metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2D8, D20⋊9D4, C42.73D10, C4⋊2(D4⋊D5), C4⋊1D4⋊1D5, C5⋊4(C4⋊D8), C4.53(D4×D5), (C4×D20)⋊22C2, C20⋊3C8⋊31C2, C20.30(C2×D4), C10.57(C2×D8), (C2×D4).55D10, (C2×C20).147D4, C20.76(C4○D4), D4⋊Dic5⋊22C2, C4.3(D4⋊2D5), C2.12(C20⋊2D4), C10.94(C8⋊C22), (C2×C20).390C23, (C4×C20).120C22, (D4×C10).71C22, C10.103(C4⋊D4), (C2×D20).254C22, C4⋊Dic5.344C22, C2.15(D4.D10), (C2×D4⋊D5)⋊13C2, (C5×C4⋊1D4)⋊1C2, C2.12(C2×D4⋊D5), (C2×C10).521(C2×D4), (C2×C4).185(C5⋊D4), (C2×C4).488(C22×D5), C22.194(C2×C5⋊D4), (C2×C5⋊2C8).130C22, SmallGroup(320,699)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊2D8
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >
Subgroups: 582 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊D8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C4×C20, C2×C4×D5, C2×D20, D4×C10, D4×C10, C20⋊3C8, D4⋊Dic5, C4×D20, C2×D4⋊D5, C5×C4⋊1D4, C20⋊2D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C4⋊D8, D4⋊D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C2×D4⋊D5, D4.D10, C20⋊2D4, C20⋊2D8
►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 64 44 139 32 120 95 157)(2 63 45 138 33 119 96 156)(3 62 46 137 34 118 97 155)(4 61 47 136 35 117 98 154)(5 80 48 135 36 116 99 153)(6 79 49 134 37 115 100 152)(7 78 50 133 38 114 81 151)(8 77 51 132 39 113 82 150)(9 76 52 131 40 112 83 149)(10 75 53 130 21 111 84 148)(11 74 54 129 22 110 85 147)(12 73 55 128 23 109 86 146)(13 72 56 127 24 108 87 145)(14 71 57 126 25 107 88 144)(15 70 58 125 26 106 89 143)(16 69 59 124 27 105 90 142)(17 68 60 123 28 104 91 141)(18 67 41 122 29 103 92 160)(19 66 42 121 30 102 93 159)(20 65 43 140 31 101 94 158)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)(41 88)(42 97)(43 86)(44 95)(45 84)(46 93)(47 82)(48 91)(49 100)(50 89)(51 98)(52 87)(53 96)(54 85)(55 94)(56 83)(57 92)(58 81)(59 90)(60 99)(61 150)(62 159)(63 148)(64 157)(65 146)(66 155)(67 144)(68 153)(69 142)(70 151)(71 160)(72 149)(73 158)(74 147)(75 156)(76 145)(77 154)(78 143)(79 152)(80 141)(101 128)(102 137)(103 126)(104 135)(105 124)(106 133)(107 122)(108 131)(109 140)(110 129)(111 138)(112 127)(113 136)(114 125)(115 134)(116 123)(117 132)(118 121)(119 130)(120 139)
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), (1,64,44,139,32,120,95,157)(2,63,45,138,33,119,96,156)(3,62,46,137,34,118,97,155)(4,61,47,136,35,117,98,154)(5,80,48,135,36,116,99,153)(6,79,49,134,37,115,100,152)(7,78,50,133,38,114,81,151)(8,77,51,132,39,113,82,150)(9,76,52,131,40,112,83,149)(10,75,53,130,21,111,84,148)(11,74,54,129,22,110,85,147)(12,73,55,128,23,109,86,146)(13,72,56,127,24,108,87,145)(14,71,57,126,25,107,88,144)(15,70,58,125,26,106,89,143)(16,69,59,124,27,105,90,142)(17,68,60,123,28,104,91,141)(18,67,41,122,29,103,92,160)(19,66,42,121,30,102,93,159)(20,65,43,140,31,101,94,158), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,88)(42,97)(43,86)(44,95)(45,84)(46,93)(47,82)(48,91)(49,100)(50,89)(51,98)(52,87)(53,96)(54,85)(55,94)(56,83)(57,92)(58,81)(59,90)(60,99)(61,150)(62,159)(63,148)(64,157)(65,146)(66,155)(67,144)(68,153)(69,142)(70,151)(71,160)(72,149)(73,158)(74,147)(75,156)(76,145)(77,154)(78,143)(79,152)(80,141)(101,128)(102,137)(103,126)(104,135)(105,124)(106,133)(107,122)(108,131)(109,140)(110,129)(111,138)(112,127)(113,136)(114,125)(115,134)(116,123)(117,132)(118,121)(119,130)(120,139)>;
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), (1,64,44,139,32,120,95,157)(2,63,45,138,33,119,96,156)(3,62,46,137,34,118,97,155)(4,61,47,136,35,117,98,154)(5,80,48,135,36,116,99,153)(6,79,49,134,37,115,100,152)(7,78,50,133,38,114,81,151)(8,77,51,132,39,113,82,150)(9,76,52,131,40,112,83,149)(10,75,53,130,21,111,84,148)(11,74,54,129,22,110,85,147)(12,73,55,128,23,109,86,146)(13,72,56,127,24,108,87,145)(14,71,57,126,25,107,88,144)(15,70,58,125,26,106,89,143)(16,69,59,124,27,105,90,142)(17,68,60,123,28,104,91,141)(18,67,41,122,29,103,92,160)(19,66,42,121,30,102,93,159)(20,65,43,140,31,101,94,158), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,88)(42,97)(43,86)(44,95)(45,84)(46,93)(47,82)(48,91)(49,100)(50,89)(51,98)(52,87)(53,96)(54,85)(55,94)(56,83)(57,92)(58,81)(59,90)(60,99)(61,150)(62,159)(63,148)(64,157)(65,146)(66,155)(67,144)(68,153)(69,142)(70,151)(71,160)(72,149)(73,158)(74,147)(75,156)(76,145)(77,154)(78,143)(79,152)(80,141)(101,128)(102,137)(103,126)(104,135)(105,124)(106,133)(107,122)(108,131)(109,140)(110,129)(111,138)(112,127)(113,136)(114,125)(115,134)(116,123)(117,132)(118,121)(119,130)(120,139) );
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)], [(1,64,44,139,32,120,95,157),(2,63,45,138,33,119,96,156),(3,62,46,137,34,118,97,155),(4,61,47,136,35,117,98,154),(5,80,48,135,36,116,99,153),(6,79,49,134,37,115,100,152),(7,78,50,133,38,114,81,151),(8,77,51,132,39,113,82,150),(9,76,52,131,40,112,83,149),(10,75,53,130,21,111,84,148),(11,74,54,129,22,110,85,147),(12,73,55,128,23,109,86,146),(13,72,56,127,24,108,87,145),(14,71,57,126,25,107,88,144),(15,70,58,125,26,106,89,143),(16,69,59,124,27,105,90,142),(17,68,60,123,28,104,91,141),(18,67,41,122,29,103,92,160),(19,66,42,121,30,102,93,159),(20,65,43,140,31,101,94,158)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39),(41,88),(42,97),(43,86),(44,95),(45,84),(46,93),(47,82),(48,91),(49,100),(50,89),(51,98),(52,87),(53,96),(54,85),(55,94),(56,83),(57,92),(58,81),(59,90),(60,99),(61,150),(62,159),(63,148),(64,157),(65,146),(66,155),(67,144),(68,153),(69,142),(70,151),(71,160),(72,149),(73,158),(74,147),(75,156),(76,145),(77,154),(78,143),(79,152),(80,141),(101,128),(102,137),(103,126),(104,135),(105,124),(106,133),(107,122),(108,131),(109,140),(110,129),(111,138),(112,127),(113,136),(114,125),(115,134),(116,123),(117,132),(118,121),(119,130),(120,139)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊D5 | D4×D5 | D4⋊2D5 | D4.D10 |
| kernel | C20⋊2D8 | C20⋊3C8 | D4⋊Dic5 | C4×D20 | C2×D4⋊D5 | C5×C4⋊1D4 | D20 | C2×C20 | C4⋊1D4 | C20 | C20 | C42 | C2×D4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C20⋊2D8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 33 |
| 0 | 0 | 0 | 0 | 32 | 3 |
| 17 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 5 |
| 0 | 0 | 0 | 0 | 35 | 20 |
| 1 | 21 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 36 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,0,38,32,0,0,0,0,33,3],[17,7,0,0,0,0,35,0,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,21,35,0,0,0,0,5,20],[1,0,0,0,0,0,21,40,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C20⋊2D8 in GAP, Magma, Sage, TeX
C_{20}\rtimes_2D_8 % in TeX
G:=Group("C20:2D8"); // GroupNames label
G:=SmallGroup(320,699);
// by ID
G=gap.SmallGroup(320,699);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations