metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D26, C52.57D4, Q8.8D26, C52.18C23, D52.12C22, Dic26.11C22, D4⋊D13⋊7C2, C4○D4⋊2D13, C13⋊5(C4○D8), Q8⋊D13⋊7C2, (C2×C26).9D4, D4.D13⋊7C2, C13⋊Q16⋊7C2, C26.60(C2×D4), (C2×C4).59D26, D52⋊5C2⋊4C2, C4.32(C13⋊D4), (C2×C52).43C22, (D4×C13).8C22, C4.18(C22×D13), (Q8×C13).8C22, C13⋊2C8.10C22, C22.1(C13⋊D4), (C2×C13⋊2C8)⋊8C2, (C13×C4○D4)⋊2C2, C2.24(C2×C13⋊D4), SmallGroup(416,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.C23
G = < a,b,c,d | a52=b2=d2=1, c2=a26, bab=a-1, ac=ca, dad=a27, bc=cb, dbd=a13b, cd=dc >
Subgroups: 368 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C13, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, D13, C26, C26 [×2], C4○D8, Dic13, C52 [×2], C52, D26, C2×C26, C2×C26, C13⋊2C8 [×2], Dic26, C4×D13, D52, C13⋊D4, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C2×C13⋊2C8, D4⋊D13, D4.D13, Q8⋊D13, C13⋊Q16, D52⋊5C2, C13×C4○D4, C52.C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, C52.C23
►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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 178)(54 177)(55 176)(56 175)(57 174)(58 173)(59 172)(60 171)(61 170)(62 169)(63 168)(64 167)(65 166)(66 165)(67 164)(68 163)(69 162)(70 161)(71 160)(72 159)(73 158)(74 157)(75 208)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 195)(89 194)(90 193)(91 192)(92 191)(93 190)(94 189)(95 188)(96 187)(97 186)(98 185)(99 184)(100 183)(101 182)(102 181)(103 180)(104 179)
(1 40 27 14)(2 41 28 15)(3 42 29 16)(4 43 30 17)(5 44 31 18)(6 45 32 19)(7 46 33 20)(8 47 34 21)(9 48 35 22)(10 49 36 23)(11 50 37 24)(12 51 38 25)(13 52 39 26)(53 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)(105 118 131 144)(106 119 132 145)(107 120 133 146)(108 121 134 147)(109 122 135 148)(110 123 136 149)(111 124 137 150)(112 125 138 151)(113 126 139 152)(114 127 140 153)(115 128 141 154)(116 129 142 155)(117 130 143 156)(157 196 183 170)(158 197 184 171)(159 198 185 172)(160 199 186 173)(161 200 187 174)(162 201 188 175)(163 202 189 176)(164 203 190 177)(165 204 191 178)(166 205 192 179)(167 206 193 180)(168 207 194 181)(169 208 195 182)
(1 68)(2 95)(3 70)(4 97)(5 72)(6 99)(7 74)(8 101)(9 76)(10 103)(11 78)(12 53)(13 80)(14 55)(15 82)(16 57)(17 84)(18 59)(19 86)(20 61)(21 88)(22 63)(23 90)(24 65)(25 92)(26 67)(27 94)(28 69)(29 96)(30 71)(31 98)(32 73)(33 100)(34 75)(35 102)(36 77)(37 104)(38 79)(39 54)(40 81)(41 56)(42 83)(43 58)(44 85)(45 60)(46 87)(47 62)(48 89)(49 64)(50 91)(51 66)(52 93)(105 172)(106 199)(107 174)(108 201)(109 176)(110 203)(111 178)(112 205)(113 180)(114 207)(115 182)(116 157)(117 184)(118 159)(119 186)(120 161)(121 188)(122 163)(123 190)(124 165)(125 192)(126 167)(127 194)(128 169)(129 196)(130 171)(131 198)(132 173)(133 200)(134 175)(135 202)(136 177)(137 204)(138 179)(139 206)(140 181)(141 208)(142 183)(143 158)(144 185)(145 160)(146 187)(147 162)(148 189)(149 164)(150 191)(151 166)(152 193)(153 168)(154 195)(155 170)(156 197)
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,182)(102,181)(103,180)(104,179), (1,40,27,14)(2,41,28,15)(3,42,29,16)(4,43,30,17)(5,44,31,18)(6,45,32,19)(7,46,33,20)(8,47,34,21)(9,48,35,22)(10,49,36,23)(11,50,37,24)(12,51,38,25)(13,52,39,26)(53,66,79,92)(54,67,80,93)(55,68,81,94)(56,69,82,95)(57,70,83,96)(58,71,84,97)(59,72,85,98)(60,73,86,99)(61,74,87,100)(62,75,88,101)(63,76,89,102)(64,77,90,103)(65,78,91,104)(105,118,131,144)(106,119,132,145)(107,120,133,146)(108,121,134,147)(109,122,135,148)(110,123,136,149)(111,124,137,150)(112,125,138,151)(113,126,139,152)(114,127,140,153)(115,128,141,154)(116,129,142,155)(117,130,143,156)(157,196,183,170)(158,197,184,171)(159,198,185,172)(160,199,186,173)(161,200,187,174)(162,201,188,175)(163,202,189,176)(164,203,190,177)(165,204,191,178)(166,205,192,179)(167,206,193,180)(168,207,194,181)(169,208,195,182), (1,68)(2,95)(3,70)(4,97)(5,72)(6,99)(7,74)(8,101)(9,76)(10,103)(11,78)(12,53)(13,80)(14,55)(15,82)(16,57)(17,84)(18,59)(19,86)(20,61)(21,88)(22,63)(23,90)(24,65)(25,92)(26,67)(27,94)(28,69)(29,96)(30,71)(31,98)(32,73)(33,100)(34,75)(35,102)(36,77)(37,104)(38,79)(39,54)(40,81)(41,56)(42,83)(43,58)(44,85)(45,60)(46,87)(47,62)(48,89)(49,64)(50,91)(51,66)(52,93)(105,172)(106,199)(107,174)(108,201)(109,176)(110,203)(111,178)(112,205)(113,180)(114,207)(115,182)(116,157)(117,184)(118,159)(119,186)(120,161)(121,188)(122,163)(123,190)(124,165)(125,192)(126,167)(127,194)(128,169)(129,196)(130,171)(131,198)(132,173)(133,200)(134,175)(135,202)(136,177)(137,204)(138,179)(139,206)(140,181)(141,208)(142,183)(143,158)(144,185)(145,160)(146,187)(147,162)(148,189)(149,164)(150,191)(151,166)(152,193)(153,168)(154,195)(155,170)(156,197)>;
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,182)(102,181)(103,180)(104,179), (1,40,27,14)(2,41,28,15)(3,42,29,16)(4,43,30,17)(5,44,31,18)(6,45,32,19)(7,46,33,20)(8,47,34,21)(9,48,35,22)(10,49,36,23)(11,50,37,24)(12,51,38,25)(13,52,39,26)(53,66,79,92)(54,67,80,93)(55,68,81,94)(56,69,82,95)(57,70,83,96)(58,71,84,97)(59,72,85,98)(60,73,86,99)(61,74,87,100)(62,75,88,101)(63,76,89,102)(64,77,90,103)(65,78,91,104)(105,118,131,144)(106,119,132,145)(107,120,133,146)(108,121,134,147)(109,122,135,148)(110,123,136,149)(111,124,137,150)(112,125,138,151)(113,126,139,152)(114,127,140,153)(115,128,141,154)(116,129,142,155)(117,130,143,156)(157,196,183,170)(158,197,184,171)(159,198,185,172)(160,199,186,173)(161,200,187,174)(162,201,188,175)(163,202,189,176)(164,203,190,177)(165,204,191,178)(166,205,192,179)(167,206,193,180)(168,207,194,181)(169,208,195,182), (1,68)(2,95)(3,70)(4,97)(5,72)(6,99)(7,74)(8,101)(9,76)(10,103)(11,78)(12,53)(13,80)(14,55)(15,82)(16,57)(17,84)(18,59)(19,86)(20,61)(21,88)(22,63)(23,90)(24,65)(25,92)(26,67)(27,94)(28,69)(29,96)(30,71)(31,98)(32,73)(33,100)(34,75)(35,102)(36,77)(37,104)(38,79)(39,54)(40,81)(41,56)(42,83)(43,58)(44,85)(45,60)(46,87)(47,62)(48,89)(49,64)(50,91)(51,66)(52,93)(105,172)(106,199)(107,174)(108,201)(109,176)(110,203)(111,178)(112,205)(113,180)(114,207)(115,182)(116,157)(117,184)(118,159)(119,186)(120,161)(121,188)(122,163)(123,190)(124,165)(125,192)(126,167)(127,194)(128,169)(129,196)(130,171)(131,198)(132,173)(133,200)(134,175)(135,202)(136,177)(137,204)(138,179)(139,206)(140,181)(141,208)(142,183)(143,158)(144,185)(145,160)(146,187)(147,162)(148,189)(149,164)(150,191)(151,166)(152,193)(153,168)(154,195)(155,170)(156,197) );
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)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,120),(17,119),(18,118),(19,117),(20,116),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,109),(28,108),(29,107),(30,106),(31,105),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,140),(49,139),(50,138),(51,137),(52,136),(53,178),(54,177),(55,176),(56,175),(57,174),(58,173),(59,172),(60,171),(61,170),(62,169),(63,168),(64,167),(65,166),(66,165),(67,164),(68,163),(69,162),(70,161),(71,160),(72,159),(73,158),(74,157),(75,208),(76,207),(77,206),(78,205),(79,204),(80,203),(81,202),(82,201),(83,200),(84,199),(85,198),(86,197),(87,196),(88,195),(89,194),(90,193),(91,192),(92,191),(93,190),(94,189),(95,188),(96,187),(97,186),(98,185),(99,184),(100,183),(101,182),(102,181),(103,180),(104,179)], [(1,40,27,14),(2,41,28,15),(3,42,29,16),(4,43,30,17),(5,44,31,18),(6,45,32,19),(7,46,33,20),(8,47,34,21),(9,48,35,22),(10,49,36,23),(11,50,37,24),(12,51,38,25),(13,52,39,26),(53,66,79,92),(54,67,80,93),(55,68,81,94),(56,69,82,95),(57,70,83,96),(58,71,84,97),(59,72,85,98),(60,73,86,99),(61,74,87,100),(62,75,88,101),(63,76,89,102),(64,77,90,103),(65,78,91,104),(105,118,131,144),(106,119,132,145),(107,120,133,146),(108,121,134,147),(109,122,135,148),(110,123,136,149),(111,124,137,150),(112,125,138,151),(113,126,139,152),(114,127,140,153),(115,128,141,154),(116,129,142,155),(117,130,143,156),(157,196,183,170),(158,197,184,171),(159,198,185,172),(160,199,186,173),(161,200,187,174),(162,201,188,175),(163,202,189,176),(164,203,190,177),(165,204,191,178),(166,205,192,179),(167,206,193,180),(168,207,194,181),(169,208,195,182)], [(1,68),(2,95),(3,70),(4,97),(5,72),(6,99),(7,74),(8,101),(9,76),(10,103),(11,78),(12,53),(13,80),(14,55),(15,82),(16,57),(17,84),(18,59),(19,86),(20,61),(21,88),(22,63),(23,90),(24,65),(25,92),(26,67),(27,94),(28,69),(29,96),(30,71),(31,98),(32,73),(33,100),(34,75),(35,102),(36,77),(37,104),(38,79),(39,54),(40,81),(41,56),(42,83),(43,58),(44,85),(45,60),(46,87),(47,62),(48,89),(49,64),(50,91),(51,66),(52,93),(105,172),(106,199),(107,174),(108,201),(109,176),(110,203),(111,178),(112,205),(113,180),(114,207),(115,182),(116,157),(117,184),(118,159),(119,186),(120,161),(121,188),(122,163),(123,190),(124,165),(125,192),(126,167),(127,194),(128,169),(129,196),(130,171),(131,198),(132,173),(133,200),(134,175),(135,202),(136,177),(137,204),(138,179),(139,206),(140,181),(141,208),(142,183),(143,158),(144,185),(145,160),(146,187),(147,162),(148,189),(149,164),(150,191),(151,166),(152,193),(153,168),(154,195),(155,170),(156,197)])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 52 | 1 | 1 | 2 | 4 | 52 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | C4○D8 | D26 | D26 | D26 | C13⋊D4 | C13⋊D4 | C52.C23 |
| kernel | C52.C23 | C2×C13⋊2C8 | D4⋊D13 | D4.D13 | Q8⋊D13 | C13⋊Q16 | D52⋊5C2 | C13×C4○D4 | C52 | C2×C26 | C4○D4 | C13 | C2×C4 | D4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 4 | 6 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C52.C23 ►in GL4(𝔽313) generated by
| 151 | 288 | 0 | 0 |
| 134 | 239 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 18 | 288 |
| 146 | 165 | 0 | 0 |
| 70 | 167 | 0 | 0 |
| 0 | 0 | 5 | 160 |
| 0 | 0 | 172 | 308 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 0 | 288 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 312 | 281 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(313))| [151,134,0,0,288,239,0,0,0,0,25,18,0,0,0,288],[146,70,0,0,165,167,0,0,0,0,5,172,0,0,160,308],[1,0,0,0,0,1,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,281,1] >;
C52.C23 in GAP, Magma, Sage, TeX
C_{52}.C_2^3 % in TeX
G:=Group("C52.C2^3"); // GroupNames label
G:=SmallGroup(416,171);
// by ID
G=gap.SmallGroup(416,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,579,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^52=b^2=d^2=1,c^2=a^26,b*a*b=a^-1,a*c=c*a,d*a*d=a^27,b*c=c*b,d*b*d=a^13*b,c*d=d*c>;
// generators/relations