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 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×8], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C40 [×2], C40, C2×C20, C2×C20 [×2], C2×C20 [×6], C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], D8⋊C4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C22×C20 [×2], D4×C10 [×2], C5×C8⋊C4, C5×D4⋊C4 [×2], C5×C4.Q8, D4×C20 [×2], C10×D8, C5×D8⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C8⋊C22 [×2], C2×C20 [×6], C5×D4 [×2], C22×C10, D8⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8⋊C22 [×2], 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 85 54 48 93)(26 86 55 41 94)(27 87 56 42 95)(28 88 49 43 96)(29 81 50 44 89)(30 82 51 45 90)(31 83 52 46 91)(32 84 53 47 92)(57 75 107 67 102)(58 76 108 68 103)(59 77 109 69 104)(60 78 110 70 97)(61 79 111 71 98)(62 80 112 72 99)(63 73 105 65 100)(64 74 106 66 101)
(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 26)(27 32)(28 31)(29 30)(33 38)(34 37)(35 36)(39 40)(41 48)(42 47)(43 46)(44 45)(49 52)(50 51)(53 56)(54 55)(57 60)(58 59)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 102)(98 101)(99 100)(103 104)(105 112)(106 111)(107 110)(108 109)(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 69)(2 116 42 66)(3 113 43 71)(4 118 44 68)(5 115 45 65)(6 120 46 70)(7 117 47 67)(8 114 48 72)(9 56 106 152)(10 53 107 149)(11 50 108 146)(12 55 109 151)(13 52 110 148)(14 49 111 145)(15 54 112 150)(16 51 105 147)(17 27 64 160)(18 32 57 157)(19 29 58 154)(20 26 59 159)(21 31 60 156)(22 28 61 153)(23 25 62 158)(24 30 63 155)(33 136 83 78)(34 133 84 75)(35 130 85 80)(36 135 86 77)(37 132 87 74)(38 129 88 79)(39 134 81 76)(40 131 82 73)(89 103 122 142)(90 100 123 139)(91 97 124 144)(92 102 125 141)(93 99 126 138)(94 104 127 143)(95 101 128 140)(96 98 121 137)
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)(25,85,54,48,93)(26,86,55,41,94)(27,87,56,42,95)(28,88,49,43,96)(29,81,50,44,89)(30,82,51,45,90)(31,83,52,46,91)(32,84,53,47,92)(57,75,107,67,102)(58,76,108,68,103)(59,77,109,69,104)(60,78,110,70,97)(61,79,111,71,98)(62,80,112,72,99)(63,73,105,65,100)(64,74,106,66,101), (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)(25,26)(27,32)(28,31)(29,30)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(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), (1,119,41,69)(2,116,42,66)(3,113,43,71)(4,118,44,68)(5,115,45,65)(6,120,46,70)(7,117,47,67)(8,114,48,72)(9,56,106,152)(10,53,107,149)(11,50,108,146)(12,55,109,151)(13,52,110,148)(14,49,111,145)(15,54,112,150)(16,51,105,147)(17,27,64,160)(18,32,57,157)(19,29,58,154)(20,26,59,159)(21,31,60,156)(22,28,61,153)(23,25,62,158)(24,30,63,155)(33,136,83,78)(34,133,84,75)(35,130,85,80)(36,135,86,77)(37,132,87,74)(38,129,88,79)(39,134,81,76)(40,131,82,73)(89,103,122,142)(90,100,123,139)(91,97,124,144)(92,102,125,141)(93,99,126,138)(94,104,127,143)(95,101,128,140)(96,98,121,137)>;
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)(25,85,54,48,93)(26,86,55,41,94)(27,87,56,42,95)(28,88,49,43,96)(29,81,50,44,89)(30,82,51,45,90)(31,83,52,46,91)(32,84,53,47,92)(57,75,107,67,102)(58,76,108,68,103)(59,77,109,69,104)(60,78,110,70,97)(61,79,111,71,98)(62,80,112,72,99)(63,73,105,65,100)(64,74,106,66,101), (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)(25,26)(27,32)(28,31)(29,30)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(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), (1,119,41,69)(2,116,42,66)(3,113,43,71)(4,118,44,68)(5,115,45,65)(6,120,46,70)(7,117,47,67)(8,114,48,72)(9,56,106,152)(10,53,107,149)(11,50,108,146)(12,55,109,151)(13,52,110,148)(14,49,111,145)(15,54,112,150)(16,51,105,147)(17,27,64,160)(18,32,57,157)(19,29,58,154)(20,26,59,159)(21,31,60,156)(22,28,61,153)(23,25,62,158)(24,30,63,155)(33,136,83,78)(34,133,84,75)(35,130,85,80)(36,135,86,77)(37,132,87,74)(38,129,88,79)(39,134,81,76)(40,131,82,73)(89,103,122,142)(90,100,123,139)(91,97,124,144)(92,102,125,141)(93,99,126,138)(94,104,127,143)(95,101,128,140)(96,98,121,137) );
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),(25,85,54,48,93),(26,86,55,41,94),(27,87,56,42,95),(28,88,49,43,96),(29,81,50,44,89),(30,82,51,45,90),(31,83,52,46,91),(32,84,53,47,92),(57,75,107,67,102),(58,76,108,68,103),(59,77,109,69,104),(60,78,110,70,97),(61,79,111,71,98),(62,80,112,72,99),(63,73,105,65,100),(64,74,106,66,101)], [(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),(25,26),(27,32),(28,31),(29,30),(33,38),(34,37),(35,36),(39,40),(41,48),(42,47),(43,46),(44,45),(49,52),(50,51),(53,56),(54,55),(57,60),(58,59),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(73,80),(74,79),(75,78),(76,77),(81,82),(83,88),(84,87),(85,86),(89,90),(91,96),(92,95),(93,94),(97,102),(98,101),(99,100),(103,104),(105,112),(106,111),(107,110),(108,109),(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)], [(1,119,41,69),(2,116,42,66),(3,113,43,71),(4,118,44,68),(5,115,45,65),(6,120,46,70),(7,117,47,67),(8,114,48,72),(9,56,106,152),(10,53,107,149),(11,50,108,146),(12,55,109,151),(13,52,110,148),(14,49,111,145),(15,54,112,150),(16,51,105,147),(17,27,64,160),(18,32,57,157),(19,29,58,154),(20,26,59,159),(21,31,60,156),(22,28,61,153),(23,25,62,158),(24,30,63,155),(33,136,83,78),(34,133,84,75),(35,130,85,80),(36,135,86,77),(37,132,87,74),(38,129,88,79),(39,134,81,76),(40,131,82,73),(89,103,122,142),(90,100,123,139),(91,97,124,144),(92,102,125,141),(93,99,126,138),(94,104,127,143),(95,101,128,140),(96,98,121,137)])
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