metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.2Q8, C4.8(Q8×D5), C2.D8⋊9D5, C5⋊5(D4.Q8), (C2×C8).32D10, C20.24(C2×Q8), C4⋊C4.174D10, C4.Dic10⋊7C2, D20⋊6C4.9C2, D20⋊8C4.9C2, C4.85(C4○D20), C10.77(C4○D8), C20.Q8⋊23C2, C20.8Q8⋊26C2, (C2×Dic5).60D4, C22.236(D4×D5), D20⋊5C4.11C2, C20.173(C4○D4), C2.25(D8⋊D5), C10.45(C8⋊C22), (C2×C40).246C22, (C2×C20).307C23, (C2×D20).91C22, C10.42(C22⋊Q8), C2.15(Q8.D10), C2.19(D10⋊Q8), C4⋊Dic5.129C22, (C4×Dic5).45C22, (C5×C2.D8)⋊16C2, (C2×C10).312(C2×D4), (C5×C4⋊C4).100C22, (C2×C5⋊2C8).76C22, (C2×C4).410(C22×D5), SmallGroup(320,518)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.2Q8
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=a10c3 >
Subgroups: 454 in 102 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D4.Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C20.Q8, D20⋊6C4, C20.8Q8, D20⋊5C4, C5×C2.D8, C4.Dic10, D20⋊8C4, D20.2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, C22×D5, D4.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D8⋊D5, Q8.D10, D20.2Q8
►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 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 100)(20 99)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 160)(29 159)(30 158)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)
(1 49 99 137 11 59 89 127)(2 50 100 138 12 60 90 128)(3 51 81 139 13 41 91 129)(4 52 82 140 14 42 92 130)(5 53 83 121 15 43 93 131)(6 54 84 122 16 44 94 132)(7 55 85 123 17 45 95 133)(8 56 86 124 18 46 96 134)(9 57 87 125 19 47 97 135)(10 58 88 126 20 48 98 136)(21 118 143 66 31 108 153 76)(22 119 144 67 32 109 154 77)(23 120 145 68 33 110 155 78)(24 101 146 69 34 111 156 79)(25 102 147 70 35 112 157 80)(26 103 148 71 36 113 158 61)(27 104 149 72 37 114 159 62)(28 105 150 73 38 115 160 63)(29 106 151 74 39 116 141 64)(30 107 152 75 40 117 142 65)
(1 76 94 113)(2 67 95 104)(3 78 96 115)(4 69 97 106)(5 80 98 117)(6 71 99 108)(7 62 100 119)(8 73 81 110)(9 64 82 101)(10 75 83 112)(11 66 84 103)(12 77 85 114)(13 68 86 105)(14 79 87 116)(15 70 88 107)(16 61 89 118)(17 72 90 109)(18 63 91 120)(19 74 92 111)(20 65 93 102)(21 44 158 127)(22 55 159 138)(23 46 160 129)(24 57 141 140)(25 48 142 131)(26 59 143 122)(27 50 144 133)(28 41 145 124)(29 52 146 135)(30 43 147 126)(31 54 148 137)(32 45 149 128)(33 56 150 139)(34 47 151 130)(35 58 152 121)(36 49 153 132)(37 60 154 123)(38 51 155 134)(39 42 156 125)(40 53 157 136)
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), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,160)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113), (1,49,99,137,11,59,89,127)(2,50,100,138,12,60,90,128)(3,51,81,139,13,41,91,129)(4,52,82,140,14,42,92,130)(5,53,83,121,15,43,93,131)(6,54,84,122,16,44,94,132)(7,55,85,123,17,45,95,133)(8,56,86,124,18,46,96,134)(9,57,87,125,19,47,97,135)(10,58,88,126,20,48,98,136)(21,118,143,66,31,108,153,76)(22,119,144,67,32,109,154,77)(23,120,145,68,33,110,155,78)(24,101,146,69,34,111,156,79)(25,102,147,70,35,112,157,80)(26,103,148,71,36,113,158,61)(27,104,149,72,37,114,159,62)(28,105,150,73,38,115,160,63)(29,106,151,74,39,116,141,64)(30,107,152,75,40,117,142,65), (1,76,94,113)(2,67,95,104)(3,78,96,115)(4,69,97,106)(5,80,98,117)(6,71,99,108)(7,62,100,119)(8,73,81,110)(9,64,82,101)(10,75,83,112)(11,66,84,103)(12,77,85,114)(13,68,86,105)(14,79,87,116)(15,70,88,107)(16,61,89,118)(17,72,90,109)(18,63,91,120)(19,74,92,111)(20,65,93,102)(21,44,158,127)(22,55,159,138)(23,46,160,129)(24,57,141,140)(25,48,142,131)(26,59,143,122)(27,50,144,133)(28,41,145,124)(29,52,146,135)(30,43,147,126)(31,54,148,137)(32,45,149,128)(33,56,150,139)(34,47,151,130)(35,58,152,121)(36,49,153,132)(37,60,154,123)(38,51,155,134)(39,42,156,125)(40,53,157,136)>;
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), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,160)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113), (1,49,99,137,11,59,89,127)(2,50,100,138,12,60,90,128)(3,51,81,139,13,41,91,129)(4,52,82,140,14,42,92,130)(5,53,83,121,15,43,93,131)(6,54,84,122,16,44,94,132)(7,55,85,123,17,45,95,133)(8,56,86,124,18,46,96,134)(9,57,87,125,19,47,97,135)(10,58,88,126,20,48,98,136)(21,118,143,66,31,108,153,76)(22,119,144,67,32,109,154,77)(23,120,145,68,33,110,155,78)(24,101,146,69,34,111,156,79)(25,102,147,70,35,112,157,80)(26,103,148,71,36,113,158,61)(27,104,149,72,37,114,159,62)(28,105,150,73,38,115,160,63)(29,106,151,74,39,116,141,64)(30,107,152,75,40,117,142,65), (1,76,94,113)(2,67,95,104)(3,78,96,115)(4,69,97,106)(5,80,98,117)(6,71,99,108)(7,62,100,119)(8,73,81,110)(9,64,82,101)(10,75,83,112)(11,66,84,103)(12,77,85,114)(13,68,86,105)(14,79,87,116)(15,70,88,107)(16,61,89,118)(17,72,90,109)(18,63,91,120)(19,74,92,111)(20,65,93,102)(21,44,158,127)(22,55,159,138)(23,46,160,129)(24,57,141,140)(25,48,142,131)(26,59,143,122)(27,50,144,133)(28,41,145,124)(29,52,146,135)(30,43,147,126)(31,54,148,137)(32,45,149,128)(33,56,150,139)(34,47,151,130)(35,58,152,121)(36,49,153,132)(37,60,154,123)(38,51,155,134)(39,42,156,125)(40,53,157,136) );
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)], [(1,98),(2,97),(3,96),(4,95),(5,94),(6,93),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,100),(20,99),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,160),(29,159),(30,158),(31,157),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,133),(58,132),(59,131),(60,130),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113)], [(1,49,99,137,11,59,89,127),(2,50,100,138,12,60,90,128),(3,51,81,139,13,41,91,129),(4,52,82,140,14,42,92,130),(5,53,83,121,15,43,93,131),(6,54,84,122,16,44,94,132),(7,55,85,123,17,45,95,133),(8,56,86,124,18,46,96,134),(9,57,87,125,19,47,97,135),(10,58,88,126,20,48,98,136),(21,118,143,66,31,108,153,76),(22,119,144,67,32,109,154,77),(23,120,145,68,33,110,155,78),(24,101,146,69,34,111,156,79),(25,102,147,70,35,112,157,80),(26,103,148,71,36,113,158,61),(27,104,149,72,37,114,159,62),(28,105,150,73,38,115,160,63),(29,106,151,74,39,116,141,64),(30,107,152,75,40,117,142,65)], [(1,76,94,113),(2,67,95,104),(3,78,96,115),(4,69,97,106),(5,80,98,117),(6,71,99,108),(7,62,100,119),(8,73,81,110),(9,64,82,101),(10,75,83,112),(11,66,84,103),(12,77,85,114),(13,68,86,105),(14,79,87,116),(15,70,88,107),(16,61,89,118),(17,72,90,109),(18,63,91,120),(19,74,92,111),(20,65,93,102),(21,44,158,127),(22,55,159,138),(23,46,160,129),(24,57,141,140),(25,48,142,131),(26,59,143,122),(27,50,144,133),(28,41,145,124),(29,52,146,135),(30,43,147,126),(31,54,148,137),(32,45,149,128),(33,56,150,139),(34,47,151,130),(35,58,152,121),(36,49,153,132),(37,60,154,123),(38,51,155,134),(39,42,156,125),(40,53,157,136)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | Q8×D5 | D4×D5 | D8⋊D5 | Q8.D10 |
| kernel | D20.2Q8 | C20.Q8 | D20⋊6C4 | C20.8Q8 | D20⋊5C4 | C5×C2.D8 | C4.Dic10 | D20⋊8C4 | D20 | C2×Dic5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.2Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 20 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 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 | 6 |
| 0 | 0 | 0 | 0 | 34 | 17 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 16 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,20,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,0,0,0,0,6,17],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,16,9] >;
D20.2Q8 in GAP, Magma, Sage, TeX
D_{20}._2Q_8 % in TeX
G:=Group("D20.2Q8"); // GroupNames label
G:=SmallGroup(320,518);
// by ID
G=gap.SmallGroup(320,518);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,100,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^10*c^3>;
// generators/relations