direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C13×SD32, D8.C26, C16⋊2C26, C208⋊6C2, Q16⋊1C26, C26.16D8, C52.37D4, C104.25C22, C8.3(C2×C26), C4.2(D4×C13), C2.4(C13×D8), (C13×Q16)⋊5C2, (C13×D8).2C2, SmallGroup(416,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×SD32
G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C13×SD32
►On 208 points
Generators in S208
(1 103 129 96 56 178 161 195 19 127 43 71 150)(2 104 130 81 57 179 162 196 20 128 44 72 151)(3 105 131 82 58 180 163 197 21 113 45 73 152)(4 106 132 83 59 181 164 198 22 114 46 74 153)(5 107 133 84 60 182 165 199 23 115 47 75 154)(6 108 134 85 61 183 166 200 24 116 48 76 155)(7 109 135 86 62 184 167 201 25 117 33 77 156)(8 110 136 87 63 185 168 202 26 118 34 78 157)(9 111 137 88 64 186 169 203 27 119 35 79 158)(10 112 138 89 49 187 170 204 28 120 36 80 159)(11 97 139 90 50 188 171 205 29 121 37 65 160)(12 98 140 91 51 189 172 206 30 122 38 66 145)(13 99 141 92 52 190 173 207 31 123 39 67 146)(14 100 142 93 53 191 174 208 32 124 40 68 147)(15 101 143 94 54 192 175 193 17 125 41 69 148)(16 102 144 95 55 177 176 194 18 126 42 70 149)
(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)(177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 37)(34 44)(36 42)(38 40)(39 47)(41 45)(46 48)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)(81 87)(82 94)(83 85)(84 92)(86 90)(89 95)(91 93)(97 109)(98 100)(99 107)(101 105)(102 112)(104 110)(106 108)(113 125)(114 116)(115 123)(117 121)(118 128)(120 126)(122 124)(130 136)(131 143)(132 134)(133 141)(135 139)(138 144)(140 142)(145 147)(146 154)(148 152)(149 159)(151 157)(153 155)(156 160)(162 168)(163 175)(164 166)(165 173)(167 171)(170 176)(172 174)(177 187)(179 185)(180 192)(181 183)(182 190)(184 188)(189 191)(193 197)(194 204)(196 202)(198 200)(199 207)(201 205)(206 208)
G:=sub<Sym(208)| (1,103,129,96,56,178,161,195,19,127,43,71,150)(2,104,130,81,57,179,162,196,20,128,44,72,151)(3,105,131,82,58,180,163,197,21,113,45,73,152)(4,106,132,83,59,181,164,198,22,114,46,74,153)(5,107,133,84,60,182,165,199,23,115,47,75,154)(6,108,134,85,61,183,166,200,24,116,48,76,155)(7,109,135,86,62,184,167,201,25,117,33,77,156)(8,110,136,87,63,185,168,202,26,118,34,78,157)(9,111,137,88,64,186,169,203,27,119,35,79,158)(10,112,138,89,49,187,170,204,28,120,36,80,159)(11,97,139,90,50,188,171,205,29,121,37,65,160)(12,98,140,91,51,189,172,206,30,122,38,66,145)(13,99,141,92,52,190,173,207,31,123,39,67,146)(14,100,142,93,53,191,174,208,32,124,40,68,147)(15,101,143,94,54,192,175,193,17,125,41,69,148)(16,102,144,95,55,177,176,194,18,126,42,70,149), (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,87)(82,94)(83,85)(84,92)(86,90)(89,95)(91,93)(97,109)(98,100)(99,107)(101,105)(102,112)(104,110)(106,108)(113,125)(114,116)(115,123)(117,121)(118,128)(120,126)(122,124)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,147)(146,154)(148,152)(149,159)(151,157)(153,155)(156,160)(162,168)(163,175)(164,166)(165,173)(167,171)(170,176)(172,174)(177,187)(179,185)(180,192)(181,183)(182,190)(184,188)(189,191)(193,197)(194,204)(196,202)(198,200)(199,207)(201,205)(206,208)>;
G:=Group( (1,103,129,96,56,178,161,195,19,127,43,71,150)(2,104,130,81,57,179,162,196,20,128,44,72,151)(3,105,131,82,58,180,163,197,21,113,45,73,152)(4,106,132,83,59,181,164,198,22,114,46,74,153)(5,107,133,84,60,182,165,199,23,115,47,75,154)(6,108,134,85,61,183,166,200,24,116,48,76,155)(7,109,135,86,62,184,167,201,25,117,33,77,156)(8,110,136,87,63,185,168,202,26,118,34,78,157)(9,111,137,88,64,186,169,203,27,119,35,79,158)(10,112,138,89,49,187,170,204,28,120,36,80,159)(11,97,139,90,50,188,171,205,29,121,37,65,160)(12,98,140,91,51,189,172,206,30,122,38,66,145)(13,99,141,92,52,190,173,207,31,123,39,67,146)(14,100,142,93,53,191,174,208,32,124,40,68,147)(15,101,143,94,54,192,175,193,17,125,41,69,148)(16,102,144,95,55,177,176,194,18,126,42,70,149), (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(81,87)(82,94)(83,85)(84,92)(86,90)(89,95)(91,93)(97,109)(98,100)(99,107)(101,105)(102,112)(104,110)(106,108)(113,125)(114,116)(115,123)(117,121)(118,128)(120,126)(122,124)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,147)(146,154)(148,152)(149,159)(151,157)(153,155)(156,160)(162,168)(163,175)(164,166)(165,173)(167,171)(170,176)(172,174)(177,187)(179,185)(180,192)(181,183)(182,190)(184,188)(189,191)(193,197)(194,204)(196,202)(198,200)(199,207)(201,205)(206,208) );
G=PermutationGroup([[(1,103,129,96,56,178,161,195,19,127,43,71,150),(2,104,130,81,57,179,162,196,20,128,44,72,151),(3,105,131,82,58,180,163,197,21,113,45,73,152),(4,106,132,83,59,181,164,198,22,114,46,74,153),(5,107,133,84,60,182,165,199,23,115,47,75,154),(6,108,134,85,61,183,166,200,24,116,48,76,155),(7,109,135,86,62,184,167,201,25,117,33,77,156),(8,110,136,87,63,185,168,202,26,118,34,78,157),(9,111,137,88,64,186,169,203,27,119,35,79,158),(10,112,138,89,49,187,170,204,28,120,36,80,159),(11,97,139,90,50,188,171,205,29,121,37,65,160),(12,98,140,91,51,189,172,206,30,122,38,66,145),(13,99,141,92,52,190,173,207,31,123,39,67,146),(14,100,142,93,53,191,174,208,32,124,40,68,147),(15,101,143,94,54,192,175,193,17,125,41,69,148),(16,102,144,95,55,177,176,194,18,126,42,70,149)], [(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),(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,21),(18,28),(20,26),(22,24),(23,31),(25,29),(30,32),(33,37),(34,44),(36,42),(38,40),(39,47),(41,45),(46,48),(49,55),(50,62),(51,53),(52,60),(54,58),(57,63),(59,61),(65,77),(66,68),(67,75),(69,73),(70,80),(72,78),(74,76),(81,87),(82,94),(83,85),(84,92),(86,90),(89,95),(91,93),(97,109),(98,100),(99,107),(101,105),(102,112),(104,110),(106,108),(113,125),(114,116),(115,123),(117,121),(118,128),(120,126),(122,124),(130,136),(131,143),(132,134),(133,141),(135,139),(138,144),(140,142),(145,147),(146,154),(148,152),(149,159),(151,157),(153,155),(156,160),(162,168),(163,175),(164,166),(165,173),(167,171),(170,176),(172,174),(177,187),(179,185),(180,192),(181,183),(182,190),(184,188),(189,191),(193,197),(194,204),(196,202),(198,200),(199,207),(201,205),(206,208)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13L | 16A | 16B | 16C | 16D | 26A | ··· | 26L | 26M | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52X | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 8 | 2 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | D4 | D8 | SD32 | D4×C13 | C13×D8 | C13×SD32 |
| kernel | C13×SD32 | C208 | C13×D8 | C13×Q16 | SD32 | C16 | D8 | Q16 | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 1 | 2 | 4 | 12 | 24 | 48 |
Matrix representation of C13×SD32 ►in GL2(𝔽1249) generated by
| 240 | 0 |
| 0 | 240 |
| 276 | 1034 |
| 215 | 276 |
| 1 | 0 |
| 0 | 1248 |
G:=sub<GL(2,GF(1249))| [240,0,0,240],[276,215,1034,276],[1,0,0,1248] >;
C13×SD32 in GAP, Magma, Sage, TeX
C_{13}\times {\rm SD}_{32} % in TeX
G:=Group("C13xSD32"); // GroupNames label
G:=SmallGroup(416,62);
// by ID
G=gap.SmallGroup(416,62);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,1248,649,3747,1881,165,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
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