Copied to
clipboard

## G = C2×Q8×D13order 416 = 25·13

### Direct product of C2, Q8 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×Q8×D13
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C2×C4×D13 — C2×Q8×D13
 Lower central C13 — C26 — C2×Q8×D13
 Upper central C1 — C22 — C2×Q8

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

Smallest permutation representation of 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

׿
×
𝔽