metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊1D4, C8⋊1D20, C42.18D10, C8⋊C4⋊3D5, C5⋊1(C8⋊3D4), (C2×D40)⋊10C2, C20⋊4D4⋊2C2, (C2×C4).25D20, (C2×C8).55D10, C4.34(C2×D20), (C2×C20).36D4, C20.277(C2×D4), C4.D20⋊2C2, (C4×C20).3C22, C2.7(C8⋊D10), C2.9(C20⋊4D4), C10.7(C4⋊1D4), C10.4(C8⋊C22), (C2×C40).56C22, (C2×D20).8C22, C22.97(C2×D20), (C2×C20).733C23, (C2×Dic10).9C22, (C5×C8⋊C4)⋊4C2, (C2×C40⋊C2)⋊1C2, (C2×C10).116(C2×D4), (C2×C4).677(C22×D5), SmallGroup(320,339)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D20
G = < a,b,c | a8=b20=c2=1, bab-1=a5, cac=a3, cbc=b-1 >
Subgroups: 878 in 144 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8⋊3D4, C40⋊C2, D40, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×D20, C2×D20, C2×D20, C5×C8⋊C4, C20⋊4D4, C4.D20, C2×C40⋊C2, C2×D40, C8⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8⋊C22, D20, C22×D5, C8⋊3D4, C2×D20, C20⋊4D4, C8⋊D10, C8⋊D20
►On 160 points
Generators in S160
(1 61 118 45 81 34 127 148)(2 35 119 149 82 62 128 46)(3 63 120 47 83 36 129 150)(4 37 101 151 84 64 130 48)(5 65 102 49 85 38 131 152)(6 39 103 153 86 66 132 50)(7 67 104 51 87 40 133 154)(8 21 105 155 88 68 134 52)(9 69 106 53 89 22 135 156)(10 23 107 157 90 70 136 54)(11 71 108 55 91 24 137 158)(12 25 109 159 92 72 138 56)(13 73 110 57 93 26 139 160)(14 27 111 141 94 74 140 58)(15 75 112 59 95 28 121 142)(16 29 113 143 96 76 122 60)(17 77 114 41 97 30 123 144)(18 31 115 145 98 78 124 42)(19 79 116 43 99 32 125 146)(20 33 117 147 100 80 126 44)
(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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 160)(9 159)(10 158)(11 157)(12 156)(13 155)(14 154)(15 153)(16 152)(17 151)(18 150)(19 149)(20 148)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 84)(42 83)(43 82)(44 81)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 140)(68 139)(69 138)(70 137)(71 136)(72 135)(73 134)(74 133)(75 132)(76 131)(77 130)(78 129)(79 128)(80 127)
G:=sub<Sym(160)| (1,61,118,45,81,34,127,148)(2,35,119,149,82,62,128,46)(3,63,120,47,83,36,129,150)(4,37,101,151,84,64,130,48)(5,65,102,49,85,38,131,152)(6,39,103,153,86,66,132,50)(7,67,104,51,87,40,133,154)(8,21,105,155,88,68,134,52)(9,69,106,53,89,22,135,156)(10,23,107,157,90,70,136,54)(11,71,108,55,91,24,137,158)(12,25,109,159,92,72,138,56)(13,73,110,57,93,26,139,160)(14,27,111,141,94,74,140,58)(15,75,112,59,95,28,121,142)(16,29,113,143,96,76,122,60)(17,77,114,41,97,30,123,144)(18,31,115,145,98,78,124,42)(19,79,116,43,99,32,125,146)(20,33,117,147,100,80,126,44), (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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,160)(9,159)(10,158)(11,157)(12,156)(13,155)(14,154)(15,153)(16,152)(17,151)(18,150)(19,149)(20,148)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,84)(42,83)(43,82)(44,81)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)>;
G:=Group( (1,61,118,45,81,34,127,148)(2,35,119,149,82,62,128,46)(3,63,120,47,83,36,129,150)(4,37,101,151,84,64,130,48)(5,65,102,49,85,38,131,152)(6,39,103,153,86,66,132,50)(7,67,104,51,87,40,133,154)(8,21,105,155,88,68,134,52)(9,69,106,53,89,22,135,156)(10,23,107,157,90,70,136,54)(11,71,108,55,91,24,137,158)(12,25,109,159,92,72,138,56)(13,73,110,57,93,26,139,160)(14,27,111,141,94,74,140,58)(15,75,112,59,95,28,121,142)(16,29,113,143,96,76,122,60)(17,77,114,41,97,30,123,144)(18,31,115,145,98,78,124,42)(19,79,116,43,99,32,125,146)(20,33,117,147,100,80,126,44), (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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,160)(9,159)(10,158)(11,157)(12,156)(13,155)(14,154)(15,153)(16,152)(17,151)(18,150)(19,149)(20,148)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,84)(42,83)(43,82)(44,81)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127) );
G=PermutationGroup([[(1,61,118,45,81,34,127,148),(2,35,119,149,82,62,128,46),(3,63,120,47,83,36,129,150),(4,37,101,151,84,64,130,48),(5,65,102,49,85,38,131,152),(6,39,103,153,86,66,132,50),(7,67,104,51,87,40,133,154),(8,21,105,155,88,68,134,52),(9,69,106,53,89,22,135,156),(10,23,107,157,90,70,136,54),(11,71,108,55,91,24,137,158),(12,25,109,159,92,72,138,56),(13,73,110,57,93,26,139,160),(14,27,111,141,94,74,140,58),(15,75,112,59,95,28,121,142),(16,29,113,143,96,76,122,60),(17,77,114,41,97,30,123,144),(18,31,115,145,98,78,124,42),(19,79,116,43,99,32,125,146),(20,33,117,147,100,80,126,44)], [(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)], [(1,147),(2,146),(3,145),(4,144),(5,143),(6,142),(7,141),(8,160),(9,159),(10,158),(11,157),(12,156),(13,155),(14,154),(15,153),(16,152),(17,151),(18,150),(19,149),(20,148),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,102),(30,101),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,84),(42,83),(43,82),(44,81),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,140),(68,139),(69,138),(70,137),(71,136),(72,135),(73,134),(74,133),(75,132),(76,131),(77,130),(78,129),(79,128),(80,127)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D20 | D20 | C8⋊C22 | C8⋊D10 |
| kernel | C8⋊D20 | C5×C8⋊C4 | C20⋊4D4 | C4.D20 | C2×C40⋊C2 | C2×D40 | C40 | C2×C20 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 16 | 8 | 2 | 8 |
Matrix representation of C8⋊D20 ►in GL6(𝔽41)
| 2 | 28 | 0 | 0 | 0 | 0 |
| 13 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 6 | 6 | 27 |
| 0 | 0 | 30 | 39 | 12 | 35 |
| 0 | 0 | 3 | 0 | 13 | 35 |
| 0 | 0 | 0 | 3 | 11 | 2 |
| 16 | 39 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 33 | 13 | 34 |
| 0 | 0 | 1 | 34 | 6 | 7 |
| 0 | 0 | 2 | 33 | 6 | 8 |
| 0 | 0 | 1 | 1 | 40 | 7 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [2,13,0,0,0,0,28,39,0,0,0,0,0,0,28,30,3,0,0,0,6,39,0,3,0,0,6,12,13,11,0,0,27,35,35,2],[16,2,0,0,0,0,39,28,0,0,0,0,0,0,35,1,2,1,0,0,33,34,33,1,0,0,13,6,6,40,0,0,34,7,8,7],[1,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,40,5,0,0,0,0,0,1,0,0] >;
C8⋊D20 in GAP, Magma, Sage, TeX
C_8\rtimes D_{20} % in TeX
G:=Group("C8:D20"); // GroupNames label
G:=SmallGroup(320,339);
// by ID
G=gap.SmallGroup(320,339);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations