metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8⋊8D4, C5⋊D4⋊C8, C5⋊1(C8×D4), Dic5⋊(C2×C8), C2.1(D4×F5), D10⋊1(C2×C8), C10.1(C4×D4), C22⋊C4.8F5, C22⋊1(D5⋊C8), C10.2(C8○D4), C10.5(C22×C8), D10⋊C8⋊11C2, C2.2(D4.F5), C23.24(C2×F5), D10⋊C4.5C4, Dic5.65(C2×D4), C10.D4.6C4, C23.2F5⋊3C2, Dic5⋊C8⋊12C2, Dic5.50(C4○D4), Dic5⋊4D4.10C2, C22.34(C22×F5), (C2×Dic5).322C23, (C4×Dic5).247C22, (C22×Dic5).177C22, (C4×C5⋊C8)⋊11C2, (C2×C10)⋊1(C2×C8), (C2×D5⋊C8)⋊8C2, (C22×C5⋊C8)⋊1C2, C2.7(C2×D5⋊C8), (C2×C5⋊D4).3C4, (C2×C4).57(C2×F5), (C2×C20).90(C2×C4), (C5×C22⋊C4).8C4, (C2×C5⋊C8).20C22, (C2×C4×D5).286C22, (C2×C10).30(C22×C4), (C22×C10).13(C2×C4), (C2×Dic5).47(C2×C4), (C22×D5).39(C2×C4), SmallGroup(320,1030)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 134 conjugacy classes, 56 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22, C22 [×2], C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C8×D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, D5⋊C8 [×2], C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C5⋊C8, D10⋊C8, Dic5⋊C8, C23.2F5, Dic5⋊4D4, C2×D5⋊C8, C22×C5⋊C8, C5⋊C8⋊8D4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C2×F5 [×3], C8×D4, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D4.F5, D4×F5, C5⋊C8⋊8D4
Generators and relations
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a3, cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 12 132 143 38)(2 144 13 39 133)(3 40 137 134 14)(4 135 33 15 138)(5 16 136 139 34)(6 140 9 35 129)(7 36 141 130 10)(8 131 37 11 142)(17 65 128 88 51)(18 81 66 52 121)(19 53 82 122 67)(20 123 54 68 83)(21 69 124 84 55)(22 85 70 56 125)(23 49 86 126 71)(24 127 50 72 87)(25 116 90 109 102)(26 110 117 103 91)(27 104 111 92 118)(28 93 97 119 112)(29 120 94 105 98)(30 106 113 99 95)(31 100 107 96 114)(32 89 101 115 108)(41 149 73 63 157)(42 64 150 158 74)(43 159 57 75 151)(44 76 160 152 58)(45 145 77 59 153)(46 60 146 154 78)(47 155 61 79 147)(48 80 156 148 62)
(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 98 57 52)(2 99 58 53)(3 100 59 54)(4 101 60 55)(5 102 61 56)(6 103 62 49)(7 104 63 50)(8 97 64 51)(9 110 80 71)(10 111 73 72)(11 112 74 65)(12 105 75 66)(13 106 76 67)(14 107 77 68)(15 108 78 69)(16 109 79 70)(17 142 119 42)(18 143 120 43)(19 144 113 44)(20 137 114 45)(21 138 115 46)(22 139 116 47)(23 140 117 48)(24 141 118 41)(25 155 125 34)(26 156 126 35)(27 157 127 36)(28 158 128 37)(29 159 121 38)(30 160 122 39)(31 153 123 40)(32 154 124 33)(81 132 94 151)(82 133 95 152)(83 134 96 145)(84 135 89 146)(85 136 90 147)(86 129 91 148)(87 130 92 149)(88 131 93 150)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 129)(24 130)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(41 92)(42 93)(43 94)(44 95)(45 96)(46 89)(47 90)(48 91)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 97)(81 143)(82 144)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(105 159)(106 160)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 152)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)
G:=sub<Sym(160)| (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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), (1,98,57,52)(2,99,58,53)(3,100,59,54)(4,101,60,55)(5,102,61,56)(6,103,62,49)(7,104,63,50)(8,97,64,51)(9,110,80,71)(10,111,73,72)(11,112,74,65)(12,105,75,66)(13,106,76,67)(14,107,77,68)(15,108,78,69)(16,109,79,70)(17,142,119,42)(18,143,120,43)(19,144,113,44)(20,137,114,45)(21,138,115,46)(22,139,116,47)(23,140,117,48)(24,141,118,41)(25,155,125,34)(26,156,126,35)(27,157,127,36)(28,158,128,37)(29,159,121,38)(30,160,122,39)(31,153,123,40)(32,154,124,33)(81,132,94,151)(82,133,95,152)(83,134,96,145)(84,135,89,146)(85,136,90,147)(86,129,91,148)(87,130,92,149)(88,131,93,150), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,129)(24,130)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,92)(42,93)(43,94)(44,95)(45,96)(46,89)(47,90)(48,91)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(81,143)(82,144)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(105,159)(106,160)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,152)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)>;
G:=Group( (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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), (1,98,57,52)(2,99,58,53)(3,100,59,54)(4,101,60,55)(5,102,61,56)(6,103,62,49)(7,104,63,50)(8,97,64,51)(9,110,80,71)(10,111,73,72)(11,112,74,65)(12,105,75,66)(13,106,76,67)(14,107,77,68)(15,108,78,69)(16,109,79,70)(17,142,119,42)(18,143,120,43)(19,144,113,44)(20,137,114,45)(21,138,115,46)(22,139,116,47)(23,140,117,48)(24,141,118,41)(25,155,125,34)(26,156,126,35)(27,157,127,36)(28,158,128,37)(29,159,121,38)(30,160,122,39)(31,153,123,40)(32,154,124,33)(81,132,94,151)(82,133,95,152)(83,134,96,145)(84,135,89,146)(85,136,90,147)(86,129,91,148)(87,130,92,149)(88,131,93,150), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,129)(24,130)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(41,92)(42,93)(43,94)(44,95)(45,96)(46,89)(47,90)(48,91)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,97)(81,143)(82,144)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(105,159)(106,160)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,152)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151) );
G=PermutationGroup([(1,12,132,143,38),(2,144,13,39,133),(3,40,137,134,14),(4,135,33,15,138),(5,16,136,139,34),(6,140,9,35,129),(7,36,141,130,10),(8,131,37,11,142),(17,65,128,88,51),(18,81,66,52,121),(19,53,82,122,67),(20,123,54,68,83),(21,69,124,84,55),(22,85,70,56,125),(23,49,86,126,71),(24,127,50,72,87),(25,116,90,109,102),(26,110,117,103,91),(27,104,111,92,118),(28,93,97,119,112),(29,120,94,105,98),(30,106,113,99,95),(31,100,107,96,114),(32,89,101,115,108),(41,149,73,63,157),(42,64,150,158,74),(43,159,57,75,151),(44,76,160,152,58),(45,145,77,59,153),(46,60,146,154,78),(47,155,61,79,147),(48,80,156,148,62)], [(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)], [(1,98,57,52),(2,99,58,53),(3,100,59,54),(4,101,60,55),(5,102,61,56),(6,103,62,49),(7,104,63,50),(8,97,64,51),(9,110,80,71),(10,111,73,72),(11,112,74,65),(12,105,75,66),(13,106,76,67),(14,107,77,68),(15,108,78,69),(16,109,79,70),(17,142,119,42),(18,143,120,43),(19,144,113,44),(20,137,114,45),(21,138,115,46),(22,139,116,47),(23,140,117,48),(24,141,118,41),(25,155,125,34),(26,156,126,35),(27,157,127,36),(28,158,128,37),(29,159,121,38),(30,160,122,39),(31,153,123,40),(32,154,124,33),(81,132,94,151),(82,133,95,152),(83,134,96,145),(84,135,89,146),(85,136,90,147),(86,129,91,148),(87,130,92,149),(88,131,93,150)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,126),(10,127),(11,128),(12,121),(13,122),(14,123),(15,124),(16,125),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,129),(24,130),(25,79),(26,80),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,69),(34,70),(35,71),(36,72),(37,65),(38,66),(39,67),(40,68),(41,92),(42,93),(43,94),(44,95),(45,96),(46,89),(47,90),(48,91),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,97),(81,143),(82,144),(83,137),(84,138),(85,139),(86,140),(87,141),(88,142),(105,159),(106,160),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,152),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 22 | 21 |
| 0 | 0 | 0 | 21 | 1 | 40 |
| 0 | 0 | 20 | 40 | 1 | 21 |
| 0 | 0 | 20 | 21 | 22 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,19,0,20,20,0,0,0,21,40,21,0,0,22,1,1,22,0,0,21,40,21,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40,0,0,0,0,40,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | F5 | C2×F5 | C2×F5 | D5⋊C8 | D4.F5 | D4×F5 |
| kernel | C5⋊C8⋊8D4 | C4×C5⋊C8 | D10⋊C8 | Dic5⋊C8 | C23.2F5 | Dic5⋊4D4 | C2×D5⋊C8 | C22×C5⋊C8 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C5⋊D4 | C5⋊D4 | C5⋊C8 | Dic5 | C10 | C22⋊C4 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_5\rtimes C_8\rtimes_8D_4
% in TeX
G:=Group("C5:C8:8D4"); // GroupNames label
G:=SmallGroup(320,1030);
// by ID
G=gap.SmallGroup(320,1030);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^3,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀