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G = C2×Q8×D13order 416 = 25·13

Direct product of C2, Q8 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q8×D13, C26.8C24, C52.22C23, Dic269C22, D26.17C23, Dic13.5C23, C262(C2×Q8), (Q8×C26)⋊5C2, C132(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

C1C26 — C2×Q8×D13
C1C13C26D26C22×D13C2×C4×D13 — C2×Q8×D13
C13C26 — C2×Q8×D13
C1C22C2×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 [×2], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, C13, C22×C4 [×3], C2×Q8, C2×Q8 [×11], D13 [×4], C26, C26 [×2], C22×Q8, Dic13 [×6], C52 [×6], D26 [×6], C2×C26, Dic26 [×12], C4×D13 [×12], C2×Dic13 [×3], C2×C52 [×3], Q8×C13 [×4], C22×D13, C2×Dic26 [×3], C2×C4×D13 [×3], Q8×D13 [×8], Q8×C26, C2×Q8×D13
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, D13, C22×Q8, D26 [×7], C22×D13 [×7], Q8×D13 [×2], C23×D13, C2×Q8×D13

Smallest permutation representation of C2×Q8×D13
On 208 points
Generators in S208
(1 115)(2 116)(3 117)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 118)(26 119)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 143)(38 131)(39 132)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 162)(54 163)(55 164)(56 165)(57 166)(58 167)(59 168)(60 169)(61 157)(62 158)(63 159)(64 160)(65 161)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 194)(80 195)(81 183)(82 184)(83 185)(84 186)(85 187)(86 188)(87 189)(88 190)(89 191)(90 192)(91 193)(92 202)(93 203)(94 204)(95 205)(96 206)(97 207)(98 208)(99 196)(100 197)(101 198)(102 199)(103 200)(104 201)
(1 74 22 54)(2 75 23 55)(3 76 24 56)(4 77 25 57)(5 78 26 58)(6 66 14 59)(7 67 15 60)(8 68 16 61)(9 69 17 62)(10 70 18 63)(11 71 19 64)(12 72 20 65)(13 73 21 53)(27 100 46 90)(28 101 47 91)(29 102 48 79)(30 103 49 80)(31 104 50 81)(32 92 51 82)(33 93 52 83)(34 94 40 84)(35 95 41 85)(36 96 42 86)(37 97 43 87)(38 98 44 88)(39 99 45 89)(105 181 118 166)(106 182 119 167)(107 170 120 168)(108 171 121 169)(109 172 122 157)(110 173 123 158)(111 174 124 159)(112 175 125 160)(113 176 126 161)(114 177 127 162)(115 178 128 163)(116 179 129 164)(117 180 130 165)(131 208 155 190)(132 196 156 191)(133 197 144 192)(134 198 145 193)(135 199 146 194)(136 200 147 195)(137 201 148 183)(138 202 149 184)(139 203 150 185)(140 204 151 186)(141 205 152 187)(142 206 153 188)(143 207 154 189)
(1 153 22 142)(2 154 23 143)(3 155 24 131)(4 156 25 132)(5 144 26 133)(6 145 14 134)(7 146 15 135)(8 147 16 136)(9 148 17 137)(10 149 18 138)(11 150 19 139)(12 151 20 140)(13 152 21 141)(27 106 46 119)(28 107 47 120)(29 108 48 121)(30 109 49 122)(31 110 50 123)(32 111 51 124)(33 112 52 125)(34 113 40 126)(35 114 41 127)(36 115 42 128)(37 116 43 129)(38 117 44 130)(39 105 45 118)(53 187 73 205)(54 188 74 206)(55 189 75 207)(56 190 76 208)(57 191 77 196)(58 192 78 197)(59 193 66 198)(60 194 67 199)(61 195 68 200)(62 183 69 201)(63 184 70 202)(64 185 71 203)(65 186 72 204)(79 171 102 169)(80 172 103 157)(81 173 104 158)(82 174 92 159)(83 175 93 160)(84 176 94 161)(85 177 95 162)(86 178 96 163)(87 179 97 164)(88 180 98 165)(89 181 99 166)(90 182 100 167)(91 170 101 168)
(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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 120)(9 119)(10 118)(11 130)(12 129)(13 128)(14 109)(15 108)(16 107)(17 106)(18 105)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 148)(28 147)(29 146)(30 145)(31 144)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 143)(41 142)(42 141)(43 140)(44 139)(45 138)(46 137)(47 136)(48 135)(49 134)(50 133)(51 132)(52 131)(53 178)(54 177)(55 176)(56 175)(57 174)(58 173)(59 172)(60 171)(61 170)(62 182)(63 181)(64 180)(65 179)(66 157)(67 169)(68 168)(69 167)(70 166)(71 165)(72 164)(73 163)(74 162)(75 161)(76 160)(77 159)(78 158)(79 199)(80 198)(81 197)(82 196)(83 208)(84 207)(85 206)(86 205)(87 204)(88 203)(89 202)(90 201)(91 200)(92 191)(93 190)(94 189)(95 188)(96 187)(97 186)(98 185)(99 184)(100 183)(101 195)(102 194)(103 193)(104 192)

G:=sub<Sym(208)| (1,115)(2,116)(3,117)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,129)(24,130)(25,118)(26,119)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,143)(38,131)(39,132)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,157)(62,158)(63,159)(64,160)(65,161)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,181)(78,182)(79,194)(80,195)(81,183)(82,184)(83,185)(84,186)(85,187)(86,188)(87,189)(88,190)(89,191)(90,192)(91,193)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201), (1,74,22,54)(2,75,23,55)(3,76,24,56)(4,77,25,57)(5,78,26,58)(6,66,14,59)(7,67,15,60)(8,68,16,61)(9,69,17,62)(10,70,18,63)(11,71,19,64)(12,72,20,65)(13,73,21,53)(27,100,46,90)(28,101,47,91)(29,102,48,79)(30,103,49,80)(31,104,50,81)(32,92,51,82)(33,93,52,83)(34,94,40,84)(35,95,41,85)(36,96,42,86)(37,97,43,87)(38,98,44,88)(39,99,45,89)(105,181,118,166)(106,182,119,167)(107,170,120,168)(108,171,121,169)(109,172,122,157)(110,173,123,158)(111,174,124,159)(112,175,125,160)(113,176,126,161)(114,177,127,162)(115,178,128,163)(116,179,129,164)(117,180,130,165)(131,208,155,190)(132,196,156,191)(133,197,144,192)(134,198,145,193)(135,199,146,194)(136,200,147,195)(137,201,148,183)(138,202,149,184)(139,203,150,185)(140,204,151,186)(141,205,152,187)(142,206,153,188)(143,207,154,189), (1,153,22,142)(2,154,23,143)(3,155,24,131)(4,156,25,132)(5,144,26,133)(6,145,14,134)(7,146,15,135)(8,147,16,136)(9,148,17,137)(10,149,18,138)(11,150,19,139)(12,151,20,140)(13,152,21,141)(27,106,46,119)(28,107,47,120)(29,108,48,121)(30,109,49,122)(31,110,50,123)(32,111,51,124)(33,112,52,125)(34,113,40,126)(35,114,41,127)(36,115,42,128)(37,116,43,129)(38,117,44,130)(39,105,45,118)(53,187,73,205)(54,188,74,206)(55,189,75,207)(56,190,76,208)(57,191,77,196)(58,192,78,197)(59,193,66,198)(60,194,67,199)(61,195,68,200)(62,183,69,201)(63,184,70,202)(64,185,71,203)(65,186,72,204)(79,171,102,169)(80,172,103,157)(81,173,104,158)(82,174,92,159)(83,175,93,160)(84,176,94,161)(85,177,95,162)(86,178,96,163)(87,179,97,164)(88,180,98,165)(89,181,99,166)(90,182,100,167)(91,170,101,168), (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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,130)(12,129)(13,128)(14,109)(15,108)(16,107)(17,106)(18,105)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,148)(28,147)(29,146)(30,145)(31,144)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,182)(63,181)(64,180)(65,179)(66,157)(67,169)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,199)(80,198)(81,197)(82,196)(83,208)(84,207)(85,206)(86,205)(87,204)(88,203)(89,202)(90,201)(91,200)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,195)(102,194)(103,193)(104,192)>;

G:=Group( (1,115)(2,116)(3,117)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,129)(24,130)(25,118)(26,119)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,143)(38,131)(39,132)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,157)(62,158)(63,159)(64,160)(65,161)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,181)(78,182)(79,194)(80,195)(81,183)(82,184)(83,185)(84,186)(85,187)(86,188)(87,189)(88,190)(89,191)(90,192)(91,193)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201), (1,74,22,54)(2,75,23,55)(3,76,24,56)(4,77,25,57)(5,78,26,58)(6,66,14,59)(7,67,15,60)(8,68,16,61)(9,69,17,62)(10,70,18,63)(11,71,19,64)(12,72,20,65)(13,73,21,53)(27,100,46,90)(28,101,47,91)(29,102,48,79)(30,103,49,80)(31,104,50,81)(32,92,51,82)(33,93,52,83)(34,94,40,84)(35,95,41,85)(36,96,42,86)(37,97,43,87)(38,98,44,88)(39,99,45,89)(105,181,118,166)(106,182,119,167)(107,170,120,168)(108,171,121,169)(109,172,122,157)(110,173,123,158)(111,174,124,159)(112,175,125,160)(113,176,126,161)(114,177,127,162)(115,178,128,163)(116,179,129,164)(117,180,130,165)(131,208,155,190)(132,196,156,191)(133,197,144,192)(134,198,145,193)(135,199,146,194)(136,200,147,195)(137,201,148,183)(138,202,149,184)(139,203,150,185)(140,204,151,186)(141,205,152,187)(142,206,153,188)(143,207,154,189), (1,153,22,142)(2,154,23,143)(3,155,24,131)(4,156,25,132)(5,144,26,133)(6,145,14,134)(7,146,15,135)(8,147,16,136)(9,148,17,137)(10,149,18,138)(11,150,19,139)(12,151,20,140)(13,152,21,141)(27,106,46,119)(28,107,47,120)(29,108,48,121)(30,109,49,122)(31,110,50,123)(32,111,51,124)(33,112,52,125)(34,113,40,126)(35,114,41,127)(36,115,42,128)(37,116,43,129)(38,117,44,130)(39,105,45,118)(53,187,73,205)(54,188,74,206)(55,189,75,207)(56,190,76,208)(57,191,77,196)(58,192,78,197)(59,193,66,198)(60,194,67,199)(61,195,68,200)(62,183,69,201)(63,184,70,202)(64,185,71,203)(65,186,72,204)(79,171,102,169)(80,172,103,157)(81,173,104,158)(82,174,92,159)(83,175,93,160)(84,176,94,161)(85,177,95,162)(86,178,96,163)(87,179,97,164)(88,180,98,165)(89,181,99,166)(90,182,100,167)(91,170,101,168), (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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,130)(12,129)(13,128)(14,109)(15,108)(16,107)(17,106)(18,105)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,148)(28,147)(29,146)(30,145)(31,144)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,182)(63,181)(64,180)(65,179)(66,157)(67,169)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,199)(80,198)(81,197)(82,196)(83,208)(84,207)(85,206)(86,205)(87,204)(88,203)(89,202)(90,201)(91,200)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,195)(102,194)(103,193)(104,192) );

G=PermutationGroup([(1,115),(2,116),(3,117),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,120),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,127),(22,128),(23,129),(24,130),(25,118),(26,119),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,141),(36,142),(37,143),(38,131),(39,132),(40,151),(41,152),(42,153),(43,154),(44,155),(45,156),(46,144),(47,145),(48,146),(49,147),(50,148),(51,149),(52,150),(53,162),(54,163),(55,164),(56,165),(57,166),(58,167),(59,168),(60,169),(61,157),(62,158),(63,159),(64,160),(65,161),(66,170),(67,171),(68,172),(69,173),(70,174),(71,175),(72,176),(73,177),(74,178),(75,179),(76,180),(77,181),(78,182),(79,194),(80,195),(81,183),(82,184),(83,185),(84,186),(85,187),(86,188),(87,189),(88,190),(89,191),(90,192),(91,193),(92,202),(93,203),(94,204),(95,205),(96,206),(97,207),(98,208),(99,196),(100,197),(101,198),(102,199),(103,200),(104,201)], [(1,74,22,54),(2,75,23,55),(3,76,24,56),(4,77,25,57),(5,78,26,58),(6,66,14,59),(7,67,15,60),(8,68,16,61),(9,69,17,62),(10,70,18,63),(11,71,19,64),(12,72,20,65),(13,73,21,53),(27,100,46,90),(28,101,47,91),(29,102,48,79),(30,103,49,80),(31,104,50,81),(32,92,51,82),(33,93,52,83),(34,94,40,84),(35,95,41,85),(36,96,42,86),(37,97,43,87),(38,98,44,88),(39,99,45,89),(105,181,118,166),(106,182,119,167),(107,170,120,168),(108,171,121,169),(109,172,122,157),(110,173,123,158),(111,174,124,159),(112,175,125,160),(113,176,126,161),(114,177,127,162),(115,178,128,163),(116,179,129,164),(117,180,130,165),(131,208,155,190),(132,196,156,191),(133,197,144,192),(134,198,145,193),(135,199,146,194),(136,200,147,195),(137,201,148,183),(138,202,149,184),(139,203,150,185),(140,204,151,186),(141,205,152,187),(142,206,153,188),(143,207,154,189)], [(1,153,22,142),(2,154,23,143),(3,155,24,131),(4,156,25,132),(5,144,26,133),(6,145,14,134),(7,146,15,135),(8,147,16,136),(9,148,17,137),(10,149,18,138),(11,150,19,139),(12,151,20,140),(13,152,21,141),(27,106,46,119),(28,107,47,120),(29,108,48,121),(30,109,49,122),(31,110,50,123),(32,111,51,124),(33,112,52,125),(34,113,40,126),(35,114,41,127),(36,115,42,128),(37,116,43,129),(38,117,44,130),(39,105,45,118),(53,187,73,205),(54,188,74,206),(55,189,75,207),(56,190,76,208),(57,191,77,196),(58,192,78,197),(59,193,66,198),(60,194,67,199),(61,195,68,200),(62,183,69,201),(63,184,70,202),(64,185,71,203),(65,186,72,204),(79,171,102,169),(80,172,103,157),(81,173,104,158),(82,174,92,159),(83,175,93,160),(84,176,94,161),(85,177,95,162),(86,178,96,163),(87,179,97,164),(88,180,98,165),(89,181,99,166),(90,182,100,167),(91,170,101,168)], [(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)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,120),(9,119),(10,118),(11,130),(12,129),(13,128),(14,109),(15,108),(16,107),(17,106),(18,105),(19,117),(20,116),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,148),(28,147),(29,146),(30,145),(31,144),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,143),(41,142),(42,141),(43,140),(44,139),(45,138),(46,137),(47,136),(48,135),(49,134),(50,133),(51,132),(52,131),(53,178),(54,177),(55,176),(56,175),(57,174),(58,173),(59,172),(60,171),(61,170),(62,182),(63,181),(64,180),(65,179),(66,157),(67,169),(68,168),(69,167),(70,166),(71,165),(72,164),(73,163),(74,162),(75,161),(76,160),(77,159),(78,158),(79,199),(80,198),(81,197),(82,196),(83,208),(84,207),(85,206),(86,205),(87,204),(88,203),(89,202),(90,201),(91,200),(92,191),(93,190),(94,189),(95,188),(96,187),(97,186),(98,185),(99,184),(100,183),(101,195),(102,194),(103,193),(104,192)])

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L13A···13F26A···26R52A···52AJ
order122222224···44···413···1326···2652···52
size1111131313132···226···262···22···24···4

80 irreducible representations

dim1111122224
type+++++-+++-
imageC1C2C2C2C2Q8D13D26D26Q8×D13
kernelC2×Q8×D13C2×Dic26C2×C4×D13Q8×D13Q8×C26D26C2×Q8C2×C4Q8C2
# reps1338146182412

Matrix representation of C2×Q8×D13 in GL4(𝔽53) generated by

52000
05200
00520
00052
,
52000
05200
001010
002743
,
1000
0100
0037
004450
,
23100
414100
0010
0001
,
362200
111700
0010
0001
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