direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×D13, C26⋊C14, C182⋊2C2, C91⋊3C22, C13⋊(C2×C14), SmallGroup(364,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C14×D13 |
Generators and relations for C14×D13
G = < a,b,c | a14=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C14×D13
►On 182 points
Generators in S182
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)
(1 74 147 37 22 121 102 159 43 60 130 88 176)(2 75 148 38 23 122 103 160 44 61 131 89 177)(3 76 149 39 24 123 104 161 45 62 132 90 178)(4 77 150 40 25 124 105 162 46 63 133 91 179)(5 78 151 41 26 125 106 163 47 64 134 92 180)(6 79 152 42 27 126 107 164 48 65 135 93 181)(7 80 153 29 28 113 108 165 49 66 136 94 182)(8 81 154 30 15 114 109 166 50 67 137 95 169)(9 82 141 31 16 115 110 167 51 68 138 96 170)(10 83 142 32 17 116 111 168 52 69 139 97 171)(11 84 143 33 18 117 112 155 53 70 140 98 172)(12 71 144 34 19 118 99 156 54 57 127 85 173)(13 72 145 35 20 119 100 157 55 58 128 86 174)(14 73 146 36 21 120 101 158 56 59 129 87 175)
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 57)(42 58)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 155)(125 156)(126 157)(127 151)(128 152)(129 153)(130 154)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)
G:=sub<Sym(182)| (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182), (1,74,147,37,22,121,102,159,43,60,130,88,176)(2,75,148,38,23,122,103,160,44,61,131,89,177)(3,76,149,39,24,123,104,161,45,62,132,90,178)(4,77,150,40,25,124,105,162,46,63,133,91,179)(5,78,151,41,26,125,106,163,47,64,134,92,180)(6,79,152,42,27,126,107,164,48,65,135,93,181)(7,80,153,29,28,113,108,165,49,66,136,94,182)(8,81,154,30,15,114,109,166,50,67,137,95,169)(9,82,141,31,16,115,110,167,51,68,138,96,170)(10,83,142,32,17,116,111,168,52,69,139,97,171)(11,84,143,33,18,117,112,155,53,70,140,98,172)(12,71,144,34,19,118,99,156,54,57,127,85,173)(13,72,145,35,20,119,100,157,55,58,128,86,174)(14,73,146,36,21,120,101,158,56,59,129,87,175), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,57)(42,58)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,155)(125,156)(126,157)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)>;
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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182), (1,74,147,37,22,121,102,159,43,60,130,88,176)(2,75,148,38,23,122,103,160,44,61,131,89,177)(3,76,149,39,24,123,104,161,45,62,132,90,178)(4,77,150,40,25,124,105,162,46,63,133,91,179)(5,78,151,41,26,125,106,163,47,64,134,92,180)(6,79,152,42,27,126,107,164,48,65,135,93,181)(7,80,153,29,28,113,108,165,49,66,136,94,182)(8,81,154,30,15,114,109,166,50,67,137,95,169)(9,82,141,31,16,115,110,167,51,68,138,96,170)(10,83,142,32,17,116,111,168,52,69,139,97,171)(11,84,143,33,18,117,112,155,53,70,140,98,172)(12,71,144,34,19,118,99,156,54,57,127,85,173)(13,72,145,35,20,119,100,157,55,58,128,86,174)(14,73,146,36,21,120,101,158,56,59,129,87,175), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,57)(42,58)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,155)(125,156)(126,157)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150) );
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),(155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182)], [(1,74,147,37,22,121,102,159,43,60,130,88,176),(2,75,148,38,23,122,103,160,44,61,131,89,177),(3,76,149,39,24,123,104,161,45,62,132,90,178),(4,77,150,40,25,124,105,162,46,63,133,91,179),(5,78,151,41,26,125,106,163,47,64,134,92,180),(6,79,152,42,27,126,107,164,48,65,135,93,181),(7,80,153,29,28,113,108,165,49,66,136,94,182),(8,81,154,30,15,114,109,166,50,67,137,95,169),(9,82,141,31,16,115,110,167,51,68,138,96,170),(10,83,142,32,17,116,111,168,52,69,139,97,171),(11,84,143,33,18,117,112,155,53,70,140,98,172),(12,71,144,34,19,118,99,156,54,57,127,85,173),(13,72,145,35,20,119,100,157,55,58,128,86,174),(14,73,146,36,21,120,101,158,56,59,129,87,175)], [(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,181),(14,182),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,57),(42,58),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,155),(125,156),(126,157),(127,151),(128,152),(129,153),(130,154),(131,141),(132,142),(133,143),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | ··· | 7F | 13A | ··· | 13F | 14A | ··· | 14F | 14G | ··· | 14R | 26A | ··· | 26F | 91A | ··· | 91AJ | 182A | ··· | 182AJ |
| order | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 13 | ··· | 13 | 14 | ··· | 14 | 14 | ··· | 14 | 26 | ··· | 26 | 91 | ··· | 91 | 182 | ··· | 182 |
| size | 1 | 1 | 13 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D13 | D26 | C7×D13 | C14×D13 |
| kernel | C14×D13 | C7×D13 | C182 | D26 | D13 | C26 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 6 | 6 | 36 | 36 |
Matrix representation of C14×D13 ►in GL2(𝔽547) generated by
| 466 | 0 |
| 0 | 466 |
| 0 | 1 |
| 546 | 64 |
| 0 | 546 |
| 546 | 0 |
G:=sub<GL(2,GF(547))| [466,0,0,466],[0,546,1,64],[0,546,546,0] >;
C14×D13 in GAP, Magma, Sage, TeX
C_{14}\times D_{13} % in TeX
G:=Group("C14xD13"); // GroupNames label
G:=SmallGroup(364,8);
// by ID
G=gap.SmallGroup(364,8);
# by ID
G:=PCGroup([4,-2,-2,-7,-13,5379]);
// Polycyclic
G:=Group<a,b,c|a^14=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export