direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D8⋊C4, D8⋊3C20, C8⋊3(C2×C20), C40⋊31(C2×C4), (C5×D8)⋊15C4, D4⋊2(C2×C20), (C4×D4)⋊2C10, C8⋊C4⋊2C10, C4.Q8⋊3C10, (D4×C20)⋊31C2, (C2×D8).6C10, C2.17(D4×C20), (C10×D8).13C2, (C2×C20).458D4, C10.149(C4×D4), D4⋊C4⋊16C10, C42.10(C2×C10), C4.14(C22×C20), C22.56(D4×C10), C20.261(C4○D4), (C4×C20).251C22, (C2×C40).331C22, C20.218(C22×C4), (C2×C20).909C23, C10.130(C8⋊C22), (D4×C10).293C22, C4.6(C5×C4○D4), (C5×D4)⋊25(C2×C4), (C5×C8⋊C4)⋊11C2, (C5×C4.Q8)⋊12C2, C2.5(C5×C8⋊C22), C4⋊C4.50(C2×C10), (C2×C8).20(C2×C10), (C2×C4).104(C5×D4), (C5×D4⋊C4)⋊39C2, (C2×D4).51(C2×C10), (C2×C10).632(C2×D4), (C5×C4⋊C4).371C22, (C2×C4).84(C22×C10), SmallGroup(320,943)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D8⋊C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >
Subgroups: 250 in 132 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C40, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D8⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C22×C20, D4×C10, C5×C8⋊C4, C5×D4⋊C4, C5×C4.Q8, D4×C20, C10×D8, C5×D8⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C8⋊C22, C2×C20, C5×D4, C22×C10, D8⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8⋊C22, C5×D8⋊C4
►On 160 points
Generators in S160
(1 127 159 36 151)(2 128 160 37 152)(3 121 153 38 145)(4 122 154 39 146)(5 123 155 40 147)(6 124 156 33 148)(7 125 157 34 149)(8 126 158 35 150)(9 116 140 17 132)(10 117 141 18 133)(11 118 142 19 134)(12 119 143 20 135)(13 120 144 21 136)(14 113 137 22 129)(15 114 138 23 130)(16 115 139 24 131)(25 86 46 91 51)(26 87 47 92 52)(27 88 48 93 53)(28 81 41 94 54)(29 82 42 95 55)(30 83 43 96 56)(31 84 44 89 49)(32 85 45 90 50)(57 80 112 71 102)(58 73 105 72 103)(59 74 106 65 104)(60 75 107 66 97)(61 76 108 67 98)(62 77 109 68 99)(63 78 110 69 100)(64 79 111 70 101)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 22)(18 21)(19 20)(23 24)(25 30)(26 29)(27 28)(31 32)(33 38)(34 37)(35 36)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 56)(52 55)(53 54)(57 62)(58 61)(59 60)(63 64)(65 66)(67 72)(68 71)(69 70)(73 76)(74 75)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 90)(91 96)(92 95)(93 94)(97 104)(98 103)(99 102)(100 101)(105 108)(106 107)(109 112)(110 111)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)
(1 119 41 111)(2 116 42 108)(3 113 43 105)(4 118 44 110)(5 115 45 107)(6 120 46 112)(7 117 47 109)(8 114 48 106)(9 82 76 152)(10 87 77 149)(11 84 78 146)(12 81 79 151)(13 86 80 148)(14 83 73 145)(15 88 74 150)(16 85 75 147)(17 55 98 160)(18 52 99 157)(19 49 100 154)(20 54 101 159)(21 51 102 156)(22 56 103 153)(23 53 104 158)(24 50 97 155)(25 57 33 136)(26 62 34 133)(27 59 35 130)(28 64 36 135)(29 61 37 132)(30 58 38 129)(31 63 39 134)(32 60 40 131)(65 126 138 93)(66 123 139 90)(67 128 140 95)(68 125 141 92)(69 122 142 89)(70 127 143 94)(71 124 144 91)(72 121 137 96)
G:=sub<Sym(160)| (1,127,159,36,151)(2,128,160,37,152)(3,121,153,38,145)(4,122,154,39,146)(5,123,155,40,147)(6,124,156,33,148)(7,125,157,34,149)(8,126,158,35,150)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,86,46,91,51)(26,87,47,92,52)(27,88,48,93,53)(28,81,41,94,54)(29,82,42,95,55)(30,83,43,96,56)(31,84,44,89,49)(32,85,45,90,50)(57,80,112,71,102)(58,73,105,72,103)(59,74,106,65,104)(60,75,107,66,97)(61,76,108,67,98)(62,77,109,68,99)(63,78,110,69,100)(64,79,111,70,101), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,56)(52,55)(53,54)(57,62)(58,61)(59,60)(63,64)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,88)(82,87)(83,86)(84,85)(89,90)(91,96)(92,95)(93,94)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159), (1,119,41,111)(2,116,42,108)(3,113,43,105)(4,118,44,110)(5,115,45,107)(6,120,46,112)(7,117,47,109)(8,114,48,106)(9,82,76,152)(10,87,77,149)(11,84,78,146)(12,81,79,151)(13,86,80,148)(14,83,73,145)(15,88,74,150)(16,85,75,147)(17,55,98,160)(18,52,99,157)(19,49,100,154)(20,54,101,159)(21,51,102,156)(22,56,103,153)(23,53,104,158)(24,50,97,155)(25,57,33,136)(26,62,34,133)(27,59,35,130)(28,64,36,135)(29,61,37,132)(30,58,38,129)(31,63,39,134)(32,60,40,131)(65,126,138,93)(66,123,139,90)(67,128,140,95)(68,125,141,92)(69,122,142,89)(70,127,143,94)(71,124,144,91)(72,121,137,96)>;
G:=Group( (1,127,159,36,151)(2,128,160,37,152)(3,121,153,38,145)(4,122,154,39,146)(5,123,155,40,147)(6,124,156,33,148)(7,125,157,34,149)(8,126,158,35,150)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,86,46,91,51)(26,87,47,92,52)(27,88,48,93,53)(28,81,41,94,54)(29,82,42,95,55)(30,83,43,96,56)(31,84,44,89,49)(32,85,45,90,50)(57,80,112,71,102)(58,73,105,72,103)(59,74,106,65,104)(60,75,107,66,97)(61,76,108,67,98)(62,77,109,68,99)(63,78,110,69,100)(64,79,111,70,101), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,56)(52,55)(53,54)(57,62)(58,61)(59,60)(63,64)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,88)(82,87)(83,86)(84,85)(89,90)(91,96)(92,95)(93,94)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159), (1,119,41,111)(2,116,42,108)(3,113,43,105)(4,118,44,110)(5,115,45,107)(6,120,46,112)(7,117,47,109)(8,114,48,106)(9,82,76,152)(10,87,77,149)(11,84,78,146)(12,81,79,151)(13,86,80,148)(14,83,73,145)(15,88,74,150)(16,85,75,147)(17,55,98,160)(18,52,99,157)(19,49,100,154)(20,54,101,159)(21,51,102,156)(22,56,103,153)(23,53,104,158)(24,50,97,155)(25,57,33,136)(26,62,34,133)(27,59,35,130)(28,64,36,135)(29,61,37,132)(30,58,38,129)(31,63,39,134)(32,60,40,131)(65,126,138,93)(66,123,139,90)(67,128,140,95)(68,125,141,92)(69,122,142,89)(70,127,143,94)(71,124,144,91)(72,121,137,96) );
G=PermutationGroup([[(1,127,159,36,151),(2,128,160,37,152),(3,121,153,38,145),(4,122,154,39,146),(5,123,155,40,147),(6,124,156,33,148),(7,125,157,34,149),(8,126,158,35,150),(9,116,140,17,132),(10,117,141,18,133),(11,118,142,19,134),(12,119,143,20,135),(13,120,144,21,136),(14,113,137,22,129),(15,114,138,23,130),(16,115,139,24,131),(25,86,46,91,51),(26,87,47,92,52),(27,88,48,93,53),(28,81,41,94,54),(29,82,42,95,55),(30,83,43,96,56),(31,84,44,89,49),(32,85,45,90,50),(57,80,112,71,102),(58,73,105,72,103),(59,74,106,65,104),(60,75,107,66,97),(61,76,108,67,98),(62,77,109,68,99),(63,78,110,69,100),(64,79,111,70,101)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,22),(18,21),(19,20),(23,24),(25,30),(26,29),(27,28),(31,32),(33,38),(34,37),(35,36),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,56),(52,55),(53,54),(57,62),(58,61),(59,60),(63,64),(65,66),(67,72),(68,71),(69,70),(73,76),(74,75),(77,80),(78,79),(81,88),(82,87),(83,86),(84,85),(89,90),(91,96),(92,95),(93,94),(97,104),(98,103),(99,102),(100,101),(105,108),(106,107),(109,112),(110,111),(113,116),(114,115),(117,120),(118,119),(121,124),(122,123),(125,128),(126,127),(129,132),(130,131),(133,136),(134,135),(137,140),(138,139),(141,144),(142,143),(145,148),(146,147),(149,152),(150,151),(153,156),(154,155),(157,160),(158,159)], [(1,119,41,111),(2,116,42,108),(3,113,43,105),(4,118,44,110),(5,115,45,107),(6,120,46,112),(7,117,47,109),(8,114,48,106),(9,82,76,152),(10,87,77,149),(11,84,78,146),(12,81,79,151),(13,86,80,148),(14,83,73,145),(15,88,74,150),(16,85,75,147),(17,55,98,160),(18,52,99,157),(19,49,100,154),(20,54,101,159),(21,51,102,156),(22,56,103,153),(23,53,104,158),(24,50,97,155),(25,57,33,136),(26,62,34,133),(27,59,35,130),(28,64,36,135),(29,61,37,132),(30,58,38,129),(31,63,39,134),(32,60,40,131),(65,126,138,93),(66,123,139,90),(67,128,140,95),(68,125,141,92),(69,122,142,89),(70,127,143,94),(71,124,144,91),(72,121,137,96)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×D8⋊C4 | C5×C8⋊C4 | C5×D4⋊C4 | C5×C4.Q8 | D4×C20 | C10×D8 | C5×D8 | D8⋊C4 | C8⋊C4 | D4⋊C4 | C4.Q8 | C4×D4 | C2×D8 | D8 | C2×C20 | C20 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×D8⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 12 | 0 | 31 |
| 0 | 0 | 21 | 29 | 31 | 10 |
| 0 | 0 | 36 | 14 | 0 | 26 |
| 0 | 0 | 37 | 40 | 29 | 18 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 12 | 0 | 31 |
| 0 | 0 | 20 | 12 | 10 | 31 |
| 0 | 0 | 36 | 14 | 0 | 26 |
| 0 | 0 | 7 | 40 | 12 | 35 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 26 | 8 | 1 | 39 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 17 | 33 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,35,21,36,37,0,0,12,29,14,40,0,0,0,31,0,29,0,0,31,10,26,18],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,35,20,36,7,0,0,12,12,14,40,0,0,0,10,0,12,0,0,31,31,26,35],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,26,1,2,0,0,0,8,0,11,0,0,1,1,0,17,0,0,0,39,0,33] >;
C5×D8⋊C4 in GAP, Magma, Sage, TeX
C_5\times D_8\rtimes C_4
% in TeX
G:=Group("C5xD8:C4"); // GroupNames label
G:=SmallGroup(320,943);
// by ID
G=gap.SmallGroup(320,943);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1276,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations