,348,0,1,322],[348,0,0,0,0,348,0,348,0] >;
C6×D29 in GAP, Magma, Sage, TeX
C_6\times D_{29} % in TeX
G:=Group("C6xD29"); // GroupNames label
G:=SmallGroup(348,9);
// by ID
G=gap.SmallGroup(348,9);
# by ID
G:=PCGroup([4,-2,-2,-3,-29,5379]);
// Polycyclic
G:=Group<a,b,c|a^6=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export