direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C22.D4, C4⋊C4⋊4C26, C2.7(D4×C26), C22⋊C4⋊4C26, (C22×C4)⋊3C26, (C22×C52)⋊5C2, (C2×D4).4C26, C26.70(C2×D4), (C2×C26).23D4, (D4×C26).11C2, C22.4(D4×C13), C26.43(C4○D4), (C2×C52).65C22, C23.10(C2×C26), (C2×C26).78C23, (C22×C26).29C22, C22.13(C22×C26), (C13×C4⋊C4)⋊13C2, (C2×C4).5(C2×C26), C2.6(C13×C4○D4), (C13×C22⋊C4)⋊12C2, SmallGroup(416,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C22.D4
G = < a,b,c,d,e | a13=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 116 in 78 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C2×C4, C2×C4 [×4], C2×C4 [×2], D4 [×2], C23 [×2], C13, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C26, C26 [×2], C26 [×3], C22.D4, C52 [×5], C2×C26, C2×C26 [×2], C2×C26 [×5], C2×C52, C2×C52 [×4], C2×C52 [×2], D4×C13 [×2], C22×C26 [×2], C13×C22⋊C4, C13×C22⋊C4 [×2], C13×C4⋊C4 [×2], C22×C52, D4×C26, C13×C22.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C4○D4 [×2], C26 [×7], C22.D4, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C4○D4 [×2], C13×C22.D4
►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 52)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 27)(53 124)(54 125)(55 126)(56 127)(57 128)(58 129)(59 130)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 193)(67 194)(68 195)(69 183)(70 184)(71 185)(72 186)(73 187)(74 188)(75 189)(76 190)(77 191)(78 192)(79 167)(80 168)(81 169)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 144)(100 145)(101 146)(102 147)(103 148)(104 149)(105 143)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(170 207)(171 208)(172 196)(173 197)(174 198)(175 199)(176 200)(177 201)(178 202)(179 203)(180 204)(181 205)(182 206)
(1 129)(2 130)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 138)(15 139)(16 140)(17 141)(18 142)(19 143)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 53)(48 54)(49 55)(50 56)(51 57)(52 58)(66 175)(67 176)(68 177)(69 178)(70 179)(71 180)(72 181)(73 182)(74 170)(75 171)(76 172)(77 173)(78 174)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(144 167)(145 168)(146 169)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)(183 202)(184 203)(185 204)(186 205)(187 206)(188 207)(189 208)(190 196)(191 197)(192 198)(193 199)(194 200)(195 201)
(1 156 58 98)(2 144 59 99)(3 145 60 100)(4 146 61 101)(5 147 62 102)(6 148 63 103)(7 149 64 104)(8 150 65 92)(9 151 53 93)(10 152 54 94)(11 153 55 95)(12 154 56 96)(13 155 57 97)(14 76 28 196)(15 77 29 197)(16 78 30 198)(17 66 31 199)(18 67 32 200)(19 68 33 201)(20 69 34 202)(21 70 35 203)(22 71 36 204)(23 72 37 205)(24 73 38 206)(25 74 39 207)(26 75 27 208)(40 79 130 167)(41 80 118 168)(42 81 119 169)(43 82 120 157)(44 83 121 158)(45 84 122 159)(46 85 123 160)(47 86 124 161)(48 87 125 162)(49 88 126 163)(50 89 127 164)(51 90 128 165)(52 91 129 166)(105 195 143 177)(106 183 131 178)(107 184 132 179)(108 185 133 180)(109 186 134 181)(110 187 135 182)(111 188 136 170)(112 189 137 171)(113 190 138 172)(114 191 139 173)(115 192 140 174)(116 193 141 175)(117 194 142 176)
(1 116)(2 117)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 125)(28 126)(29 127)(30 128)(31 129)(32 130)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(53 136)(54 137)(55 138)(56 139)(57 140)(58 141)(59 142)(60 143)(61 131)(62 132)(63 133)(64 134)(65 135)(66 156)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(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)(92 206)(93 207)(94 208)(95 196)(96 197)(97 198)(98 199)(99 200)(100 201)(101 202)(102 203)(103 204)(104 205)(157 179)(158 180)(159 181)(160 182)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)
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), (1,52)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,27)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,193)(67,194)(68,195)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,167)(80,168)(81,169)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,143)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(170,207)(171,208)(172,196)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206), (1,129)(2,130)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,138)(15,139)(16,140)(17,141)(18,142)(19,143)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,170)(75,171)(76,172)(77,173)(78,174)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(144,167)(145,168)(146,169)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(183,202)(184,203)(185,204)(186,205)(187,206)(188,207)(189,208)(190,196)(191,197)(192,198)(193,199)(194,200)(195,201), (1,156,58,98)(2,144,59,99)(3,145,60,100)(4,146,61,101)(5,147,62,102)(6,148,63,103)(7,149,64,104)(8,150,65,92)(9,151,53,93)(10,152,54,94)(11,153,55,95)(12,154,56,96)(13,155,57,97)(14,76,28,196)(15,77,29,197)(16,78,30,198)(17,66,31,199)(18,67,32,200)(19,68,33,201)(20,69,34,202)(21,70,35,203)(22,71,36,204)(23,72,37,205)(24,73,38,206)(25,74,39,207)(26,75,27,208)(40,79,130,167)(41,80,118,168)(42,81,119,169)(43,82,120,157)(44,83,121,158)(45,84,122,159)(46,85,123,160)(47,86,124,161)(48,87,125,162)(49,88,126,163)(50,89,127,164)(51,90,128,165)(52,91,129,166)(105,195,143,177)(106,183,131,178)(107,184,132,179)(108,185,133,180)(109,186,134,181)(110,187,135,182)(111,188,136,170)(112,189,137,171)(113,190,138,172)(114,191,139,173)(115,192,140,174)(116,193,141,175)(117,194,142,176), (1,116)(2,117)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,131)(62,132)(63,133)(64,134)(65,135)(66,156)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(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)(92,206)(93,207)(94,208)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,205)(157,179)(158,180)(159,181)(160,182)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)>;
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), (1,52)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,27)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,193)(67,194)(68,195)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,167)(80,168)(81,169)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,143)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(170,207)(171,208)(172,196)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206), (1,129)(2,130)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,138)(15,139)(16,140)(17,141)(18,142)(19,143)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,170)(75,171)(76,172)(77,173)(78,174)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(144,167)(145,168)(146,169)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(183,202)(184,203)(185,204)(186,205)(187,206)(188,207)(189,208)(190,196)(191,197)(192,198)(193,199)(194,200)(195,201), (1,156,58,98)(2,144,59,99)(3,145,60,100)(4,146,61,101)(5,147,62,102)(6,148,63,103)(7,149,64,104)(8,150,65,92)(9,151,53,93)(10,152,54,94)(11,153,55,95)(12,154,56,96)(13,155,57,97)(14,76,28,196)(15,77,29,197)(16,78,30,198)(17,66,31,199)(18,67,32,200)(19,68,33,201)(20,69,34,202)(21,70,35,203)(22,71,36,204)(23,72,37,205)(24,73,38,206)(25,74,39,207)(26,75,27,208)(40,79,130,167)(41,80,118,168)(42,81,119,169)(43,82,120,157)(44,83,121,158)(45,84,122,159)(46,85,123,160)(47,86,124,161)(48,87,125,162)(49,88,126,163)(50,89,127,164)(51,90,128,165)(52,91,129,166)(105,195,143,177)(106,183,131,178)(107,184,132,179)(108,185,133,180)(109,186,134,181)(110,187,135,182)(111,188,136,170)(112,189,137,171)(113,190,138,172)(114,191,139,173)(115,192,140,174)(116,193,141,175)(117,194,142,176), (1,116)(2,117)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,131)(62,132)(63,133)(64,134)(65,135)(66,156)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(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)(92,206)(93,207)(94,208)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(103,204)(104,205)(157,179)(158,180)(159,181)(160,182)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178) );
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)], [(1,52),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,27),(53,124),(54,125),(55,126),(56,127),(57,128),(58,129),(59,130),(60,118),(61,119),(62,120),(63,121),(64,122),(65,123),(66,193),(67,194),(68,195),(69,183),(70,184),(71,185),(72,186),(73,187),(74,188),(75,189),(76,190),(77,191),(78,192),(79,167),(80,168),(81,169),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,150),(93,151),(94,152),(95,153),(96,154),(97,155),(98,156),(99,144),(100,145),(101,146),(102,147),(103,148),(104,149),(105,143),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(170,207),(171,208),(172,196),(173,197),(174,198),(175,199),(176,200),(177,201),(178,202),(179,203),(180,204),(181,205),(182,206)], [(1,129),(2,130),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,138),(15,139),(16,140),(17,141),(18,142),(19,143),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,53),(48,54),(49,55),(50,56),(51,57),(52,58),(66,175),(67,176),(68,177),(69,178),(70,179),(71,180),(72,181),(73,182),(74,170),(75,171),(76,172),(77,173),(78,174),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(144,167),(145,168),(146,169),(147,157),(148,158),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(155,165),(156,166),(183,202),(184,203),(185,204),(186,205),(187,206),(188,207),(189,208),(190,196),(191,197),(192,198),(193,199),(194,200),(195,201)], [(1,156,58,98),(2,144,59,99),(3,145,60,100),(4,146,61,101),(5,147,62,102),(6,148,63,103),(7,149,64,104),(8,150,65,92),(9,151,53,93),(10,152,54,94),(11,153,55,95),(12,154,56,96),(13,155,57,97),(14,76,28,196),(15,77,29,197),(16,78,30,198),(17,66,31,199),(18,67,32,200),(19,68,33,201),(20,69,34,202),(21,70,35,203),(22,71,36,204),(23,72,37,205),(24,73,38,206),(25,74,39,207),(26,75,27,208),(40,79,130,167),(41,80,118,168),(42,81,119,169),(43,82,120,157),(44,83,121,158),(45,84,122,159),(46,85,123,160),(47,86,124,161),(48,87,125,162),(49,88,126,163),(50,89,127,164),(51,90,128,165),(52,91,129,166),(105,195,143,177),(106,183,131,178),(107,184,132,179),(108,185,133,180),(109,186,134,181),(110,187,135,182),(111,188,136,170),(112,189,137,171),(113,190,138,172),(114,191,139,173),(115,192,140,174),(116,193,141,175),(117,194,142,176)], [(1,116),(2,117),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,115),(14,49),(15,50),(16,51),(17,52),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,125),(28,126),(29,127),(30,128),(31,129),(32,130),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(53,136),(54,137),(55,138),(56,139),(57,140),(58,141),(59,142),(60,143),(61,131),(62,132),(63,133),(64,134),(65,135),(66,156),(67,144),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(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),(92,206),(93,207),(94,208),(95,196),(96,197),(97,198),(98,199),(99,200),(100,201),(101,202),(102,203),(103,204),(104,205),(157,179),(158,180),(159,181),(160,182),(161,170),(162,171),(163,172),(164,173),(165,174),(166,175),(167,176),(168,177),(169,178)])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 26BI | ··· | 26BT | 52A | ··· | 52AV | 52AW | ··· | 52CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | D4 | C4○D4 | D4×C13 | C13×C4○D4 |
| kernel | C13×C22.D4 | C13×C22⋊C4 | C13×C4⋊C4 | C22×C52 | D4×C26 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C26 | C26 | C22 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 12 | 36 | 24 | 12 | 12 | 2 | 4 | 24 | 48 |
Matrix representation of C13×C22.D4 ►in GL4(𝔽53) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 30 | 0 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 0 | 17 | 5 |
| 0 | 0 | 48 | 36 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 48 | 36 |
| 0 | 0 | 17 | 5 |
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[0,30,0,0,30,0,0,0,0,0,17,48,0,0,5,36],[0,1,0,0,1,0,0,0,0,0,48,17,0,0,36,5] >;
C13×C22.D4 in GAP, Magma, Sage, TeX
C_{13}\times C_2^2.D_4 % in TeX
G:=Group("C13xC2^2.D4"); // GroupNames label
G:=SmallGroup(416,184);
// by ID
G=gap.SmallGroup(416,184);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818,482]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations