direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4⋊D4, C52⋊9D4, C4⋊C4⋊2C26, (C2×C26)⋊4D4, C4⋊2(D4×C13), (C2×D4)⋊2C26, C2.5(D4×C26), (D4×C26)⋊11C2, C22⋊C4⋊3C26, (C22×C4)⋊4C26, C26.68(C2×D4), C22⋊1(D4×C13), (C22×C52)⋊11C2, C23.8(C2×C26), C26.41(C4○D4), (C2×C26).76C23, (C2×C52).123C22, (C22×C26).27C22, C22.11(C22×C26), (C13×C4⋊C4)⋊11C2, (C2×C4).3(C2×C26), C2.4(C13×C4○D4), (C13×C22⋊C4)⋊11C2, SmallGroup(416,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4⋊D4
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C26, C26, C4⋊D4, C52, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, C2×C52, D4×C13, C22×C26, C22×C26, C13×C22⋊C4, C13×C4⋊C4, C22×C52, D4×C26, D4×C26, C13×C4⋊D4
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C4○D4, C26, C4⋊D4, C2×C26, D4×C13, C22×C26, D4×C26, C13×C4○D4, C13×C4⋊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)
(1 174 74 56)(2 175 75 57)(3 176 76 58)(4 177 77 59)(5 178 78 60)(6 179 66 61)(7 180 67 62)(8 181 68 63)(9 182 69 64)(10 170 70 65)(11 171 71 53)(12 172 72 54)(13 173 73 55)(14 204 147 166)(15 205 148 167)(16 206 149 168)(17 207 150 169)(18 208 151 157)(19 196 152 158)(20 197 153 159)(21 198 154 160)(22 199 155 161)(23 200 156 162)(24 201 144 163)(25 202 145 164)(26 203 146 165)(27 106 133 121)(28 107 134 122)(29 108 135 123)(30 109 136 124)(31 110 137 125)(32 111 138 126)(33 112 139 127)(34 113 140 128)(35 114 141 129)(36 115 142 130)(37 116 143 118)(38 117 131 119)(39 105 132 120)(40 183 92 83)(41 184 93 84)(42 185 94 85)(43 186 95 86)(44 187 96 87)(45 188 97 88)(46 189 98 89)(47 190 99 90)(48 191 100 91)(49 192 101 79)(50 193 102 80)(51 194 103 81)(52 195 104 82)
(1 154 139 103)(2 155 140 104)(3 156 141 92)(4 144 142 93)(5 145 143 94)(6 146 131 95)(7 147 132 96)(8 148 133 97)(9 149 134 98)(10 150 135 99)(11 151 136 100)(12 152 137 101)(13 153 138 102)(14 39 44 67)(15 27 45 68)(16 28 46 69)(17 29 47 70)(18 30 48 71)(19 31 49 72)(20 32 50 73)(21 33 51 74)(22 34 52 75)(23 35 40 76)(24 36 41 77)(25 37 42 78)(26 38 43 66)(53 157 109 91)(54 158 110 79)(55 159 111 80)(56 160 112 81)(57 161 113 82)(58 162 114 83)(59 163 115 84)(60 164 116 85)(61 165 117 86)(62 166 105 87)(63 167 106 88)(64 168 107 89)(65 169 108 90)(118 185 178 202)(119 186 179 203)(120 187 180 204)(121 188 181 205)(122 189 182 206)(123 190 170 207)(124 191 171 208)(125 192 172 196)(126 193 173 197)(127 194 174 198)(128 195 175 199)(129 183 176 200)(130 184 177 201)
(1 103)(2 104)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 39)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(40 76)(41 77)(42 78)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 91)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(105 166)(106 167)(107 168)(108 169)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 208)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 144)(143 145)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)
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), (1,174,74,56)(2,175,75,57)(3,176,76,58)(4,177,77,59)(5,178,78,60)(6,179,66,61)(7,180,67,62)(8,181,68,63)(9,182,69,64)(10,170,70,65)(11,171,71,53)(12,172,72,54)(13,173,73,55)(14,204,147,166)(15,205,148,167)(16,206,149,168)(17,207,150,169)(18,208,151,157)(19,196,152,158)(20,197,153,159)(21,198,154,160)(22,199,155,161)(23,200,156,162)(24,201,144,163)(25,202,145,164)(26,203,146,165)(27,106,133,121)(28,107,134,122)(29,108,135,123)(30,109,136,124)(31,110,137,125)(32,111,138,126)(33,112,139,127)(34,113,140,128)(35,114,141,129)(36,115,142,130)(37,116,143,118)(38,117,131,119)(39,105,132,120)(40,183,92,83)(41,184,93,84)(42,185,94,85)(43,186,95,86)(44,187,96,87)(45,188,97,88)(46,189,98,89)(47,190,99,90)(48,191,100,91)(49,192,101,79)(50,193,102,80)(51,194,103,81)(52,195,104,82), (1,154,139,103)(2,155,140,104)(3,156,141,92)(4,144,142,93)(5,145,143,94)(6,146,131,95)(7,147,132,96)(8,148,133,97)(9,149,134,98)(10,150,135,99)(11,151,136,100)(12,152,137,101)(13,153,138,102)(14,39,44,67)(15,27,45,68)(16,28,46,69)(17,29,47,70)(18,30,48,71)(19,31,49,72)(20,32,50,73)(21,33,51,74)(22,34,52,75)(23,35,40,76)(24,36,41,77)(25,37,42,78)(26,38,43,66)(53,157,109,91)(54,158,110,79)(55,159,111,80)(56,160,112,81)(57,161,113,82)(58,162,114,83)(59,163,115,84)(60,164,116,85)(61,165,117,86)(62,166,105,87)(63,167,106,88)(64,168,107,89)(65,169,108,90)(118,185,178,202)(119,186,179,203)(120,187,180,204)(121,188,181,205)(122,189,182,206)(123,190,170,207)(124,191,171,208)(125,192,172,196)(126,193,173,197)(127,194,174,198)(128,195,175,199)(129,183,176,200)(130,184,177,201), (1,103)(2,104)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,39)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(40,76)(41,77)(42,78)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,91)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(105,166)(106,167)(107,168)(108,169)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,144)(143,145)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)>;
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), (1,174,74,56)(2,175,75,57)(3,176,76,58)(4,177,77,59)(5,178,78,60)(6,179,66,61)(7,180,67,62)(8,181,68,63)(9,182,69,64)(10,170,70,65)(11,171,71,53)(12,172,72,54)(13,173,73,55)(14,204,147,166)(15,205,148,167)(16,206,149,168)(17,207,150,169)(18,208,151,157)(19,196,152,158)(20,197,153,159)(21,198,154,160)(22,199,155,161)(23,200,156,162)(24,201,144,163)(25,202,145,164)(26,203,146,165)(27,106,133,121)(28,107,134,122)(29,108,135,123)(30,109,136,124)(31,110,137,125)(32,111,138,126)(33,112,139,127)(34,113,140,128)(35,114,141,129)(36,115,142,130)(37,116,143,118)(38,117,131,119)(39,105,132,120)(40,183,92,83)(41,184,93,84)(42,185,94,85)(43,186,95,86)(44,187,96,87)(45,188,97,88)(46,189,98,89)(47,190,99,90)(48,191,100,91)(49,192,101,79)(50,193,102,80)(51,194,103,81)(52,195,104,82), (1,154,139,103)(2,155,140,104)(3,156,141,92)(4,144,142,93)(5,145,143,94)(6,146,131,95)(7,147,132,96)(8,148,133,97)(9,149,134,98)(10,150,135,99)(11,151,136,100)(12,152,137,101)(13,153,138,102)(14,39,44,67)(15,27,45,68)(16,28,46,69)(17,29,47,70)(18,30,48,71)(19,31,49,72)(20,32,50,73)(21,33,51,74)(22,34,52,75)(23,35,40,76)(24,36,41,77)(25,37,42,78)(26,38,43,66)(53,157,109,91)(54,158,110,79)(55,159,111,80)(56,160,112,81)(57,161,113,82)(58,162,114,83)(59,163,115,84)(60,164,116,85)(61,165,117,86)(62,166,105,87)(63,167,106,88)(64,168,107,89)(65,169,108,90)(118,185,178,202)(119,186,179,203)(120,187,180,204)(121,188,181,205)(122,189,182,206)(123,190,170,207)(124,191,171,208)(125,192,172,196)(126,193,173,197)(127,194,174,198)(128,195,175,199)(129,183,176,200)(130,184,177,201), (1,103)(2,104)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,39)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(40,76)(41,77)(42,78)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,91)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(105,166)(106,167)(107,168)(108,169)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,144)(143,145)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189) );
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)], [(1,174,74,56),(2,175,75,57),(3,176,76,58),(4,177,77,59),(5,178,78,60),(6,179,66,61),(7,180,67,62),(8,181,68,63),(9,182,69,64),(10,170,70,65),(11,171,71,53),(12,172,72,54),(13,173,73,55),(14,204,147,166),(15,205,148,167),(16,206,149,168),(17,207,150,169),(18,208,151,157),(19,196,152,158),(20,197,153,159),(21,198,154,160),(22,199,155,161),(23,200,156,162),(24,201,144,163),(25,202,145,164),(26,203,146,165),(27,106,133,121),(28,107,134,122),(29,108,135,123),(30,109,136,124),(31,110,137,125),(32,111,138,126),(33,112,139,127),(34,113,140,128),(35,114,141,129),(36,115,142,130),(37,116,143,118),(38,117,131,119),(39,105,132,120),(40,183,92,83),(41,184,93,84),(42,185,94,85),(43,186,95,86),(44,187,96,87),(45,188,97,88),(46,189,98,89),(47,190,99,90),(48,191,100,91),(49,192,101,79),(50,193,102,80),(51,194,103,81),(52,195,104,82)], [(1,154,139,103),(2,155,140,104),(3,156,141,92),(4,144,142,93),(5,145,143,94),(6,146,131,95),(7,147,132,96),(8,148,133,97),(9,149,134,98),(10,150,135,99),(11,151,136,100),(12,152,137,101),(13,153,138,102),(14,39,44,67),(15,27,45,68),(16,28,46,69),(17,29,47,70),(18,30,48,71),(19,31,49,72),(20,32,50,73),(21,33,51,74),(22,34,52,75),(23,35,40,76),(24,36,41,77),(25,37,42,78),(26,38,43,66),(53,157,109,91),(54,158,110,79),(55,159,111,80),(56,160,112,81),(57,161,113,82),(58,162,114,83),(59,163,115,84),(60,164,116,85),(61,165,117,86),(62,166,105,87),(63,167,106,88),(64,168,107,89),(65,169,108,90),(118,185,178,202),(119,186,179,203),(120,187,180,204),(121,188,181,205),(122,189,182,206),(123,190,170,207),(124,191,171,208),(125,192,172,196),(126,193,173,197),(127,194,174,198),(128,195,175,199),(129,183,176,200),(130,184,177,201)], [(1,103),(2,104),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,39),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(40,76),(41,77),(42,78),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,91),(54,79),(55,80),(56,81),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(105,166),(106,167),(107,168),(108,169),(109,157),(110,158),(111,159),(112,160),(113,161),(114,162),(115,163),(116,164),(117,165),(118,202),(119,203),(120,204),(121,205),(122,206),(123,207),(124,208),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155),(141,156),(142,144),(143,145),(170,190),(171,191),(172,192),(173,193),(174,194),(175,195),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 26BI | ··· | 26CF | 52A | ··· | 52AV | 52AW | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 2 | 2 | 2 | 2 | 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | D4 | D4 | C4○D4 | D4×C13 | D4×C13 | C13×C4○D4 |
| kernel | C13×C4⋊D4 | C13×C22⋊C4 | C13×C4⋊C4 | C22×C52 | D4×C26 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 12 | 24 | 12 | 12 | 36 | 2 | 2 | 2 | 24 | 24 | 24 |
Matrix representation of C13×C4⋊D4 ►in GL4(𝔽53) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 23 | 44 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 52 | 0 |
| 24 | 47 | 0 | 0 |
| 52 | 29 | 0 | 0 |
| 0 | 0 | 10 | 22 |
| 0 | 0 | 22 | 43 |
| 29 | 8 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 10 | 22 |
| 0 | 0 | 22 | 43 |
G:=sub<GL(4,GF(53))| [28,0,0,0,0,28,0,0,0,0,42,0,0,0,0,42],[23,0,0,0,44,30,0,0,0,0,0,52,0,0,1,0],[24,52,0,0,47,29,0,0,0,0,10,22,0,0,22,43],[29,1,0,0,8,24,0,0,0,0,10,22,0,0,22,43] >;
C13×C4⋊D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4\rtimes D_4 % in TeX
G:=Group("C13xC4:D4"); // GroupNames label
G:=SmallGroup(416,182);
// by ID
G=gap.SmallGroup(416,182);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,631,3818]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations