metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D13, (C4×C52)⋊9C2, (C4×D13)⋊3C4, C52.46(C2×C4), D26.8(C2×C4), C4.22(C4×D13), (C2×C4).96D26, (C4×Dic13)⋊8C2, C26.3(C4○D4), C13⋊2(C42⋊C2), D26⋊C4.7C2, C26.D4⋊17C2, C26.16(C22×C4), (C2×C26).13C23, (C2×C52).73C22, C2.2(D52⋊5C2), Dic13.11(C2×C4), C22.10(C22×D13), (C2×Dic13).27C22, (C22×D13).17C22, C2.5(C2×C4×D13), (C2×C4×D13).10C2, SmallGroup(416,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D13
G = < a,b,c,d | a4=b4=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 464 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, C23, C13, C42, C42, C22⋊C4, C4⋊C4, C22×C4, D13, C26, C26, C42⋊C2, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×D13, C4×Dic13, C26.D4, D26⋊C4, C4×C52, C2×C4×D13, C42⋊D13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, D13, C42⋊C2, D26, C4×D13, C22×D13, C2×C4×D13, D52⋊5C2, C42⋊D13
►On 208 points
Generators in S208
(1 120 24 110)(2 121 25 111)(3 122 26 112)(4 123 14 113)(5 124 15 114)(6 125 16 115)(7 126 17 116)(8 127 18 117)(9 128 19 105)(10 129 20 106)(11 130 21 107)(12 118 22 108)(13 119 23 109)(27 153 43 132)(28 154 44 133)(29 155 45 134)(30 156 46 135)(31 144 47 136)(32 145 48 137)(33 146 49 138)(34 147 50 139)(35 148 51 140)(36 149 52 141)(37 150 40 142)(38 151 41 143)(39 152 42 131)(53 174 67 159)(54 175 68 160)(55 176 69 161)(56 177 70 162)(57 178 71 163)(58 179 72 164)(59 180 73 165)(60 181 74 166)(61 182 75 167)(62 170 76 168)(63 171 77 169)(64 172 78 157)(65 173 66 158)(79 203 94 183)(80 204 95 184)(81 205 96 185)(82 206 97 186)(83 207 98 187)(84 208 99 188)(85 196 100 189)(86 197 101 190)(87 198 102 191)(88 199 103 192)(89 200 104 193)(90 201 92 194)(91 202 93 195)
(1 86 38 57)(2 87 39 58)(3 88 27 59)(4 89 28 60)(5 90 29 61)(6 91 30 62)(7 79 31 63)(8 80 32 64)(9 81 33 65)(10 82 34 53)(11 83 35 54)(12 84 36 55)(13 85 37 56)(14 104 44 74)(15 92 45 75)(16 93 46 76)(17 94 47 77)(18 95 48 78)(19 96 49 66)(20 97 50 67)(21 98 51 68)(22 99 52 69)(23 100 40 70)(24 101 41 71)(25 102 42 72)(26 103 43 73)(105 185 138 158)(106 186 139 159)(107 187 140 160)(108 188 141 161)(109 189 142 162)(110 190 143 163)(111 191 131 164)(112 192 132 165)(113 193 133 166)(114 194 134 167)(115 195 135 168)(116 183 136 169)(117 184 137 157)(118 208 149 176)(119 196 150 177)(120 197 151 178)(121 198 152 179)(122 199 153 180)(123 200 154 181)(124 201 155 182)(125 202 156 170)(126 203 144 171)(127 204 145 172)(128 205 146 173)(129 206 147 174)(130 207 148 175)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 39)(13 38)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 52)(26 51)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(78 91)(105 134)(106 133)(107 132)(108 131)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 156)(128 155)(129 154)(130 153)(157 202)(158 201)(159 200)(160 199)(161 198)(162 197)(163 196)(164 208)(165 207)(166 206)(167 205)(168 204)(169 203)(170 184)(171 183)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)
G:=sub<Sym(208)| (1,120,24,110)(2,121,25,111)(3,122,26,112)(4,123,14,113)(5,124,15,114)(6,125,16,115)(7,126,17,116)(8,127,18,117)(9,128,19,105)(10,129,20,106)(11,130,21,107)(12,118,22,108)(13,119,23,109)(27,153,43,132)(28,154,44,133)(29,155,45,134)(30,156,46,135)(31,144,47,136)(32,145,48,137)(33,146,49,138)(34,147,50,139)(35,148,51,140)(36,149,52,141)(37,150,40,142)(38,151,41,143)(39,152,42,131)(53,174,67,159)(54,175,68,160)(55,176,69,161)(56,177,70,162)(57,178,71,163)(58,179,72,164)(59,180,73,165)(60,181,74,166)(61,182,75,167)(62,170,76,168)(63,171,77,169)(64,172,78,157)(65,173,66,158)(79,203,94,183)(80,204,95,184)(81,205,96,185)(82,206,97,186)(83,207,98,187)(84,208,99,188)(85,196,100,189)(86,197,101,190)(87,198,102,191)(88,199,103,192)(89,200,104,193)(90,201,92,194)(91,202,93,195), (1,86,38,57)(2,87,39,58)(3,88,27,59)(4,89,28,60)(5,90,29,61)(6,91,30,62)(7,79,31,63)(8,80,32,64)(9,81,33,65)(10,82,34,53)(11,83,35,54)(12,84,36,55)(13,85,37,56)(14,104,44,74)(15,92,45,75)(16,93,46,76)(17,94,47,77)(18,95,48,78)(19,96,49,66)(20,97,50,67)(21,98,51,68)(22,99,52,69)(23,100,40,70)(24,101,41,71)(25,102,42,72)(26,103,43,73)(105,185,138,158)(106,186,139,159)(107,187,140,160)(108,188,141,161)(109,189,142,162)(110,190,143,163)(111,191,131,164)(112,192,132,165)(113,193,133,166)(114,194,134,167)(115,195,135,168)(116,183,136,169)(117,184,137,157)(118,208,149,176)(119,196,150,177)(120,197,151,178)(121,198,152,179)(122,199,153,180)(123,200,154,181)(124,201,155,182)(125,202,156,170)(126,203,144,171)(127,204,145,172)(128,205,146,173)(129,206,147,174)(130,207,148,175), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,39)(13,38)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,52)(26,51)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(78,91)(105,134)(106,133)(107,132)(108,131)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,156)(128,155)(129,154)(130,153)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,184)(171,183)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)>;
G:=Group( (1,120,24,110)(2,121,25,111)(3,122,26,112)(4,123,14,113)(5,124,15,114)(6,125,16,115)(7,126,17,116)(8,127,18,117)(9,128,19,105)(10,129,20,106)(11,130,21,107)(12,118,22,108)(13,119,23,109)(27,153,43,132)(28,154,44,133)(29,155,45,134)(30,156,46,135)(31,144,47,136)(32,145,48,137)(33,146,49,138)(34,147,50,139)(35,148,51,140)(36,149,52,141)(37,150,40,142)(38,151,41,143)(39,152,42,131)(53,174,67,159)(54,175,68,160)(55,176,69,161)(56,177,70,162)(57,178,71,163)(58,179,72,164)(59,180,73,165)(60,181,74,166)(61,182,75,167)(62,170,76,168)(63,171,77,169)(64,172,78,157)(65,173,66,158)(79,203,94,183)(80,204,95,184)(81,205,96,185)(82,206,97,186)(83,207,98,187)(84,208,99,188)(85,196,100,189)(86,197,101,190)(87,198,102,191)(88,199,103,192)(89,200,104,193)(90,201,92,194)(91,202,93,195), (1,86,38,57)(2,87,39,58)(3,88,27,59)(4,89,28,60)(5,90,29,61)(6,91,30,62)(7,79,31,63)(8,80,32,64)(9,81,33,65)(10,82,34,53)(11,83,35,54)(12,84,36,55)(13,85,37,56)(14,104,44,74)(15,92,45,75)(16,93,46,76)(17,94,47,77)(18,95,48,78)(19,96,49,66)(20,97,50,67)(21,98,51,68)(22,99,52,69)(23,100,40,70)(24,101,41,71)(25,102,42,72)(26,103,43,73)(105,185,138,158)(106,186,139,159)(107,187,140,160)(108,188,141,161)(109,189,142,162)(110,190,143,163)(111,191,131,164)(112,192,132,165)(113,193,133,166)(114,194,134,167)(115,195,135,168)(116,183,136,169)(117,184,137,157)(118,208,149,176)(119,196,150,177)(120,197,151,178)(121,198,152,179)(122,199,153,180)(123,200,154,181)(124,201,155,182)(125,202,156,170)(126,203,144,171)(127,204,145,172)(128,205,146,173)(129,206,147,174)(130,207,148,175), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,39)(13,38)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,52)(26,51)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(78,91)(105,134)(106,133)(107,132)(108,131)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,156)(128,155)(129,154)(130,153)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,184)(171,183)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185) );
G=PermutationGroup([[(1,120,24,110),(2,121,25,111),(3,122,26,112),(4,123,14,113),(5,124,15,114),(6,125,16,115),(7,126,17,116),(8,127,18,117),(9,128,19,105),(10,129,20,106),(11,130,21,107),(12,118,22,108),(13,119,23,109),(27,153,43,132),(28,154,44,133),(29,155,45,134),(30,156,46,135),(31,144,47,136),(32,145,48,137),(33,146,49,138),(34,147,50,139),(35,148,51,140),(36,149,52,141),(37,150,40,142),(38,151,41,143),(39,152,42,131),(53,174,67,159),(54,175,68,160),(55,176,69,161),(56,177,70,162),(57,178,71,163),(58,179,72,164),(59,180,73,165),(60,181,74,166),(61,182,75,167),(62,170,76,168),(63,171,77,169),(64,172,78,157),(65,173,66,158),(79,203,94,183),(80,204,95,184),(81,205,96,185),(82,206,97,186),(83,207,98,187),(84,208,99,188),(85,196,100,189),(86,197,101,190),(87,198,102,191),(88,199,103,192),(89,200,104,193),(90,201,92,194),(91,202,93,195)], [(1,86,38,57),(2,87,39,58),(3,88,27,59),(4,89,28,60),(5,90,29,61),(6,91,30,62),(7,79,31,63),(8,80,32,64),(9,81,33,65),(10,82,34,53),(11,83,35,54),(12,84,36,55),(13,85,37,56),(14,104,44,74),(15,92,45,75),(16,93,46,76),(17,94,47,77),(18,95,48,78),(19,96,49,66),(20,97,50,67),(21,98,51,68),(22,99,52,69),(23,100,40,70),(24,101,41,71),(25,102,42,72),(26,103,43,73),(105,185,138,158),(106,186,139,159),(107,187,140,160),(108,188,141,161),(109,189,142,162),(110,190,143,163),(111,191,131,164),(112,192,132,165),(113,193,133,166),(114,194,134,167),(115,195,135,168),(116,183,136,169),(117,184,137,157),(118,208,149,176),(119,196,150,177),(120,197,151,178),(121,198,152,179),(122,199,153,180),(123,200,154,181),(124,201,155,182),(125,202,156,170),(126,203,144,171),(127,204,145,172),(128,205,146,173),(129,206,147,174),(130,207,148,175)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,39),(13,38),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,52),(26,51),(53,104),(54,103),(55,102),(56,101),(57,100),(58,99),(59,98),(60,97),(61,96),(62,95),(63,94),(64,93),(65,92),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(78,91),(105,134),(106,133),(107,132),(108,131),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,135),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,156),(128,155),(129,154),(130,153),(157,202),(158,201),(159,200),(160,199),(161,198),(162,197),(163,196),(164,208),(165,207),(166,206),(167,205),(168,204),(169,203),(170,184),(171,183),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187),(181,186),(182,185)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | D13 | D26 | C4×D13 | D52⋊5C2 |
| kernel | C42⋊D13 | C4×Dic13 | C26.D4 | D26⋊C4 | C4×C52 | C2×C4×D13 | C4×D13 | C26 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 4 | 6 | 18 | 24 | 48 |
Matrix representation of C42⋊D13 ►in GL5(𝔽53)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 0 | 30 |
| 23 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 1 | 0 | 0 |
| 0 | 25 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 |
| 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 52 |
G:=sub<GL(5,GF(53))| [1,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,30,0,0,0,0,0,30],[23,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,52,25,0,0,0,1,27,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,52,25,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,52] >;
C42⋊D13 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_{13} % in TeX
G:=Group("C4^2:D13"); // GroupNames label
G:=SmallGroup(416,93);
// by ID
G=gap.SmallGroup(416,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,50,13829]);
// 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,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations