metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊26D4, C5⋊7(C8⋊9D4), C22⋊C8⋊15D5, C40⋊8C4⋊14C2, C4.199(D4×D5), C10.59(C4×D4), C20.358(C2×D4), (C2×C8).164D10, (C2×C10)⋊6M4(2), D10⋊1C8⋊21C2, C23.24(C4×D5), C10.49(C8○D4), C22⋊1(C8⋊D5), C20.8Q8⋊21C2, C23.D5.14C4, D10⋊C4.20C4, C20.300(C4○D4), (C2×C40).174C22, (C2×C20).825C23, C10.D4.20C4, (C22×C4).305D10, C10.40(C2×M4(2)), C4.126(D4⋊2D5), C2.13(Dic5⋊4D4), C2.11(D20.2C4), (C22×C20).339C22, (C4×Dic5).203C22, (C2×C4).64(C4×D5), (C2×C8⋊D5)⋊14C2, (C5×C22⋊C8)⋊19C2, (C4×C5⋊D4).14C2, C2.10(C2×C8⋊D5), (C2×C5⋊D4).16C4, C22.107(C2×C4×D5), (C2×C20).328(C2×C4), (C22×C5⋊2C8)⋊17C2, (C2×C4×D5).231C22, (C2×Dic5).20(C2×C4), (C22×D5).19(C2×C4), (C2×C4).767(C22×D5), (C22×C10).111(C2×C4), (C2×C10).181(C22×C4), (C2×C5⋊2C8).310C22, SmallGroup(320,357)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊2C8⋊26D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >
Subgroups: 398 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊9D4, C8⋊D5, C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C40⋊8C4, D10⋊1C8, C5×C22⋊C8, C2×C8⋊D5, C22×C5⋊2C8, C4×C5⋊D4, C5⋊2C8⋊26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C22×D5, C8⋊9D4, C8⋊D5, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, C2×C8⋊D5, D20.2C4, C5⋊2C8⋊26D4
►On 160 points
Generators in S160
(1 139 41 36 147)(2 148 37 42 140)(3 141 43 38 149)(4 150 39 44 142)(5 143 45 40 151)(6 152 33 46 144)(7 137 47 34 145)(8 146 35 48 138)(9 63 126 94 118)(10 119 95 127 64)(11 57 128 96 120)(12 113 89 121 58)(13 59 122 90 114)(14 115 91 123 60)(15 61 124 92 116)(16 117 93 125 62)(17 134 84 76 53)(18 54 77 85 135)(19 136 86 78 55)(20 56 79 87 129)(21 130 88 80 49)(22 50 73 81 131)(23 132 82 74 51)(24 52 75 83 133)(25 160 109 98 72)(26 65 99 110 153)(27 154 111 100 66)(28 67 101 112 155)(29 156 105 102 68)(30 69 103 106 157)(31 158 107 104 70)(32 71 97 108 159)
(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)
(1 52 16 111)(2 49 9 108)(3 54 10 105)(4 51 11 110)(5 56 12 107)(6 53 13 112)(7 50 14 109)(8 55 15 106)(17 114 155 144)(18 119 156 141)(19 116 157 138)(20 113 158 143)(21 118 159 140)(22 115 160 137)(23 120 153 142)(24 117 154 139)(25 47 131 91)(26 44 132 96)(27 41 133 93)(28 46 134 90)(29 43 135 95)(30 48 136 92)(31 45 129 89)(32 42 130 94)(33 84 122 67)(34 81 123 72)(35 86 124 69)(36 83 125 66)(37 88 126 71)(38 85 127 68)(39 82 128 65)(40 87 121 70)(57 99 150 74)(58 104 151 79)(59 101 152 76)(60 98 145 73)(61 103 146 78)(62 100 147 75)(63 97 148 80)(64 102 149 77)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 112)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 129)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)(57 157)(58 158)(59 159)(60 160)(61 153)(62 154)(63 155)(64 156)(65 92)(66 93)(67 94)(68 95)(69 96)(70 89)(71 90)(72 91)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)
G:=sub<Sym(160)| (1,139,41,36,147)(2,148,37,42,140)(3,141,43,38,149)(4,150,39,44,142)(5,143,45,40,151)(6,152,33,46,144)(7,137,47,34,145)(8,146,35,48,138)(9,63,126,94,118)(10,119,95,127,64)(11,57,128,96,120)(12,113,89,121,58)(13,59,122,90,114)(14,115,91,123,60)(15,61,124,92,116)(16,117,93,125,62)(17,134,84,76,53)(18,54,77,85,135)(19,136,86,78,55)(20,56,79,87,129)(21,130,88,80,49)(22,50,73,81,131)(23,132,82,74,51)(24,52,75,83,133)(25,160,109,98,72)(26,65,99,110,153)(27,154,111,100,66)(28,67,101,112,155)(29,156,105,102,68)(30,69,103,106,157)(31,158,107,104,70)(32,71,97,108,159), (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), (1,52,16,111)(2,49,9,108)(3,54,10,105)(4,51,11,110)(5,56,12,107)(6,53,13,112)(7,50,14,109)(8,55,15,106)(17,114,155,144)(18,119,156,141)(19,116,157,138)(20,113,158,143)(21,118,159,140)(22,115,160,137)(23,120,153,142)(24,117,154,139)(25,47,131,91)(26,44,132,96)(27,41,133,93)(28,46,134,90)(29,43,135,95)(30,48,136,92)(31,45,129,89)(32,42,130,94)(33,84,122,67)(34,81,123,72)(35,86,124,69)(36,83,125,66)(37,88,126,71)(38,85,127,68)(39,82,128,65)(40,87,121,70)(57,99,150,74)(58,104,151,79)(59,101,152,76)(60,98,145,73)(61,103,146,78)(62,100,147,75)(63,97,148,80)(64,102,149,77), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,129)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,92)(66,93)(67,94)(68,95)(69,96)(70,89)(71,90)(72,91)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113)>;
G:=Group( (1,139,41,36,147)(2,148,37,42,140)(3,141,43,38,149)(4,150,39,44,142)(5,143,45,40,151)(6,152,33,46,144)(7,137,47,34,145)(8,146,35,48,138)(9,63,126,94,118)(10,119,95,127,64)(11,57,128,96,120)(12,113,89,121,58)(13,59,122,90,114)(14,115,91,123,60)(15,61,124,92,116)(16,117,93,125,62)(17,134,84,76,53)(18,54,77,85,135)(19,136,86,78,55)(20,56,79,87,129)(21,130,88,80,49)(22,50,73,81,131)(23,132,82,74,51)(24,52,75,83,133)(25,160,109,98,72)(26,65,99,110,153)(27,154,111,100,66)(28,67,101,112,155)(29,156,105,102,68)(30,69,103,106,157)(31,158,107,104,70)(32,71,97,108,159), (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), (1,52,16,111)(2,49,9,108)(3,54,10,105)(4,51,11,110)(5,56,12,107)(6,53,13,112)(7,50,14,109)(8,55,15,106)(17,114,155,144)(18,119,156,141)(19,116,157,138)(20,113,158,143)(21,118,159,140)(22,115,160,137)(23,120,153,142)(24,117,154,139)(25,47,131,91)(26,44,132,96)(27,41,133,93)(28,46,134,90)(29,43,135,95)(30,48,136,92)(31,45,129,89)(32,42,130,94)(33,84,122,67)(34,81,123,72)(35,86,124,69)(36,83,125,66)(37,88,126,71)(38,85,127,68)(39,82,128,65)(40,87,121,70)(57,99,150,74)(58,104,151,79)(59,101,152,76)(60,98,145,73)(61,103,146,78)(62,100,147,75)(63,97,148,80)(64,102,149,77), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,129)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,92)(66,93)(67,94)(68,95)(69,96)(70,89)(71,90)(72,91)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113) );
G=PermutationGroup([[(1,139,41,36,147),(2,148,37,42,140),(3,141,43,38,149),(4,150,39,44,142),(5,143,45,40,151),(6,152,33,46,144),(7,137,47,34,145),(8,146,35,48,138),(9,63,126,94,118),(10,119,95,127,64),(11,57,128,96,120),(12,113,89,121,58),(13,59,122,90,114),(14,115,91,123,60),(15,61,124,92,116),(16,117,93,125,62),(17,134,84,76,53),(18,54,77,85,135),(19,136,86,78,55),(20,56,79,87,129),(21,130,88,80,49),(22,50,73,81,131),(23,132,82,74,51),(24,52,75,83,133),(25,160,109,98,72),(26,65,99,110,153),(27,154,111,100,66),(28,67,101,112,155),(29,156,105,102,68),(30,69,103,106,157),(31,158,107,104,70),(32,71,97,108,159)], [(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)], [(1,52,16,111),(2,49,9,108),(3,54,10,105),(4,51,11,110),(5,56,12,107),(6,53,13,112),(7,50,14,109),(8,55,15,106),(17,114,155,144),(18,119,156,141),(19,116,157,138),(20,113,158,143),(21,118,159,140),(22,115,160,137),(23,120,153,142),(24,117,154,139),(25,47,131,91),(26,44,132,96),(27,41,133,93),(28,46,134,90),(29,43,135,95),(30,48,136,92),(31,45,129,89),(32,42,130,94),(33,84,122,67),(34,81,123,72),(35,86,124,69),(36,83,125,66),(37,88,126,71),(38,85,127,68),(39,82,128,65),(40,87,121,70),(57,99,150,74),(58,104,151,79),(59,101,152,76),(60,98,145,73),(61,103,146,78),(62,100,147,75),(63,97,148,80),(64,102,149,77)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,112),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,148),(18,149),(19,150),(20,151),(21,152),(22,145),(23,146),(24,147),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,121),(32,122),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,129),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(57,157),(58,158),(59,159),(60,160),(61,153),(62,154),(63,155),(64,156),(65,92),(66,93),(67,94),(68,95),(69,96),(70,89),(71,90),(72,91),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,113)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | C8⋊D5 | D4×D5 | D4⋊2D5 | D20.2C4 |
| kernel | C5⋊2C8⋊26D4 | C20.8Q8 | C40⋊8C4 | D10⋊1C8 | C5×C22⋊C8 | C2×C8⋊D5 | C22×C5⋊2C8 | C4×C5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C5⋊2C8 | C22⋊C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C5⋊2C8⋊26D4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 36 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 22 | 10 |
| 0 | 0 | 4 | 19 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 6 | 35 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[40,0,0,0,0,40,0,0,0,0,22,4,0,0,10,19],[0,1,0,0,40,0,0,0,0,0,6,6,0,0,1,35],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
C5⋊2C8⋊26D4 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_8\rtimes_{26}D_4 % in TeX
G:=Group("C5:2C8:26D4"); // GroupNames label
G:=SmallGroup(320,357);
// by ID
G=gap.SmallGroup(320,357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;
// generators/relations