direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×2- 1+4, C26.20C24, C52.52C23, C4○D4⋊4C26, (C2×Q8)⋊5C26, (Q8×C26)⋊12C2, D4.4(C2×C26), Q8.4(C2×C26), C2.5(C23×C26), (C2×C26).8C23, (C2×C52).71C22, C4.10(C22×C26), (D4×C13).14C22, C22.3(C22×C26), (Q8×C13).15C22, (C13×C4○D4)⋊9C2, (C2×C4).12(C2×C26), SmallGroup(416,232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×2- 1+4
G = < a,b,c,d,e | a13=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C2×C4, D4, Q8, C13, C2×Q8, C4○D4, C26, C26, 2- 1+4, C52, C2×C26, C2×C52, D4×C13, Q8×C13, Q8×C26, C13×C4○D4, C13×2- 1+4
Quotients: C1, C2, C22, C23, C13, C24, C26, 2- 1+4, C2×C26, C22×C26, C23×C26, C13×2- 1+4
►On 208 points
Generators in S208
(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)
(1 33 192 168)(2 34 193 169)(3 35 194 157)(4 36 195 158)(5 37 183 159)(6 38 184 160)(7 39 185 161)(8 27 186 162)(9 28 187 163)(10 29 188 164)(11 30 189 165)(12 31 190 166)(13 32 191 167)(14 123 102 62)(15 124 103 63)(16 125 104 64)(17 126 92 65)(18 127 93 53)(19 128 94 54)(20 129 95 55)(21 130 96 56)(22 118 97 57)(23 119 98 58)(24 120 99 59)(25 121 100 60)(26 122 101 61)(40 133 85 106)(41 134 86 107)(42 135 87 108)(43 136 88 109)(44 137 89 110)(45 138 90 111)(46 139 91 112)(47 140 79 113)(48 141 80 114)(49 142 81 115)(50 143 82 116)(51 131 83 117)(52 132 84 105)(66 175 200 156)(67 176 201 144)(68 177 202 145)(69 178 203 146)(70 179 204 147)(71 180 205 148)(72 181 206 149)(73 182 207 150)(74 170 208 151)(75 171 196 152)(76 172 197 153)(77 173 198 154)(78 174 199 155)
(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 169)(35 157)(36 158)(37 159)(38 160)(39 161)(53 127)(54 128)(55 129)(56 130)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 131)(144 176)(145 177)(146 178)(147 179)(148 180)(149 181)(150 182)(151 170)(152 171)(153 172)(154 173)(155 174)(156 175)
(1 200 192 66)(2 201 193 67)(3 202 194 68)(4 203 195 69)(5 204 183 70)(6 205 184 71)(7 206 185 72)(8 207 186 73)(9 208 187 74)(10 196 188 75)(11 197 189 76)(12 198 190 77)(13 199 191 78)(14 81 102 49)(15 82 103 50)(16 83 104 51)(17 84 92 52)(18 85 93 40)(19 86 94 41)(20 87 95 42)(21 88 96 43)(22 89 97 44)(23 90 98 45)(24 91 99 46)(25 79 100 47)(26 80 101 48)(27 150 162 182)(28 151 163 170)(29 152 164 171)(30 153 165 172)(31 154 166 173)(32 155 167 174)(33 156 168 175)(34 144 169 176)(35 145 157 177)(36 146 158 178)(37 147 159 179)(38 148 160 180)(39 149 161 181)(53 133 127 106)(54 134 128 107)(55 135 129 108)(56 136 130 109)(57 137 118 110)(58 138 119 111)(59 139 120 112)(60 140 121 113)(61 141 122 114)(62 142 123 115)(63 143 124 116)(64 131 125 117)(65 132 126 105)
(1 49 192 81)(2 50 193 82)(3 51 194 83)(4 52 195 84)(5 40 183 85)(6 41 184 86)(7 42 185 87)(8 43 186 88)(9 44 187 89)(10 45 188 90)(11 46 189 91)(12 47 190 79)(13 48 191 80)(14 200 102 66)(15 201 103 67)(16 202 104 68)(17 203 92 69)(18 204 93 70)(19 205 94 71)(20 206 95 72)(21 207 96 73)(22 208 97 74)(23 196 98 75)(24 197 99 76)(25 198 100 77)(26 199 101 78)(27 136 162 109)(28 137 163 110)(29 138 164 111)(30 139 165 112)(31 140 166 113)(32 141 167 114)(33 142 168 115)(34 143 169 116)(35 131 157 117)(36 132 158 105)(37 133 159 106)(38 134 160 107)(39 135 161 108)(53 179 127 147)(54 180 128 148)(55 181 129 149)(56 182 130 150)(57 170 118 151)(58 171 119 152)(59 172 120 153)(60 173 121 154)(61 174 122 155)(62 175 123 156)(63 176 124 144)(64 177 125 145)(65 178 126 146)
G:=sub<Sym(208)| (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), (1,33,192,168)(2,34,193,169)(3,35,194,157)(4,36,195,158)(5,37,183,159)(6,38,184,160)(7,39,185,161)(8,27,186,162)(9,28,187,163)(10,29,188,164)(11,30,189,165)(12,31,190,166)(13,32,191,167)(14,123,102,62)(15,124,103,63)(16,125,104,64)(17,126,92,65)(18,127,93,53)(19,128,94,54)(20,129,95,55)(21,130,96,56)(22,118,97,57)(23,119,98,58)(24,120,99,59)(25,121,100,60)(26,122,101,61)(40,133,85,106)(41,134,86,107)(42,135,87,108)(43,136,88,109)(44,137,89,110)(45,138,90,111)(46,139,91,112)(47,140,79,113)(48,141,80,114)(49,142,81,115)(50,143,82,116)(51,131,83,117)(52,132,84,105)(66,175,200,156)(67,176,201,144)(68,177,202,145)(69,178,203,146)(70,179,204,147)(71,180,205,148)(72,181,206,149)(73,182,207,150)(74,170,208,151)(75,171,196,152)(76,172,197,153)(77,173,198,154)(78,174,199,155), (27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,157)(36,158)(37,159)(38,160)(39,161)(53,127)(54,128)(55,129)(56,130)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,131)(144,176)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,170)(152,171)(153,172)(154,173)(155,174)(156,175), (1,200,192,66)(2,201,193,67)(3,202,194,68)(4,203,195,69)(5,204,183,70)(6,205,184,71)(7,206,185,72)(8,207,186,73)(9,208,187,74)(10,196,188,75)(11,197,189,76)(12,198,190,77)(13,199,191,78)(14,81,102,49)(15,82,103,50)(16,83,104,51)(17,84,92,52)(18,85,93,40)(19,86,94,41)(20,87,95,42)(21,88,96,43)(22,89,97,44)(23,90,98,45)(24,91,99,46)(25,79,100,47)(26,80,101,48)(27,150,162,182)(28,151,163,170)(29,152,164,171)(30,153,165,172)(31,154,166,173)(32,155,167,174)(33,156,168,175)(34,144,169,176)(35,145,157,177)(36,146,158,178)(37,147,159,179)(38,148,160,180)(39,149,161,181)(53,133,127,106)(54,134,128,107)(55,135,129,108)(56,136,130,109)(57,137,118,110)(58,138,119,111)(59,139,120,112)(60,140,121,113)(61,141,122,114)(62,142,123,115)(63,143,124,116)(64,131,125,117)(65,132,126,105), (1,49,192,81)(2,50,193,82)(3,51,194,83)(4,52,195,84)(5,40,183,85)(6,41,184,86)(7,42,185,87)(8,43,186,88)(9,44,187,89)(10,45,188,90)(11,46,189,91)(12,47,190,79)(13,48,191,80)(14,200,102,66)(15,201,103,67)(16,202,104,68)(17,203,92,69)(18,204,93,70)(19,205,94,71)(20,206,95,72)(21,207,96,73)(22,208,97,74)(23,196,98,75)(24,197,99,76)(25,198,100,77)(26,199,101,78)(27,136,162,109)(28,137,163,110)(29,138,164,111)(30,139,165,112)(31,140,166,113)(32,141,167,114)(33,142,168,115)(34,143,169,116)(35,131,157,117)(36,132,158,105)(37,133,159,106)(38,134,160,107)(39,135,161,108)(53,179,127,147)(54,180,128,148)(55,181,129,149)(56,182,130,150)(57,170,118,151)(58,171,119,152)(59,172,120,153)(60,173,121,154)(61,174,122,155)(62,175,123,156)(63,176,124,144)(64,177,125,145)(65,178,126,146)>;
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)(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), (1,33,192,168)(2,34,193,169)(3,35,194,157)(4,36,195,158)(5,37,183,159)(6,38,184,160)(7,39,185,161)(8,27,186,162)(9,28,187,163)(10,29,188,164)(11,30,189,165)(12,31,190,166)(13,32,191,167)(14,123,102,62)(15,124,103,63)(16,125,104,64)(17,126,92,65)(18,127,93,53)(19,128,94,54)(20,129,95,55)(21,130,96,56)(22,118,97,57)(23,119,98,58)(24,120,99,59)(25,121,100,60)(26,122,101,61)(40,133,85,106)(41,134,86,107)(42,135,87,108)(43,136,88,109)(44,137,89,110)(45,138,90,111)(46,139,91,112)(47,140,79,113)(48,141,80,114)(49,142,81,115)(50,143,82,116)(51,131,83,117)(52,132,84,105)(66,175,200,156)(67,176,201,144)(68,177,202,145)(69,178,203,146)(70,179,204,147)(71,180,205,148)(72,181,206,149)(73,182,207,150)(74,170,208,151)(75,171,196,152)(76,172,197,153)(77,173,198,154)(78,174,199,155), (27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,157)(36,158)(37,159)(38,160)(39,161)(53,127)(54,128)(55,129)(56,130)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,131)(144,176)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,170)(152,171)(153,172)(154,173)(155,174)(156,175), (1,200,192,66)(2,201,193,67)(3,202,194,68)(4,203,195,69)(5,204,183,70)(6,205,184,71)(7,206,185,72)(8,207,186,73)(9,208,187,74)(10,196,188,75)(11,197,189,76)(12,198,190,77)(13,199,191,78)(14,81,102,49)(15,82,103,50)(16,83,104,51)(17,84,92,52)(18,85,93,40)(19,86,94,41)(20,87,95,42)(21,88,96,43)(22,89,97,44)(23,90,98,45)(24,91,99,46)(25,79,100,47)(26,80,101,48)(27,150,162,182)(28,151,163,170)(29,152,164,171)(30,153,165,172)(31,154,166,173)(32,155,167,174)(33,156,168,175)(34,144,169,176)(35,145,157,177)(36,146,158,178)(37,147,159,179)(38,148,160,180)(39,149,161,181)(53,133,127,106)(54,134,128,107)(55,135,129,108)(56,136,130,109)(57,137,118,110)(58,138,119,111)(59,139,120,112)(60,140,121,113)(61,141,122,114)(62,142,123,115)(63,143,124,116)(64,131,125,117)(65,132,126,105), (1,49,192,81)(2,50,193,82)(3,51,194,83)(4,52,195,84)(5,40,183,85)(6,41,184,86)(7,42,185,87)(8,43,186,88)(9,44,187,89)(10,45,188,90)(11,46,189,91)(12,47,190,79)(13,48,191,80)(14,200,102,66)(15,201,103,67)(16,202,104,68)(17,203,92,69)(18,204,93,70)(19,205,94,71)(20,206,95,72)(21,207,96,73)(22,208,97,74)(23,196,98,75)(24,197,99,76)(25,198,100,77)(26,199,101,78)(27,136,162,109)(28,137,163,110)(29,138,164,111)(30,139,165,112)(31,140,166,113)(32,141,167,114)(33,142,168,115)(34,143,169,116)(35,131,157,117)(36,132,158,105)(37,133,159,106)(38,134,160,107)(39,135,161,108)(53,179,127,147)(54,180,128,148)(55,181,129,149)(56,182,130,150)(57,170,118,151)(58,171,119,152)(59,172,120,153)(60,173,121,154)(61,174,122,155)(62,175,123,156)(63,176,124,144)(64,177,125,145)(65,178,126,146) );
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),(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)], [(1,33,192,168),(2,34,193,169),(3,35,194,157),(4,36,195,158),(5,37,183,159),(6,38,184,160),(7,39,185,161),(8,27,186,162),(9,28,187,163),(10,29,188,164),(11,30,189,165),(12,31,190,166),(13,32,191,167),(14,123,102,62),(15,124,103,63),(16,125,104,64),(17,126,92,65),(18,127,93,53),(19,128,94,54),(20,129,95,55),(21,130,96,56),(22,118,97,57),(23,119,98,58),(24,120,99,59),(25,121,100,60),(26,122,101,61),(40,133,85,106),(41,134,86,107),(42,135,87,108),(43,136,88,109),(44,137,89,110),(45,138,90,111),(46,139,91,112),(47,140,79,113),(48,141,80,114),(49,142,81,115),(50,143,82,116),(51,131,83,117),(52,132,84,105),(66,175,200,156),(67,176,201,144),(68,177,202,145),(69,178,203,146),(70,179,204,147),(71,180,205,148),(72,181,206,149),(73,182,207,150),(74,170,208,151),(75,171,196,152),(76,172,197,153),(77,173,198,154),(78,174,199,155)], [(27,162),(28,163),(29,164),(30,165),(31,166),(32,167),(33,168),(34,169),(35,157),(36,158),(37,159),(38,160),(39,161),(53,127),(54,128),(55,129),(56,130),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125),(65,126),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139),(113,140),(114,141),(115,142),(116,143),(117,131),(144,176),(145,177),(146,178),(147,179),(148,180),(149,181),(150,182),(151,170),(152,171),(153,172),(154,173),(155,174),(156,175)], [(1,200,192,66),(2,201,193,67),(3,202,194,68),(4,203,195,69),(5,204,183,70),(6,205,184,71),(7,206,185,72),(8,207,186,73),(9,208,187,74),(10,196,188,75),(11,197,189,76),(12,198,190,77),(13,199,191,78),(14,81,102,49),(15,82,103,50),(16,83,104,51),(17,84,92,52),(18,85,93,40),(19,86,94,41),(20,87,95,42),(21,88,96,43),(22,89,97,44),(23,90,98,45),(24,91,99,46),(25,79,100,47),(26,80,101,48),(27,150,162,182),(28,151,163,170),(29,152,164,171),(30,153,165,172),(31,154,166,173),(32,155,167,174),(33,156,168,175),(34,144,169,176),(35,145,157,177),(36,146,158,178),(37,147,159,179),(38,148,160,180),(39,149,161,181),(53,133,127,106),(54,134,128,107),(55,135,129,108),(56,136,130,109),(57,137,118,110),(58,138,119,111),(59,139,120,112),(60,140,121,113),(61,141,122,114),(62,142,123,115),(63,143,124,116),(64,131,125,117),(65,132,126,105)], [(1,49,192,81),(2,50,193,82),(3,51,194,83),(4,52,195,84),(5,40,183,85),(6,41,184,86),(7,42,185,87),(8,43,186,88),(9,44,187,89),(10,45,188,90),(11,46,189,91),(12,47,190,79),(13,48,191,80),(14,200,102,66),(15,201,103,67),(16,202,104,68),(17,203,92,69),(18,204,93,70),(19,205,94,71),(20,206,95,72),(21,207,96,73),(22,208,97,74),(23,196,98,75),(24,197,99,76),(25,198,100,77),(26,199,101,78),(27,136,162,109),(28,137,163,110),(29,138,164,111),(30,139,165,112),(31,140,166,113),(32,141,167,114),(33,142,168,115),(34,143,169,116),(35,131,157,117),(36,132,158,105),(37,133,159,106),(38,134,160,107),(39,135,161,108),(53,179,127,147),(54,180,128,148),(55,181,129,149),(56,182,130,150),(57,170,118,151),(58,171,119,152),(59,172,120,153),(60,173,121,154),(61,174,122,155),(62,175,123,156),(63,176,124,144),(64,177,125,145),(65,178,126,146)]])
221 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 4A | ··· | 4J | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26BT | 52A | ··· | 52DP |
| order | 1 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
221 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | 2- 1+4 | C13×2- 1+4 |
| kernel | C13×2- 1+4 | Q8×C26 | C13×C4○D4 | 2- 1+4 | C2×Q8 | C4○D4 | C13 | C1 |
| # reps | 1 | 5 | 10 | 12 | 60 | 120 | 1 | 12 |
Matrix representation of C13×2- 1+4 ►in GL4(𝔽53) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 45 | 39 | 48 | 36 |
| 16 | 25 | 36 | 5 |
| 10 | 20 | 10 | 52 |
| 22 | 30 | 29 | 26 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 43 | 8 | 52 | 0 |
| 2 | 49 | 0 | 52 |
| 23 | 27 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 9 | 23 | 30 | 26 |
| 0 | 13 | 0 | 23 |
| 31 | 31 | 0 | 0 |
| 10 | 22 | 0 | 0 |
| 26 | 48 | 22 | 22 |
| 36 | 36 | 43 | 31 |
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[45,16,10,22,39,25,20,30,48,36,10,29,36,5,52,26],[1,0,43,2,0,1,8,49,0,0,52,0,0,0,0,52],[23,0,9,0,27,30,23,13,0,0,30,0,0,0,26,23],[31,10,26,36,31,22,48,36,0,0,22,43,0,0,22,31] >;
C13×2- 1+4 in GAP, Magma, Sage, TeX
C_{13}\times 2_-^{1+4} % in TeX
G:=Group("C13xES-(2,2)"); // GroupNames label
G:=SmallGroup(416,232);
// by ID
G=gap.SmallGroup(416,232);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,1255,1916,950,5187]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations