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## G = C2×Q8⋊D13order 416 = 25·13

### Direct product of C2 and Q8⋊D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×Q8⋊D13
 Chief series C1 — C13 — C26 — C52 — D52 — C2×D52 — C2×Q8⋊D13
 Lower central C13 — C26 — C52 — C2×Q8⋊D13
 Upper central C1 — C22 — C2×C4 — 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=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 528 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C13, C2×C8, SD16 [×4], C2×D4, C2×Q8, D13 [×2], C26, C26 [×2], C2×SD16, C52 [×2], C52 [×2], D26 [×4], C2×C26, C132C8 [×2], D52 [×2], D52, C2×C52, C2×C52, Q8×C13 [×2], Q8×C13, C22×D13, C2×C132C8, Q8⋊D13 [×4], C2×D52, Q8×C26, C2×Q8⋊D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, D13, C2×SD16, D26 [×3], C13⋊D4 [×2], C22×D13, Q8⋊D13 [×2], C2×C13⋊D4, C2×Q8⋊D13

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 2 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 52 52 2 2 4 4 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 SD16 D13 D26 D26 C13⋊D4 C13⋊D4 Q8⋊D13 kernel C2×Q8⋊D13 C2×C13⋊2C8 Q8⋊D13 C2×D52 Q8×C26 C52 C2×C26 C26 C2×Q8 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 4 6 6 12 12 12 12

Matrix representation of C2×Q8⋊D13 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 312 238 0 0 192 1
,
 312 0 0 0 0 312 0 0 0 0 0 133 0 0 40 0
,
 198 312 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 115 1 0 0 235 198 0 0 0 0 1 0 0 0 121 312
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,312,192,0,0,238,1],[312,0,0,0,0,312,0,0,0,0,0,40,0,0,133,0],[198,1,0,0,312,0,0,0,0,0,1,0,0,0,0,1],[115,235,0,0,1,198,0,0,0,0,1,121,0,0,0,312] >;

C2×Q8⋊D13 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_{13}
% in TeX

G:=Group("C2xQ8:D13");
// GroupNames label

G:=SmallGroup(416,162);
// by ID

G=gap.SmallGroup(416,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,86,579,159,69,13829]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
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𝔽