metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3D8, C42.218D10, C4⋊1(D4⋊D5), C4⋊1D4⋊2D5, C5⋊2C8⋊11D4, C5⋊2(C8⋊4D4), C4.14(D4×D5), C20.31(C2×D4), C10.58(C2×D8), C20⋊4D4⋊10C2, (C2×D4).56D10, (C2×C20).148D4, C10.20(C4⋊1D4), C2.11(C20⋊D4), (C4×C20).121C22, (C2×C20).391C23, (D4×C10).72C22, (C2×D20).109C22, (C4×C5⋊2C8)⋊15C2, (C2×D4⋊D5)⋊14C2, (C5×C4⋊1D4)⋊2C2, C2.13(C2×D4⋊D5), (C2×C10).522(C2×D4), (C2×C4).131(C5⋊D4), (C2×C4).489(C22×D5), C22.195(C2×C5⋊D4), (C2×C5⋊2C8).265C22, SmallGroup(320,700)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊D8
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >
Subgroups: 798 in 162 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, C42, C2×C8, D8, C2×D4, C2×D4, C20, D10, C2×C10, C2×C10, C4×C8, C4⋊1D4, C4⋊1D4, C2×D8, C5⋊2C8, D20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊4D4, C2×C5⋊2C8, D4⋊D5, C4×C20, C2×D20, C2×D20, D4×C10, D4×C10, C4×C5⋊2C8, C20⋊4D4, C2×D4⋊D5, C5×C4⋊1D4, C20⋊D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C4⋊1D4, C2×D8, C5⋊D4, C22×D5, C8⋊4D4, D4⋊D5, D4×D5, C2×C5⋊D4, C2×D4⋊D5, C20⋊D4, C20⋊D8
►On 160 points
Generators in S160
(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 86 110 78 37 129 143 56)(2 95 111 67 38 138 144 45)(3 84 112 76 39 127 145 54)(4 93 113 65 40 136 146 43)(5 82 114 74 21 125 147 52)(6 91 115 63 22 134 148 41)(7 100 116 72 23 123 149 50)(8 89 117 61 24 132 150 59)(9 98 118 70 25 121 151 48)(10 87 119 79 26 130 152 57)(11 96 120 68 27 139 153 46)(12 85 101 77 28 128 154 55)(13 94 102 66 29 137 155 44)(14 83 103 75 30 126 156 53)(15 92 104 64 31 135 157 42)(16 81 105 73 32 124 158 51)(17 90 106 62 33 133 159 60)(18 99 107 71 34 122 160 49)(19 88 108 80 35 131 141 58)(20 97 109 69 36 140 142 47)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 81)(42 100)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(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)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)
G:=sub<Sym(160)| (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), (1,86,110,78,37,129,143,56)(2,95,111,67,38,138,144,45)(3,84,112,76,39,127,145,54)(4,93,113,65,40,136,146,43)(5,82,114,74,21,125,147,52)(6,91,115,63,22,134,148,41)(7,100,116,72,23,123,149,50)(8,89,117,61,24,132,150,59)(9,98,118,70,25,121,151,48)(10,87,119,79,26,130,152,57)(11,96,120,68,27,139,153,46)(12,85,101,77,28,128,154,55)(13,94,102,66,29,137,155,44)(14,83,103,75,30,126,156,53)(15,92,104,64,31,135,157,42)(16,81,105,73,32,124,158,51)(17,90,106,62,33,133,159,60)(18,99,107,71,34,122,160,49)(19,88,108,80,35,131,141,58)(20,97,109,69,36,140,142,47), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,81)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(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)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)>;
G:=Group( (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), (1,86,110,78,37,129,143,56)(2,95,111,67,38,138,144,45)(3,84,112,76,39,127,145,54)(4,93,113,65,40,136,146,43)(5,82,114,74,21,125,147,52)(6,91,115,63,22,134,148,41)(7,100,116,72,23,123,149,50)(8,89,117,61,24,132,150,59)(9,98,118,70,25,121,151,48)(10,87,119,79,26,130,152,57)(11,96,120,68,27,139,153,46)(12,85,101,77,28,128,154,55)(13,94,102,66,29,137,155,44)(14,83,103,75,30,126,156,53)(15,92,104,64,31,135,157,42)(16,81,105,73,32,124,158,51)(17,90,106,62,33,133,159,60)(18,99,107,71,34,122,160,49)(19,88,108,80,35,131,141,58)(20,97,109,69,36,140,142,47), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,81)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,84)(59,83)(60,82)(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)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153) );
G=PermutationGroup([[(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)], [(1,86,110,78,37,129,143,56),(2,95,111,67,38,138,144,45),(3,84,112,76,39,127,145,54),(4,93,113,65,40,136,146,43),(5,82,114,74,21,125,147,52),(6,91,115,63,22,134,148,41),(7,100,116,72,23,123,149,50),(8,89,117,61,24,132,150,59),(9,98,118,70,25,121,151,48),(10,87,119,79,26,130,152,57),(11,96,120,68,27,139,153,46),(12,85,101,77,28,128,154,55),(13,94,102,66,29,137,155,44),(14,83,103,75,30,126,156,53),(15,92,104,64,31,135,157,42),(16,81,105,73,32,124,158,51),(17,90,106,62,33,133,159,60),(18,99,107,71,34,122,160,49),(19,88,108,80,35,131,141,58),(20,97,109,69,36,140,142,47)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,81),(42,100),(43,99),(44,98),(45,97),(46,96),(47,95),(48,94),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,85),(58,84),(59,83),(60,82),(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),(101,152),(102,151),(103,150),(104,149),(105,148),(106,147),(107,146),(108,145),(109,144),(110,143),(111,142),(112,141),(113,160),(114,159),(115,158),(116,157),(117,156),(118,155),(119,154),(120,153)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 40 | 40 | 2 | ··· | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D10 | D10 | C5⋊D4 | D4⋊D5 | D4×D5 |
| kernel | C20⋊D8 | C4×C5⋊2C8 | C20⋊4D4 | C2×D4⋊D5 | C5×C4⋊1D4 | C5⋊2C8 | C2×C20 | C4⋊1D4 | C20 | C42 | C2×D4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 8 | 2 | 4 | 8 | 8 | 4 |
Matrix representation of C20⋊D8 ►in GL6(𝔽41)
| 6 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 6 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,34,17,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C20⋊D8 in GAP, Magma, Sage, TeX
C_{20}\rtimes D_8 % in TeX
G:=Group("C20:D8"); // GroupNames label
G:=SmallGroup(320,700);
// by ID
G=gap.SmallGroup(320,700);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations