direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C42×D13, (C4×C52)⋊8C2, C52⋊8(C2×C4), C13⋊2(C2×C42), (C2×C4).95D26, Dic13⋊8(C2×C4), D26.18(C2×C4), (C4×Dic13)⋊17C2, C26.15(C22×C4), (C2×C26).12C23, (C2×C52).109C22, C22.9(C22×D13), (C2×Dic13).59C22, (C22×D13).41C22, C2.1(C2×C4×D13), (C2×C4×D13).19C2, SmallGroup(416,92)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C42×D13 |
Generators and relations for C42×D13
G = < a,b,c,d | a4=b4=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 576 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C42, C22×C4, D13, C26, C2×C42, Dic13, C52, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C22×D13, C4×Dic13, C4×C52, C2×C4×D13, C42×D13
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, D13, C2×C42, D26, C4×D13, C22×D13, C2×C4×D13, C42×D13
►On 208 points
Generators in S208
(1 124 19 107)(2 125 20 108)(3 126 21 109)(4 127 22 110)(5 128 23 111)(6 129 24 112)(7 130 25 113)(8 118 26 114)(9 119 14 115)(10 120 15 116)(11 121 16 117)(12 122 17 105)(13 123 18 106)(27 148 43 140)(28 149 44 141)(29 150 45 142)(30 151 46 143)(31 152 47 131)(32 153 48 132)(33 154 49 133)(34 155 50 134)(35 156 51 135)(36 144 52 136)(37 145 40 137)(38 146 41 138)(39 147 42 139)(53 180 73 160)(54 181 74 161)(55 182 75 162)(56 170 76 163)(57 171 77 164)(58 172 78 165)(59 173 66 166)(60 174 67 167)(61 175 68 168)(62 176 69 169)(63 177 70 157)(64 178 71 158)(65 179 72 159)(79 206 101 190)(80 207 102 191)(81 208 103 192)(82 196 104 193)(83 197 92 194)(84 198 93 195)(85 199 94 183)(86 200 95 184)(87 201 96 185)(88 202 97 186)(89 203 98 187)(90 204 99 188)(91 205 100 189)
(1 97 35 72)(2 98 36 73)(3 99 37 74)(4 100 38 75)(5 101 39 76)(6 102 27 77)(7 103 28 78)(8 104 29 66)(9 92 30 67)(10 93 31 68)(11 94 32 69)(12 95 33 70)(13 96 34 71)(14 83 46 60)(15 84 47 61)(16 85 48 62)(17 86 49 63)(18 87 50 64)(19 88 51 65)(20 89 52 53)(21 90 40 54)(22 91 41 55)(23 79 42 56)(24 80 43 57)(25 81 44 58)(26 82 45 59)(105 200 133 177)(106 201 134 178)(107 202 135 179)(108 203 136 180)(109 204 137 181)(110 205 138 182)(111 206 139 170)(112 207 140 171)(113 208 141 172)(114 196 142 173)(115 197 143 174)(116 198 131 175)(117 199 132 176)(118 193 150 166)(119 194 151 167)(120 195 152 168)(121 183 153 169)(122 184 154 157)(123 185 155 158)(124 186 156 159)(125 187 144 160)(126 188 145 161)(127 189 146 162)(128 190 147 163)(129 191 148 164)(130 192 149 165)
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 39)(10 38)(11 37)(12 36)(13 35)(14 42)(15 41)(16 40)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)(61 91)(62 90)(63 89)(64 88)(65 87)(66 102)(67 101)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 104)(78 103)(105 136)(106 135)(107 134)(108 133)(109 132)(110 131)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 148)(119 147)(120 146)(121 145)(122 144)(123 156)(124 155)(125 154)(126 153)(127 152)(128 151)(129 150)(130 149)(157 187)(158 186)(159 185)(160 184)(161 183)(162 195)(163 194)(164 193)(165 192)(166 191)(167 190)(168 189)(169 188)(170 197)(171 196)(172 208)(173 207)(174 206)(175 205)(176 204)(177 203)(178 202)(179 201)(180 200)(181 199)(182 198)
G:=sub<Sym(208)| (1,124,19,107)(2,125,20,108)(3,126,21,109)(4,127,22,110)(5,128,23,111)(6,129,24,112)(7,130,25,113)(8,118,26,114)(9,119,14,115)(10,120,15,116)(11,121,16,117)(12,122,17,105)(13,123,18,106)(27,148,43,140)(28,149,44,141)(29,150,45,142)(30,151,46,143)(31,152,47,131)(32,153,48,132)(33,154,49,133)(34,155,50,134)(35,156,51,135)(36,144,52,136)(37,145,40,137)(38,146,41,138)(39,147,42,139)(53,180,73,160)(54,181,74,161)(55,182,75,162)(56,170,76,163)(57,171,77,164)(58,172,78,165)(59,173,66,166)(60,174,67,167)(61,175,68,168)(62,176,69,169)(63,177,70,157)(64,178,71,158)(65,179,72,159)(79,206,101,190)(80,207,102,191)(81,208,103,192)(82,196,104,193)(83,197,92,194)(84,198,93,195)(85,199,94,183)(86,200,95,184)(87,201,96,185)(88,202,97,186)(89,203,98,187)(90,204,99,188)(91,205,100,189), (1,97,35,72)(2,98,36,73)(3,99,37,74)(4,100,38,75)(5,101,39,76)(6,102,27,77)(7,103,28,78)(8,104,29,66)(9,92,30,67)(10,93,31,68)(11,94,32,69)(12,95,33,70)(13,96,34,71)(14,83,46,60)(15,84,47,61)(16,85,48,62)(17,86,49,63)(18,87,50,64)(19,88,51,65)(20,89,52,53)(21,90,40,54)(22,91,41,55)(23,79,42,56)(24,80,43,57)(25,81,44,58)(26,82,45,59)(105,200,133,177)(106,201,134,178)(107,202,135,179)(108,203,136,180)(109,204,137,181)(110,205,138,182)(111,206,139,170)(112,207,140,171)(113,208,141,172)(114,196,142,173)(115,197,143,174)(116,198,131,175)(117,199,132,176)(118,193,150,166)(119,194,151,167)(120,195,152,168)(121,183,153,169)(122,184,154,157)(123,185,155,158)(124,186,156,159)(125,187,144,160)(126,188,145,161)(127,189,146,162)(128,190,147,163)(129,191,148,164)(130,192,149,165), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,39)(10,38)(11,37)(12,36)(13,35)(14,42)(15,41)(16,40)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,91)(62,90)(63,89)(64,88)(65,87)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,104)(78,103)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,148)(119,147)(120,146)(121,145)(122,144)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(157,187)(158,186)(159,185)(160,184)(161,183)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189)(169,188)(170,197)(171,196)(172,208)(173,207)(174,206)(175,205)(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)>;
G:=Group( (1,124,19,107)(2,125,20,108)(3,126,21,109)(4,127,22,110)(5,128,23,111)(6,129,24,112)(7,130,25,113)(8,118,26,114)(9,119,14,115)(10,120,15,116)(11,121,16,117)(12,122,17,105)(13,123,18,106)(27,148,43,140)(28,149,44,141)(29,150,45,142)(30,151,46,143)(31,152,47,131)(32,153,48,132)(33,154,49,133)(34,155,50,134)(35,156,51,135)(36,144,52,136)(37,145,40,137)(38,146,41,138)(39,147,42,139)(53,180,73,160)(54,181,74,161)(55,182,75,162)(56,170,76,163)(57,171,77,164)(58,172,78,165)(59,173,66,166)(60,174,67,167)(61,175,68,168)(62,176,69,169)(63,177,70,157)(64,178,71,158)(65,179,72,159)(79,206,101,190)(80,207,102,191)(81,208,103,192)(82,196,104,193)(83,197,92,194)(84,198,93,195)(85,199,94,183)(86,200,95,184)(87,201,96,185)(88,202,97,186)(89,203,98,187)(90,204,99,188)(91,205,100,189), (1,97,35,72)(2,98,36,73)(3,99,37,74)(4,100,38,75)(5,101,39,76)(6,102,27,77)(7,103,28,78)(8,104,29,66)(9,92,30,67)(10,93,31,68)(11,94,32,69)(12,95,33,70)(13,96,34,71)(14,83,46,60)(15,84,47,61)(16,85,48,62)(17,86,49,63)(18,87,50,64)(19,88,51,65)(20,89,52,53)(21,90,40,54)(22,91,41,55)(23,79,42,56)(24,80,43,57)(25,81,44,58)(26,82,45,59)(105,200,133,177)(106,201,134,178)(107,202,135,179)(108,203,136,180)(109,204,137,181)(110,205,138,182)(111,206,139,170)(112,207,140,171)(113,208,141,172)(114,196,142,173)(115,197,143,174)(116,198,131,175)(117,199,132,176)(118,193,150,166)(119,194,151,167)(120,195,152,168)(121,183,153,169)(122,184,154,157)(123,185,155,158)(124,186,156,159)(125,187,144,160)(126,188,145,161)(127,189,146,162)(128,190,147,163)(129,191,148,164)(130,192,149,165), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,39)(10,38)(11,37)(12,36)(13,35)(14,42)(15,41)(16,40)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,91)(62,90)(63,89)(64,88)(65,87)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,104)(78,103)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,148)(119,147)(120,146)(121,145)(122,144)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(157,187)(158,186)(159,185)(160,184)(161,183)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189)(169,188)(170,197)(171,196)(172,208)(173,207)(174,206)(175,205)(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198) );
G=PermutationGroup([[(1,124,19,107),(2,125,20,108),(3,126,21,109),(4,127,22,110),(5,128,23,111),(6,129,24,112),(7,130,25,113),(8,118,26,114),(9,119,14,115),(10,120,15,116),(11,121,16,117),(12,122,17,105),(13,123,18,106),(27,148,43,140),(28,149,44,141),(29,150,45,142),(30,151,46,143),(31,152,47,131),(32,153,48,132),(33,154,49,133),(34,155,50,134),(35,156,51,135),(36,144,52,136),(37,145,40,137),(38,146,41,138),(39,147,42,139),(53,180,73,160),(54,181,74,161),(55,182,75,162),(56,170,76,163),(57,171,77,164),(58,172,78,165),(59,173,66,166),(60,174,67,167),(61,175,68,168),(62,176,69,169),(63,177,70,157),(64,178,71,158),(65,179,72,159),(79,206,101,190),(80,207,102,191),(81,208,103,192),(82,196,104,193),(83,197,92,194),(84,198,93,195),(85,199,94,183),(86,200,95,184),(87,201,96,185),(88,202,97,186),(89,203,98,187),(90,204,99,188),(91,205,100,189)], [(1,97,35,72),(2,98,36,73),(3,99,37,74),(4,100,38,75),(5,101,39,76),(6,102,27,77),(7,103,28,78),(8,104,29,66),(9,92,30,67),(10,93,31,68),(11,94,32,69),(12,95,33,70),(13,96,34,71),(14,83,46,60),(15,84,47,61),(16,85,48,62),(17,86,49,63),(18,87,50,64),(19,88,51,65),(20,89,52,53),(21,90,40,54),(22,91,41,55),(23,79,42,56),(24,80,43,57),(25,81,44,58),(26,82,45,59),(105,200,133,177),(106,201,134,178),(107,202,135,179),(108,203,136,180),(109,204,137,181),(110,205,138,182),(111,206,139,170),(112,207,140,171),(113,208,141,172),(114,196,142,173),(115,197,143,174),(116,198,131,175),(117,199,132,176),(118,193,150,166),(119,194,151,167),(120,195,152,168),(121,183,153,169),(122,184,154,157),(123,185,155,158),(124,186,156,159),(125,187,144,160),(126,188,145,161),(127,189,146,162),(128,190,147,163),(129,191,148,164),(130,192,149,165)], [(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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,39),(10,38),(11,37),(12,36),(13,35),(14,42),(15,41),(16,40),(17,52),(18,51),(19,50),(20,49),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,80),(60,79),(61,91),(62,90),(63,89),(64,88),(65,87),(66,102),(67,101),(68,100),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93),(76,92),(77,104),(78,103),(105,136),(106,135),(107,134),(108,133),(109,132),(110,131),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,148),(119,147),(120,146),(121,145),(122,144),(123,156),(124,155),(125,154),(126,153),(127,152),(128,151),(129,150),(130,149),(157,187),(158,186),(159,185),(160,184),(161,183),(162,195),(163,194),(164,193),(165,192),(166,191),(167,190),(168,189),(169,188),(170,197),(171,196),(172,208),(173,207),(174,206),(175,205),(176,204),(177,203),(178,202),(179,201),(180,200),(181,199),(182,198)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4X | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| 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 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D13 | D26 | C4×D13 |
| kernel | C42×D13 | C4×Dic13 | C4×C52 | C2×C4×D13 | C4×D13 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 6 | 18 | 72 |
Matrix representation of C42×D13 ►in GL4(𝔽53) generated by
| 30 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 1 |
| 0 | 0 | 31 | 11 |
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 52 |
| 0 | 0 | 14 | 42 |
G:=sub<GL(4,GF(53))| [30,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,31,0,0,1,11],[52,0,0,0,0,1,0,0,0,0,11,14,0,0,52,42] >;
C42×D13 in GAP, Magma, Sage, TeX
C_4^2\times D_{13} % in TeX
G:=Group("C4^2xD13"); // GroupNames label
G:=SmallGroup(416,92);
// by ID
G=gap.SmallGroup(416,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,50,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations