direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×D13, C26.8C24, C52.22C23, Dic26⋊9C22, D26.17C23, Dic13.5C23, C26⋊2(C2×Q8), (Q8×C26)⋊5C2, C13⋊2(C22×Q8), (C2×C4).61D26, (Q8×C13)⋊5C22, C2.9(C23×D13), (C2×Dic26)⋊13C2, (C2×C26).66C23, (C2×C52).46C22, C4.22(C22×D13), (C4×D13).21C22, C22.31(C22×D13), (C2×Dic13).47C22, (C22×D13).45C22, (C2×C4×D13).6C2, SmallGroup(416,219)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×D13
G = < a,b,c,d,e | a2=b4=d13=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 864 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, Q8, C23, C13, C22×C4, C2×Q8, C2×Q8, D13, C26, C26, C22×Q8, Dic13, C52, D26, C2×C26, Dic26, C4×D13, C2×Dic13, C2×C52, Q8×C13, C22×D13, C2×Dic26, C2×C4×D13, Q8×D13, Q8×C26, C2×Q8×D13
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, D13, C22×Q8, D26, C22×D13, Q8×D13, C23×D13, C2×Q8×D13
►On 208 points
Generators in S208
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 130)(26 118)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 155)(41 156)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 167)(54 168)(55 169)(56 157)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 180)(67 181)(68 182)(69 170)(70 171)(71 172)(72 173)(73 174)(74 175)(75 176)(76 177)(77 178)(78 179)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 183)(92 206)(93 207)(94 208)(95 196)(96 197)(97 198)(98 199)(99 200)(100 201)(101 202)(102 203)(103 204)(104 205)
(1 74 24 55)(2 75 25 56)(3 76 26 57)(4 77 14 58)(5 78 15 59)(6 66 16 60)(7 67 17 61)(8 68 18 62)(9 69 19 63)(10 70 20 64)(11 71 21 65)(12 72 22 53)(13 73 23 54)(27 104 46 83)(28 92 47 84)(29 93 48 85)(30 94 49 86)(31 95 50 87)(32 96 51 88)(33 97 52 89)(34 98 40 90)(35 99 41 91)(36 100 42 79)(37 101 43 80)(38 102 44 81)(39 103 45 82)(105 181 122 162)(106 182 123 163)(107 170 124 164)(108 171 125 165)(109 172 126 166)(110 173 127 167)(111 174 128 168)(112 175 129 169)(113 176 130 157)(114 177 118 158)(115 178 119 159)(116 179 120 160)(117 180 121 161)(131 205 148 188)(132 206 149 189)(133 207 150 190)(134 208 151 191)(135 196 152 192)(136 197 153 193)(137 198 154 194)(138 199 155 195)(139 200 156 183)(140 201 144 184)(141 202 145 185)(142 203 146 186)(143 204 147 187)
(1 146 24 142)(2 147 25 143)(3 148 26 131)(4 149 14 132)(5 150 15 133)(6 151 16 134)(7 152 17 135)(8 153 18 136)(9 154 19 137)(10 155 20 138)(11 156 21 139)(12 144 22 140)(13 145 23 141)(27 114 46 118)(28 115 47 119)(29 116 48 120)(30 117 49 121)(31 105 50 122)(32 106 51 123)(33 107 52 124)(34 108 40 125)(35 109 41 126)(36 110 42 127)(37 111 43 128)(38 112 44 129)(39 113 45 130)(53 184 72 201)(54 185 73 202)(55 186 74 203)(56 187 75 204)(57 188 76 205)(58 189 77 206)(59 190 78 207)(60 191 66 208)(61 192 67 196)(62 193 68 197)(63 194 69 198)(64 195 70 199)(65 183 71 200)(79 173 100 167)(80 174 101 168)(81 175 102 169)(82 176 103 157)(83 177 104 158)(84 178 92 159)(85 179 93 160)(86 180 94 161)(87 181 95 162)(88 182 96 163)(89 170 97 164)(90 171 98 165)(91 172 99 166)
(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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 120)(10 119)(11 118)(12 130)(13 129)(14 108)(15 107)(16 106)(17 105)(18 117)(19 116)(20 115)(21 114)(22 113)(23 112)(24 111)(25 110)(26 109)(27 156)(28 155)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 132)(41 131)(42 143)(43 142)(44 141)(45 140)(46 139)(47 138)(48 137)(49 136)(50 135)(51 134)(52 133)(53 176)(54 175)(55 174)(56 173)(57 172)(58 171)(59 170)(60 182)(61 181)(62 180)(63 179)(64 178)(65 177)(66 163)(67 162)(68 161)(69 160)(70 159)(71 158)(72 157)(73 169)(74 168)(75 167)(76 166)(77 165)(78 164)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 208)(89 207)(90 206)(91 205)(92 195)(93 194)(94 193)(95 192)(96 191)(97 190)(98 189)(99 188)(100 187)(101 186)(102 185)(103 184)(104 183)
G:=sub<Sym(208)| (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,118)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,155)(41,156)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,167)(54,168)(55,169)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,180)(67,181)(68,182)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,183)(92,206)(93,207)(94,208)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,205), (1,74,24,55)(2,75,25,56)(3,76,26,57)(4,77,14,58)(5,78,15,59)(6,66,16,60)(7,67,17,61)(8,68,18,62)(9,69,19,63)(10,70,20,64)(11,71,21,65)(12,72,22,53)(13,73,23,54)(27,104,46,83)(28,92,47,84)(29,93,48,85)(30,94,49,86)(31,95,50,87)(32,96,51,88)(33,97,52,89)(34,98,40,90)(35,99,41,91)(36,100,42,79)(37,101,43,80)(38,102,44,81)(39,103,45,82)(105,181,122,162)(106,182,123,163)(107,170,124,164)(108,171,125,165)(109,172,126,166)(110,173,127,167)(111,174,128,168)(112,175,129,169)(113,176,130,157)(114,177,118,158)(115,178,119,159)(116,179,120,160)(117,180,121,161)(131,205,148,188)(132,206,149,189)(133,207,150,190)(134,208,151,191)(135,196,152,192)(136,197,153,193)(137,198,154,194)(138,199,155,195)(139,200,156,183)(140,201,144,184)(141,202,145,185)(142,203,146,186)(143,204,147,187), (1,146,24,142)(2,147,25,143)(3,148,26,131)(4,149,14,132)(5,150,15,133)(6,151,16,134)(7,152,17,135)(8,153,18,136)(9,154,19,137)(10,155,20,138)(11,156,21,139)(12,144,22,140)(13,145,23,141)(27,114,46,118)(28,115,47,119)(29,116,48,120)(30,117,49,121)(31,105,50,122)(32,106,51,123)(33,107,52,124)(34,108,40,125)(35,109,41,126)(36,110,42,127)(37,111,43,128)(38,112,44,129)(39,113,45,130)(53,184,72,201)(54,185,73,202)(55,186,74,203)(56,187,75,204)(57,188,76,205)(58,189,77,206)(59,190,78,207)(60,191,66,208)(61,192,67,196)(62,193,68,197)(63,194,69,198)(64,195,70,199)(65,183,71,200)(79,173,100,167)(80,174,101,168)(81,175,102,169)(82,176,103,157)(83,177,104,158)(84,178,92,159)(85,179,93,160)(86,180,94,161)(87,181,95,162)(88,182,96,163)(89,170,97,164)(90,171,98,165)(91,172,99,166), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,120)(10,119)(11,118)(12,130)(13,129)(14,108)(15,107)(16,106)(17,105)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,132)(41,131)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,182)(61,181)(62,180)(63,179)(64,178)(65,177)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,169)(74,168)(75,167)(76,166)(77,165)(78,164)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,208)(89,207)(90,206)(91,205)(92,195)(93,194)(94,193)(95,192)(96,191)(97,190)(98,189)(99,188)(100,187)(101,186)(102,185)(103,184)(104,183)>;
G:=Group( (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,111)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,118)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,155)(41,156)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,167)(54,168)(55,169)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,180)(67,181)(68,182)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,183)(92,206)(93,207)(94,208)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,205), (1,74,24,55)(2,75,25,56)(3,76,26,57)(4,77,14,58)(5,78,15,59)(6,66,16,60)(7,67,17,61)(8,68,18,62)(9,69,19,63)(10,70,20,64)(11,71,21,65)(12,72,22,53)(13,73,23,54)(27,104,46,83)(28,92,47,84)(29,93,48,85)(30,94,49,86)(31,95,50,87)(32,96,51,88)(33,97,52,89)(34,98,40,90)(35,99,41,91)(36,100,42,79)(37,101,43,80)(38,102,44,81)(39,103,45,82)(105,181,122,162)(106,182,123,163)(107,170,124,164)(108,171,125,165)(109,172,126,166)(110,173,127,167)(111,174,128,168)(112,175,129,169)(113,176,130,157)(114,177,118,158)(115,178,119,159)(116,179,120,160)(117,180,121,161)(131,205,148,188)(132,206,149,189)(133,207,150,190)(134,208,151,191)(135,196,152,192)(136,197,153,193)(137,198,154,194)(138,199,155,195)(139,200,156,183)(140,201,144,184)(141,202,145,185)(142,203,146,186)(143,204,147,187), (1,146,24,142)(2,147,25,143)(3,148,26,131)(4,149,14,132)(5,150,15,133)(6,151,16,134)(7,152,17,135)(8,153,18,136)(9,154,19,137)(10,155,20,138)(11,156,21,139)(12,144,22,140)(13,145,23,141)(27,114,46,118)(28,115,47,119)(29,116,48,120)(30,117,49,121)(31,105,50,122)(32,106,51,123)(33,107,52,124)(34,108,40,125)(35,109,41,126)(36,110,42,127)(37,111,43,128)(38,112,44,129)(39,113,45,130)(53,184,72,201)(54,185,73,202)(55,186,74,203)(56,187,75,204)(57,188,76,205)(58,189,77,206)(59,190,78,207)(60,191,66,208)(61,192,67,196)(62,193,68,197)(63,194,69,198)(64,195,70,199)(65,183,71,200)(79,173,100,167)(80,174,101,168)(81,175,102,169)(82,176,103,157)(83,177,104,158)(84,178,92,159)(85,179,93,160)(86,180,94,161)(87,181,95,162)(88,182,96,163)(89,170,97,164)(90,171,98,165)(91,172,99,166), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,120)(10,119)(11,118)(12,130)(13,129)(14,108)(15,107)(16,106)(17,105)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,132)(41,131)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,182)(61,181)(62,180)(63,179)(64,178)(65,177)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,169)(74,168)(75,167)(76,166)(77,165)(78,164)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,208)(89,207)(90,206)(91,205)(92,195)(93,194)(94,193)(95,192)(96,191)(97,190)(98,189)(99,188)(100,187)(101,186)(102,185)(103,184)(104,183) );
G=PermutationGroup([[(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,105),(8,106),(9,107),(10,108),(11,109),(12,110),(13,111),(14,119),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,127),(23,128),(24,129),(25,130),(26,118),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,141),(38,142),(39,143),(40,155),(41,156),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,167),(54,168),(55,169),(56,157),(57,158),(58,159),(59,160),(60,161),(61,162),(62,163),(63,164),(64,165),(65,166),(66,180),(67,181),(68,182),(69,170),(70,171),(71,172),(72,173),(73,174),(74,175),(75,176),(76,177),(77,178),(78,179),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,190),(86,191),(87,192),(88,193),(89,194),(90,195),(91,183),(92,206),(93,207),(94,208),(95,196),(96,197),(97,198),(98,199),(99,200),(100,201),(101,202),(102,203),(103,204),(104,205)], [(1,74,24,55),(2,75,25,56),(3,76,26,57),(4,77,14,58),(5,78,15,59),(6,66,16,60),(7,67,17,61),(8,68,18,62),(9,69,19,63),(10,70,20,64),(11,71,21,65),(12,72,22,53),(13,73,23,54),(27,104,46,83),(28,92,47,84),(29,93,48,85),(30,94,49,86),(31,95,50,87),(32,96,51,88),(33,97,52,89),(34,98,40,90),(35,99,41,91),(36,100,42,79),(37,101,43,80),(38,102,44,81),(39,103,45,82),(105,181,122,162),(106,182,123,163),(107,170,124,164),(108,171,125,165),(109,172,126,166),(110,173,127,167),(111,174,128,168),(112,175,129,169),(113,176,130,157),(114,177,118,158),(115,178,119,159),(116,179,120,160),(117,180,121,161),(131,205,148,188),(132,206,149,189),(133,207,150,190),(134,208,151,191),(135,196,152,192),(136,197,153,193),(137,198,154,194),(138,199,155,195),(139,200,156,183),(140,201,144,184),(141,202,145,185),(142,203,146,186),(143,204,147,187)], [(1,146,24,142),(2,147,25,143),(3,148,26,131),(4,149,14,132),(5,150,15,133),(6,151,16,134),(7,152,17,135),(8,153,18,136),(9,154,19,137),(10,155,20,138),(11,156,21,139),(12,144,22,140),(13,145,23,141),(27,114,46,118),(28,115,47,119),(29,116,48,120),(30,117,49,121),(31,105,50,122),(32,106,51,123),(33,107,52,124),(34,108,40,125),(35,109,41,126),(36,110,42,127),(37,111,43,128),(38,112,44,129),(39,113,45,130),(53,184,72,201),(54,185,73,202),(55,186,74,203),(56,187,75,204),(57,188,76,205),(58,189,77,206),(59,190,78,207),(60,191,66,208),(61,192,67,196),(62,193,68,197),(63,194,69,198),(64,195,70,199),(65,183,71,200),(79,173,100,167),(80,174,101,168),(81,175,102,169),(82,176,103,157),(83,177,104,158),(84,178,92,159),(85,179,93,160),(86,180,94,161),(87,181,95,162),(88,182,96,163),(89,170,97,164),(90,171,98,165),(91,172,99,166)], [(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)], [(1,128),(2,127),(3,126),(4,125),(5,124),(6,123),(7,122),(8,121),(9,120),(10,119),(11,118),(12,130),(13,129),(14,108),(15,107),(16,106),(17,105),(18,117),(19,116),(20,115),(21,114),(22,113),(23,112),(24,111),(25,110),(26,109),(27,156),(28,155),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,132),(41,131),(42,143),(43,142),(44,141),(45,140),(46,139),(47,138),(48,137),(49,136),(50,135),(51,134),(52,133),(53,176),(54,175),(55,174),(56,173),(57,172),(58,171),(59,170),(60,182),(61,181),(62,180),(63,179),(64,178),(65,177),(66,163),(67,162),(68,161),(69,160),(70,159),(71,158),(72,157),(73,169),(74,168),(75,167),(76,166),(77,165),(78,164),(79,204),(80,203),(81,202),(82,201),(83,200),(84,199),(85,198),(86,197),(87,196),(88,208),(89,207),(90,206),(91,205),(92,195),(93,194),(94,193),(95,192),(96,191),(97,190),(98,189),(99,188),(100,187),(101,186),(102,185),(103,184),(104,183)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D13 | D26 | D26 | Q8×D13 |
| kernel | C2×Q8×D13 | C2×Dic26 | C2×C4×D13 | Q8×D13 | Q8×C26 | D26 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 6 | 18 | 24 | 12 |
Matrix representation of C2×Q8×D13 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 27 | 43 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 7 |
| 0 | 0 | 44 | 50 |
| 23 | 1 | 0 | 0 |
| 41 | 41 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 22 | 0 | 0 |
| 11 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,10,27,0,0,10,43],[1,0,0,0,0,1,0,0,0,0,3,44,0,0,7,50],[23,41,0,0,1,41,0,0,0,0,1,0,0,0,0,1],[36,11,0,0,22,17,0,0,0,0,1,0,0,0,0,1] >;
C2×Q8×D13 in GAP, Magma, Sage, TeX
C_2\times Q_8\times D_{13} % in TeX
G:=Group("C2xQ8xD13"); // GroupNames label
G:=SmallGroup(416,219);
// by ID
G=gap.SmallGroup(416,219);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^13=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations