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 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8 [×2], C5⋊2C8, C40 [×2], C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8⋊9D4, C8⋊D5 [×2], C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], 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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C8⋊9D4, C8⋊D5 [×2], C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, C2×C8⋊D5, D20.2C4, C5⋊2C8⋊26D4
►On 160 points
Generators in S160
(1 106 33 113 147)(2 148 114 34 107)(3 108 35 115 149)(4 150 116 36 109)(5 110 37 117 151)(6 152 118 38 111)(7 112 39 119 145)(8 146 120 40 105)(9 41 63 126 94)(10 95 127 64 42)(11 43 57 128 96)(12 89 121 58 44)(13 45 59 122 90)(14 91 123 60 46)(15 47 61 124 92)(16 93 125 62 48)(17 101 132 155 65)(18 66 156 133 102)(19 103 134 157 67)(20 68 158 135 104)(21 97 136 159 69)(22 70 160 129 98)(23 99 130 153 71)(24 72 154 131 100)(25 73 142 86 49)(26 50 87 143 74)(27 75 144 88 51)(28 52 81 137 76)(29 77 138 82 53)(30 54 83 139 78)(31 79 140 84 55)(32 56 85 141 80)
(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 27 48 131)(2 32 41 136)(3 29 42 133)(4 26 43 130)(5 31 44 135)(6 28 45 132)(7 25 46 129)(8 30 47 134)(9 159 107 56)(10 156 108 53)(11 153 109 50)(12 158 110 55)(13 155 111 52)(14 160 112 49)(15 157 105 54)(16 154 106 51)(17 118 137 122)(18 115 138 127)(19 120 139 124)(20 117 140 121)(21 114 141 126)(22 119 142 123)(23 116 143 128)(24 113 144 125)(33 88 93 72)(34 85 94 69)(35 82 95 66)(36 87 96 71)(37 84 89 68)(38 81 90 65)(39 86 91 70)(40 83 92 67)(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 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 101)(10 102)(11 103)(12 104)(13 97)(14 98)(15 99)(16 100)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(33 144)(34 137)(35 138)(36 139)(37 140)(38 141)(39 142)(40 143)(41 132)(42 133)(43 134)(44 135)(45 136)(46 129)(47 130)(48 131)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 157)(58 158)(59 159)(60 160)(61 153)(62 154)(63 155)(64 156)(65 126)(66 127)(67 128)(68 121)(69 122)(70 123)(71 124)(72 125)(73 112)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 113)
G:=sub<Sym(160)| (1,106,33,113,147)(2,148,114,34,107)(3,108,35,115,149)(4,150,116,36,109)(5,110,37,117,151)(6,152,118,38,111)(7,112,39,119,145)(8,146,120,40,105)(9,41,63,126,94)(10,95,127,64,42)(11,43,57,128,96)(12,89,121,58,44)(13,45,59,122,90)(14,91,123,60,46)(15,47,61,124,92)(16,93,125,62,48)(17,101,132,155,65)(18,66,156,133,102)(19,103,134,157,67)(20,68,158,135,104)(21,97,136,159,69)(22,70,160,129,98)(23,99,130,153,71)(24,72,154,131,100)(25,73,142,86,49)(26,50,87,143,74)(27,75,144,88,51)(28,52,81,137,76)(29,77,138,82,53)(30,54,83,139,78)(31,79,140,84,55)(32,56,85,141,80), (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), (1,27,48,131)(2,32,41,136)(3,29,42,133)(4,26,43,130)(5,31,44,135)(6,28,45,132)(7,25,46,129)(8,30,47,134)(9,159,107,56)(10,156,108,53)(11,153,109,50)(12,158,110,55)(13,155,111,52)(14,160,112,49)(15,157,105,54)(16,154,106,51)(17,118,137,122)(18,115,138,127)(19,120,139,124)(20,117,140,121)(21,114,141,126)(22,119,142,123)(23,116,143,128)(24,113,144,125)(33,88,93,72)(34,85,94,69)(35,82,95,66)(36,87,96,71)(37,84,89,68)(38,81,90,65)(39,86,91,70)(40,83,92,67)(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), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,101)(10,102)(11,103)(12,104)(13,97)(14,98)(15,99)(16,100)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(33,144)(34,137)(35,138)(36,139)(37,140)(38,141)(39,142)(40,143)(41,132)(42,133)(43,134)(44,135)(45,136)(46,129)(47,130)(48,131)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,126)(66,127)(67,128)(68,121)(69,122)(70,123)(71,124)(72,125)(73,112)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113)>;
G:=Group( (1,106,33,113,147)(2,148,114,34,107)(3,108,35,115,149)(4,150,116,36,109)(5,110,37,117,151)(6,152,118,38,111)(7,112,39,119,145)(8,146,120,40,105)(9,41,63,126,94)(10,95,127,64,42)(11,43,57,128,96)(12,89,121,58,44)(13,45,59,122,90)(14,91,123,60,46)(15,47,61,124,92)(16,93,125,62,48)(17,101,132,155,65)(18,66,156,133,102)(19,103,134,157,67)(20,68,158,135,104)(21,97,136,159,69)(22,70,160,129,98)(23,99,130,153,71)(24,72,154,131,100)(25,73,142,86,49)(26,50,87,143,74)(27,75,144,88,51)(28,52,81,137,76)(29,77,138,82,53)(30,54,83,139,78)(31,79,140,84,55)(32,56,85,141,80), (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), (1,27,48,131)(2,32,41,136)(3,29,42,133)(4,26,43,130)(5,31,44,135)(6,28,45,132)(7,25,46,129)(8,30,47,134)(9,159,107,56)(10,156,108,53)(11,153,109,50)(12,158,110,55)(13,155,111,52)(14,160,112,49)(15,157,105,54)(16,154,106,51)(17,118,137,122)(18,115,138,127)(19,120,139,124)(20,117,140,121)(21,114,141,126)(22,119,142,123)(23,116,143,128)(24,113,144,125)(33,88,93,72)(34,85,94,69)(35,82,95,66)(36,87,96,71)(37,84,89,68)(38,81,90,65)(39,86,91,70)(40,83,92,67)(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), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,101)(10,102)(11,103)(12,104)(13,97)(14,98)(15,99)(16,100)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(33,144)(34,137)(35,138)(36,139)(37,140)(38,141)(39,142)(40,143)(41,132)(42,133)(43,134)(44,135)(45,136)(46,129)(47,130)(48,131)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,126)(66,127)(67,128)(68,121)(69,122)(70,123)(71,124)(72,125)(73,112)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113) );
G=PermutationGroup([(1,106,33,113,147),(2,148,114,34,107),(3,108,35,115,149),(4,150,116,36,109),(5,110,37,117,151),(6,152,118,38,111),(7,112,39,119,145),(8,146,120,40,105),(9,41,63,126,94),(10,95,127,64,42),(11,43,57,128,96),(12,89,121,58,44),(13,45,59,122,90),(14,91,123,60,46),(15,47,61,124,92),(16,93,125,62,48),(17,101,132,155,65),(18,66,156,133,102),(19,103,134,157,67),(20,68,158,135,104),(21,97,136,159,69),(22,70,160,129,98),(23,99,130,153,71),(24,72,154,131,100),(25,73,142,86,49),(26,50,87,143,74),(27,75,144,88,51),(28,52,81,137,76),(29,77,138,82,53),(30,54,83,139,78),(31,79,140,84,55),(32,56,85,141,80)], [(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)], [(1,27,48,131),(2,32,41,136),(3,29,42,133),(4,26,43,130),(5,31,44,135),(6,28,45,132),(7,25,46,129),(8,30,47,134),(9,159,107,56),(10,156,108,53),(11,153,109,50),(12,158,110,55),(13,155,111,52),(14,160,112,49),(15,157,105,54),(16,154,106,51),(17,118,137,122),(18,115,138,127),(19,120,139,124),(20,117,140,121),(21,114,141,126),(22,119,142,123),(23,116,143,128),(24,113,144,125),(33,88,93,72),(34,85,94,69),(35,82,95,66),(36,87,96,71),(37,84,89,68),(38,81,90,65),(39,86,91,70),(40,83,92,67),(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)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,101),(10,102),(11,103),(12,104),(13,97),(14,98),(15,99),(16,100),(17,94),(18,95),(19,96),(20,89),(21,90),(22,91),(23,92),(24,93),(33,144),(34,137),(35,138),(36,139),(37,140),(38,141),(39,142),(40,143),(41,132),(42,133),(43,134),(44,135),(45,136),(46,129),(47,130),(48,131),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,157),(58,158),(59,159),(60,160),(61,153),(62,154),(63,155),(64,156),(65,126),(66,127),(67,128),(68,121),(69,122),(70,123),(71,124),(72,125),(73,112),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,113)])
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