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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C4×C8, C4⋊C8, C4×D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C22×C20, D4×C10, C4×C40, C5×C4⋊C8, D4×C20, C5×D4⋊C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C10, C22⋊C4, C2×C8, M4(2), D8, SD16, C20, C2×C10, C22⋊C8, D4⋊C4, C4≀C2, C40, C2×C20, C5×D4, 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 11 47 39 31)(2 12 48 40 32)(3 13 41 33 25)(4 14 42 34 26)(5 15 43 35 27)(6 16 44 36 28)(7 9 45 37 29)(8 10 46 38 30)(17 113 105 97 96)(18 114 106 98 89)(19 115 107 99 90)(20 116 108 100 91)(21 117 109 101 92)(22 118 110 102 93)(23 119 111 103 94)(24 120 112 104 95)(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 55 127 103)(2 104 128 56)(3 49 121 97)(4 98 122 50)(5 51 123 99)(6 100 124 52)(7 53 125 101)(8 102 126 54)(9 83 157 92)(10 93 158 84)(11 85 159 94)(12 95 160 86)(13 87 153 96)(14 89 154 88)(15 81 155 90)(16 91 156 82)(17 41 73 145)(18 146 74 42)(19 43 75 147)(20 148 76 44)(21 45 77 149)(22 150 78 46)(23 47 79 151)(24 152 80 48)(25 57 129 105)(26 106 130 58)(27 59 131 107)(28 108 132 60)(29 61 133 109)(30 110 134 62)(31 63 135 111)(32 112 136 64)(33 65 137 113)(34 114 138 66)(35 67 139 115)(36 116 140 68)(37 69 141 117)(38 118 142 70)(39 71 143 119)(40 120 144 72)
(1 103)(2 128)(3 97)(4 122)(5 99)(6 124)(7 101)(8 126)(9 92)(10 158)(11 94)(12 160)(13 96)(14 154)(15 90)(16 156)(17 41)(19 43)(21 45)(23 47)(25 105)(26 130)(27 107)(28 132)(29 109)(30 134)(31 111)(32 136)(33 113)(34 138)(35 115)(36 140)(37 117)(38 142)(39 119)(40 144)(42 146)(44 148)(46 150)(48 152)(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)
(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)| (1,11,47,39,31)(2,12,48,40,32)(3,13,41,33,25)(4,14,42,34,26)(5,15,43,35,27)(6,16,44,36,28)(7,9,45,37,29)(8,10,46,38,30)(17,113,105,97,96)(18,114,106,98,89)(19,115,107,99,90)(20,116,108,100,91)(21,117,109,101,92)(22,118,110,102,93)(23,119,111,103,94)(24,120,112,104,95)(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), (1,55,127,103)(2,104,128,56)(3,49,121,97)(4,98,122,50)(5,51,123,99)(6,100,124,52)(7,53,125,101)(8,102,126,54)(9,83,157,92)(10,93,158,84)(11,85,159,94)(12,95,160,86)(13,87,153,96)(14,89,154,88)(15,81,155,90)(16,91,156,82)(17,41,73,145)(18,146,74,42)(19,43,75,147)(20,148,76,44)(21,45,77,149)(22,150,78,46)(23,47,79,151)(24,152,80,48)(25,57,129,105)(26,106,130,58)(27,59,131,107)(28,108,132,60)(29,61,133,109)(30,110,134,62)(31,63,135,111)(32,112,136,64)(33,65,137,113)(34,114,138,66)(35,67,139,115)(36,116,140,68)(37,69,141,117)(38,118,142,70)(39,71,143,119)(40,120,144,72), (1,103)(2,128)(3,97)(4,122)(5,99)(6,124)(7,101)(8,126)(9,92)(10,158)(11,94)(12,160)(13,96)(14,154)(15,90)(16,156)(17,41)(19,43)(21,45)(23,47)(25,105)(26,130)(27,107)(28,132)(29,109)(30,134)(31,111)(32,136)(33,113)(34,138)(35,115)(36,140)(37,117)(38,142)(39,119)(40,144)(42,146)(44,148)(46,150)(48,152)(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), (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( (1,11,47,39,31)(2,12,48,40,32)(3,13,41,33,25)(4,14,42,34,26)(5,15,43,35,27)(6,16,44,36,28)(7,9,45,37,29)(8,10,46,38,30)(17,113,105,97,96)(18,114,106,98,89)(19,115,107,99,90)(20,116,108,100,91)(21,117,109,101,92)(22,118,110,102,93)(23,119,111,103,94)(24,120,112,104,95)(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), (1,55,127,103)(2,104,128,56)(3,49,121,97)(4,98,122,50)(5,51,123,99)(6,100,124,52)(7,53,125,101)(8,102,126,54)(9,83,157,92)(10,93,158,84)(11,85,159,94)(12,95,160,86)(13,87,153,96)(14,89,154,88)(15,81,155,90)(16,91,156,82)(17,41,73,145)(18,146,74,42)(19,43,75,147)(20,148,76,44)(21,45,77,149)(22,150,78,46)(23,47,79,151)(24,152,80,48)(25,57,129,105)(26,106,130,58)(27,59,131,107)(28,108,132,60)(29,61,133,109)(30,110,134,62)(31,63,135,111)(32,112,136,64)(33,65,137,113)(34,114,138,66)(35,67,139,115)(36,116,140,68)(37,69,141,117)(38,118,142,70)(39,71,143,119)(40,120,144,72), (1,103)(2,128)(3,97)(4,122)(5,99)(6,124)(7,101)(8,126)(9,92)(10,158)(11,94)(12,160)(13,96)(14,154)(15,90)(16,156)(17,41)(19,43)(21,45)(23,47)(25,105)(26,130)(27,107)(28,132)(29,109)(30,134)(31,111)(32,136)(33,113)(34,138)(35,115)(36,140)(37,117)(38,142)(39,119)(40,144)(42,146)(44,148)(46,150)(48,152)(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), (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([[(1,11,47,39,31),(2,12,48,40,32),(3,13,41,33,25),(4,14,42,34,26),(5,15,43,35,27),(6,16,44,36,28),(7,9,45,37,29),(8,10,46,38,30),(17,113,105,97,96),(18,114,106,98,89),(19,115,107,99,90),(20,116,108,100,91),(21,117,109,101,92),(22,118,110,102,93),(23,119,111,103,94),(24,120,112,104,95),(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)], [(1,55,127,103),(2,104,128,56),(3,49,121,97),(4,98,122,50),(5,51,123,99),(6,100,124,52),(7,53,125,101),(8,102,126,54),(9,83,157,92),(10,93,158,84),(11,85,159,94),(12,95,160,86),(13,87,153,96),(14,89,154,88),(15,81,155,90),(16,91,156,82),(17,41,73,145),(18,146,74,42),(19,43,75,147),(20,148,76,44),(21,45,77,149),(22,150,78,46),(23,47,79,151),(24,152,80,48),(25,57,129,105),(26,106,130,58),(27,59,131,107),(28,108,132,60),(29,61,133,109),(30,110,134,62),(31,63,135,111),(32,112,136,64),(33,65,137,113),(34,114,138,66),(35,67,139,115),(36,116,140,68),(37,69,141,117),(38,118,142,70),(39,71,143,119),(40,120,144,72)], [(1,103),(2,128),(3,97),(4,122),(5,99),(6,124),(7,101),(8,126),(9,92),(10,158),(11,94),(12,160),(13,96),(14,154),(15,90),(16,156),(17,41),(19,43),(21,45),(23,47),(25,105),(26,130),(27,107),(28,132),(29,109),(30,134),(31,111),(32,136),(33,113),(34,138),(35,115),(36,140),(37,117),(38,142),(39,119),(40,144),(42,146),(44,148),(46,150),(48,152),(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)], [(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