direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4⋊C8, D4⋊C40, C20.67D8, C20.54SD16, C20.42M4(2), C4⋊C8⋊1C10, (C4×C8)⋊1C10, (C4×C40)⋊2C2, (C5×D4)⋊5C8, C4⋊C4.3C20, C4.1(C2×C40), C4.16(C5×D8), C10.32C4≀C2, C20.64(C2×C8), (C2×D4).4C20, (C4×D4).1C10, (D4×C10).27C4, (D4×C20).16C2, (C2×C20).528D4, C4.13(C5×SD16), C4.1(C5×M4(2)), C42.62(C2×C10), C10.37(C22⋊C8), (C4×C20).346C22, C10.47(D4⋊C4), (C5×C4⋊C8)⋊3C2, C2.1(C5×C4≀C2), (C5×C4⋊C4).28C4, (C2×C4).93(C5×D4), C2.5(C5×C22⋊C8), (C2×C4).38(C2×C20), C2.1(C5×D4⋊C4), (C2×C20).431(C2×C4), C22.25(C5×C22⋊C4), (C2×C10).184(C22⋊C4), SmallGroup(320,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊C8
G = < a,b,c,d | a5=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 154 in 82 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, C20 [×4], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, C4⋊C8, C4×D4, C40 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4⋊C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40 [×2], C22×C20, D4×C10, C4×C40, C5×C4⋊C8, D4×C20, C5×D4⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), D8, SD16, C20 [×2], C2×C10, C22⋊C8, D4⋊C4, C4≀C2, C40 [×2], C2×C20, C5×D4 [×2], D4⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×D8, C5×SD16, C5×C22⋊C8, C5×D4⋊C4, C5×C4≀C2, C5×D4⋊C8
►On 160 points
Generators in S160
(1 12 113 105 97)(2 13 114 106 98)(3 14 115 107 99)(4 15 116 108 100)(5 16 117 109 101)(6 9 118 110 102)(7 10 119 111 103)(8 11 120 112 104)(17 93 41 33 25)(18 94 42 34 26)(19 95 43 35 27)(20 96 44 36 28)(21 89 45 37 29)(22 90 46 38 30)(23 91 47 39 31)(24 92 48 40 32)(49 87 73 65 57)(50 88 74 66 58)(51 81 75 67 59)(52 82 76 68 60)(53 83 77 69 61)(54 84 78 70 62)(55 85 79 71 63)(56 86 80 72 64)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)
(1 55 127 17)(2 18 128 56)(3 49 121 19)(4 20 122 50)(5 51 123 21)(6 22 124 52)(7 53 125 23)(8 24 126 54)(9 90 156 82)(10 83 157 91)(11 92 158 84)(12 85 159 93)(13 94 160 86)(14 87 153 95)(15 96 154 88)(16 81 155 89)(25 97 63 135)(26 136 64 98)(27 99 57 129)(28 130 58 100)(29 101 59 131)(30 132 60 102)(31 103 61 133)(32 134 62 104)(33 105 71 143)(34 144 72 106)(35 107 65 137)(36 138 66 108)(37 109 67 139)(38 140 68 110)(39 111 69 141)(40 142 70 112)(41 113 79 151)(42 152 80 114)(43 115 73 145)(44 146 74 116)(45 117 75 147)(46 148 76 118)(47 119 77 149)(48 150 78 120)
(1 17)(2 128)(3 19)(4 122)(5 21)(6 124)(7 23)(8 126)(9 156)(10 91)(11 158)(12 93)(13 160)(14 95)(15 154)(16 89)(25 97)(27 99)(29 101)(31 103)(33 105)(35 107)(37 109)(39 111)(41 113)(43 115)(45 117)(47 119)(49 121)(51 123)(53 125)(55 127)(57 129)(59 131)(61 133)(63 135)(65 137)(67 139)(69 141)(71 143)(73 145)(75 147)(77 149)(79 151)(81 155)(83 157)(85 159)(87 153)(98 136)(100 130)(102 132)(104 134)(106 144)(108 138)(110 140)(112 142)(114 152)(116 146)(118 148)(120 150)
(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)
G:=sub<Sym(160)| (1,12,113,105,97)(2,13,114,106,98)(3,14,115,107,99)(4,15,116,108,100)(5,16,117,109,101)(6,9,118,110,102)(7,10,119,111,103)(8,11,120,112,104)(17,93,41,33,25)(18,94,42,34,26)(19,95,43,35,27)(20,96,44,36,28)(21,89,45,37,29)(22,90,46,38,30)(23,91,47,39,31)(24,92,48,40,32)(49,87,73,65,57)(50,88,74,66,58)(51,81,75,67,59)(52,82,76,68,60)(53,83,77,69,61)(54,84,78,70,62)(55,85,79,71,63)(56,86,80,72,64)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136), (1,55,127,17)(2,18,128,56)(3,49,121,19)(4,20,122,50)(5,51,123,21)(6,22,124,52)(7,53,125,23)(8,24,126,54)(9,90,156,82)(10,83,157,91)(11,92,158,84)(12,85,159,93)(13,94,160,86)(14,87,153,95)(15,96,154,88)(16,81,155,89)(25,97,63,135)(26,136,64,98)(27,99,57,129)(28,130,58,100)(29,101,59,131)(30,132,60,102)(31,103,61,133)(32,134,62,104)(33,105,71,143)(34,144,72,106)(35,107,65,137)(36,138,66,108)(37,109,67,139)(38,140,68,110)(39,111,69,141)(40,142,70,112)(41,113,79,151)(42,152,80,114)(43,115,73,145)(44,146,74,116)(45,117,75,147)(46,148,76,118)(47,119,77,149)(48,150,78,120), (1,17)(2,128)(3,19)(4,122)(5,21)(6,124)(7,23)(8,126)(9,156)(10,91)(11,158)(12,93)(13,160)(14,95)(15,154)(16,89)(25,97)(27,99)(29,101)(31,103)(33,105)(35,107)(37,109)(39,111)(41,113)(43,115)(45,117)(47,119)(49,121)(51,123)(53,125)(55,127)(57,129)(59,131)(61,133)(63,135)(65,137)(67,139)(69,141)(71,143)(73,145)(75,147)(77,149)(79,151)(81,155)(83,157)(85,159)(87,153)(98,136)(100,130)(102,132)(104,134)(106,144)(108,138)(110,140)(112,142)(114,152)(116,146)(118,148)(120,150), (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)>;
G:=Group( (1,12,113,105,97)(2,13,114,106,98)(3,14,115,107,99)(4,15,116,108,100)(5,16,117,109,101)(6,9,118,110,102)(7,10,119,111,103)(8,11,120,112,104)(17,93,41,33,25)(18,94,42,34,26)(19,95,43,35,27)(20,96,44,36,28)(21,89,45,37,29)(22,90,46,38,30)(23,91,47,39,31)(24,92,48,40,32)(49,87,73,65,57)(50,88,74,66,58)(51,81,75,67,59)(52,82,76,68,60)(53,83,77,69,61)(54,84,78,70,62)(55,85,79,71,63)(56,86,80,72,64)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136), (1,55,127,17)(2,18,128,56)(3,49,121,19)(4,20,122,50)(5,51,123,21)(6,22,124,52)(7,53,125,23)(8,24,126,54)(9,90,156,82)(10,83,157,91)(11,92,158,84)(12,85,159,93)(13,94,160,86)(14,87,153,95)(15,96,154,88)(16,81,155,89)(25,97,63,135)(26,136,64,98)(27,99,57,129)(28,130,58,100)(29,101,59,131)(30,132,60,102)(31,103,61,133)(32,134,62,104)(33,105,71,143)(34,144,72,106)(35,107,65,137)(36,138,66,108)(37,109,67,139)(38,140,68,110)(39,111,69,141)(40,142,70,112)(41,113,79,151)(42,152,80,114)(43,115,73,145)(44,146,74,116)(45,117,75,147)(46,148,76,118)(47,119,77,149)(48,150,78,120), (1,17)(2,128)(3,19)(4,122)(5,21)(6,124)(7,23)(8,126)(9,156)(10,91)(11,158)(12,93)(13,160)(14,95)(15,154)(16,89)(25,97)(27,99)(29,101)(31,103)(33,105)(35,107)(37,109)(39,111)(41,113)(43,115)(45,117)(47,119)(49,121)(51,123)(53,125)(55,127)(57,129)(59,131)(61,133)(63,135)(65,137)(67,139)(69,141)(71,143)(73,145)(75,147)(77,149)(79,151)(81,155)(83,157)(85,159)(87,153)(98,136)(100,130)(102,132)(104,134)(106,144)(108,138)(110,140)(112,142)(114,152)(116,146)(118,148)(120,150), (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) );
G=PermutationGroup([(1,12,113,105,97),(2,13,114,106,98),(3,14,115,107,99),(4,15,116,108,100),(5,16,117,109,101),(6,9,118,110,102),(7,10,119,111,103),(8,11,120,112,104),(17,93,41,33,25),(18,94,42,34,26),(19,95,43,35,27),(20,96,44,36,28),(21,89,45,37,29),(22,90,46,38,30),(23,91,47,39,31),(24,92,48,40,32),(49,87,73,65,57),(50,88,74,66,58),(51,81,75,67,59),(52,82,76,68,60),(53,83,77,69,61),(54,84,78,70,62),(55,85,79,71,63),(56,86,80,72,64),(121,153,145,137,129),(122,154,146,138,130),(123,155,147,139,131),(124,156,148,140,132),(125,157,149,141,133),(126,158,150,142,134),(127,159,151,143,135),(128,160,152,144,136)], [(1,55,127,17),(2,18,128,56),(3,49,121,19),(4,20,122,50),(5,51,123,21),(6,22,124,52),(7,53,125,23),(8,24,126,54),(9,90,156,82),(10,83,157,91),(11,92,158,84),(12,85,159,93),(13,94,160,86),(14,87,153,95),(15,96,154,88),(16,81,155,89),(25,97,63,135),(26,136,64,98),(27,99,57,129),(28,130,58,100),(29,101,59,131),(30,132,60,102),(31,103,61,133),(32,134,62,104),(33,105,71,143),(34,144,72,106),(35,107,65,137),(36,138,66,108),(37,109,67,139),(38,140,68,110),(39,111,69,141),(40,142,70,112),(41,113,79,151),(42,152,80,114),(43,115,73,145),(44,146,74,116),(45,117,75,147),(46,148,76,118),(47,119,77,149),(48,150,78,120)], [(1,17),(2,128),(3,19),(4,122),(5,21),(6,124),(7,23),(8,126),(9,156),(10,91),(11,158),(12,93),(13,160),(14,95),(15,154),(16,89),(25,97),(27,99),(29,101),(31,103),(33,105),(35,107),(37,109),(39,111),(41,113),(43,115),(45,117),(47,119),(49,121),(51,123),(53,125),(55,127),(57,129),(59,131),(61,133),(63,135),(65,137),(67,139),(69,141),(71,143),(73,145),(75,147),(77,149),(79,151),(81,155),(83,157),(85,159),(87,153),(98,136),(100,130),(102,132),(104,134),(106,144),(108,138),(110,140),(112,142),(114,152),(116,146),(118,148),(120,150)], [(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)])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | D8 | SD16 | C4≀C2 | C5×D4 | C5×M4(2) | C5×D8 | C5×SD16 | C5×C4≀C2 |
| kernel | C5×D4⋊C8 | C4×C40 | C5×C4⋊C8 | D4×C20 | C5×C4⋊C4 | D4×C10 | D4⋊C8 | C5×D4 | C4×C8 | C4⋊C8 | C4×D4 | C4⋊C4 | C2×D4 | D4 | C2×C20 | C20 | C20 | C20 | C10 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 32 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C5×D4⋊C8 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 1 | 39 |
| 0 | 1 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 39 |
| 0 | 0 | 40 |
| 27 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[1,0,0,0,1,1,0,39,40],[40,0,0,0,1,0,0,39,40],[27,0,0,0,0,4,0,8,0] >;
C5×D4⋊C8 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes C_8
% in TeX
G:=Group("C5xD4:C8"); // GroupNames label
G:=SmallGroup(320,130);
// by ID
G=gap.SmallGroup(320,130);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,1410,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations