metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.2D20, C20⋊10SD16, C42.52D10, (C4×D4).9D5, C4⋊3(D4.D5), C20⋊3C8⋊25C2, (C5×D4).19D4, C4.15(C2×D20), C20.19(C2×D4), (C2×C20).62D4, C4⋊C4.247D10, C20⋊2Q8⋊16C2, (D4×C20).10C2, C5⋊3(D4.D4), (C2×D4).194D10, C20.54(C4○D4), C4.11(C4○D20), C10.Q16⋊31C2, (C4×C20).90C22, C10.53(C2×SD16), C10.66(C4⋊D4), C2.14(C20⋊7D4), (C2×C20).341C23, C2.8(D4.9D10), (D4×C10).236C22, C10.109(C8.C22), (C2×Dic10).104C22, C2.7(C2×D4.D5), (C2×D4.D5).5C2, (C2×C10).472(C2×D4), (C2×C4).246(C5⋊D4), (C5×C4⋊C4).278C22, (C2×C5⋊2C8).96C22, (C2×C4).441(C22×D5), C22.151(C2×C5⋊D4), SmallGroup(320,646)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.2D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 406 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C5⋊2C8 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4.D4, C2×C5⋊2C8 [×2], C4⋊Dic5 [×2], D4.D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10 [×2], C22×C20, D4×C10, C20⋊3C8, C10.Q16 [×2], C20⋊2Q8, C2×D4.D5 [×2], D4×C20, D4.2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, D20 [×2], C5⋊D4 [×2], C22×D5, D4.D4, D4.D5 [×2], C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, C2×D4.D5, D4.9D10, D4.2D20
►On 160 points
Generators in S160
(1 105 56 152)(2 106 57 153)(3 107 58 154)(4 108 59 155)(5 109 60 156)(6 110 41 157)(7 111 42 158)(8 112 43 159)(9 113 44 160)(10 114 45 141)(11 115 46 142)(12 116 47 143)(13 117 48 144)(14 118 49 145)(15 119 50 146)(16 120 51 147)(17 101 52 148)(18 102 53 149)(19 103 54 150)(20 104 55 151)(21 129 83 75)(22 130 84 76)(23 131 85 77)(24 132 86 78)(25 133 87 79)(26 134 88 80)(27 135 89 61)(28 136 90 62)(29 137 91 63)(30 138 92 64)(31 139 93 65)(32 140 94 66)(33 121 95 67)(34 122 96 68)(35 123 97 69)(36 124 98 70)(37 125 99 71)(38 126 100 72)(39 127 81 73)(40 128 82 74)
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 141)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 120)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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 33 56 95)(2 32 57 94)(3 31 58 93)(4 30 59 92)(5 29 60 91)(6 28 41 90)(7 27 42 89)(8 26 43 88)(9 25 44 87)(10 24 45 86)(11 23 46 85)(12 22 47 84)(13 21 48 83)(14 40 49 82)(15 39 50 81)(16 38 51 100)(17 37 52 99)(18 36 53 98)(19 35 54 97)(20 34 55 96)(61 158 135 111)(62 157 136 110)(63 156 137 109)(64 155 138 108)(65 154 139 107)(66 153 140 106)(67 152 121 105)(68 151 122 104)(69 150 123 103)(70 149 124 102)(71 148 125 101)(72 147 126 120)(73 146 127 119)(74 145 128 118)(75 144 129 117)(76 143 130 116)(77 142 131 115)(78 141 132 114)(79 160 133 113)(80 159 134 112)
G:=sub<Sym(160)| (1,105,56,152)(2,106,57,153)(3,107,58,154)(4,108,59,155)(5,109,60,156)(6,110,41,157)(7,111,42,158)(8,112,43,159)(9,113,44,160)(10,114,45,141)(11,115,46,142)(12,116,47,143)(13,117,48,144)(14,118,49,145)(15,119,50,146)(16,120,51,147)(17,101,52,148)(18,102,53,149)(19,103,54,150)(20,104,55,151)(21,129,83,75)(22,130,84,76)(23,131,85,77)(24,132,86,78)(25,133,87,79)(26,134,88,80)(27,135,89,61)(28,136,90,62)(29,137,91,63)(30,138,92,64)(31,139,93,65)(32,140,94,66)(33,121,95,67)(34,122,96,68)(35,123,97,69)(36,124,98,70)(37,125,99,71)(38,126,100,72)(39,127,81,73)(40,128,82,74), (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,141)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,120)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,33,56,95)(2,32,57,94)(3,31,58,93)(4,30,59,92)(5,29,60,91)(6,28,41,90)(7,27,42,89)(8,26,43,88)(9,25,44,87)(10,24,45,86)(11,23,46,85)(12,22,47,84)(13,21,48,83)(14,40,49,82)(15,39,50,81)(16,38,51,100)(17,37,52,99)(18,36,53,98)(19,35,54,97)(20,34,55,96)(61,158,135,111)(62,157,136,110)(63,156,137,109)(64,155,138,108)(65,154,139,107)(66,153,140,106)(67,152,121,105)(68,151,122,104)(69,150,123,103)(70,149,124,102)(71,148,125,101)(72,147,126,120)(73,146,127,119)(74,145,128,118)(75,144,129,117)(76,143,130,116)(77,142,131,115)(78,141,132,114)(79,160,133,113)(80,159,134,112)>;
G:=Group( (1,105,56,152)(2,106,57,153)(3,107,58,154)(4,108,59,155)(5,109,60,156)(6,110,41,157)(7,111,42,158)(8,112,43,159)(9,113,44,160)(10,114,45,141)(11,115,46,142)(12,116,47,143)(13,117,48,144)(14,118,49,145)(15,119,50,146)(16,120,51,147)(17,101,52,148)(18,102,53,149)(19,103,54,150)(20,104,55,151)(21,129,83,75)(22,130,84,76)(23,131,85,77)(24,132,86,78)(25,133,87,79)(26,134,88,80)(27,135,89,61)(28,136,90,62)(29,137,91,63)(30,138,92,64)(31,139,93,65)(32,140,94,66)(33,121,95,67)(34,122,96,68)(35,123,97,69)(36,124,98,70)(37,125,99,71)(38,126,100,72)(39,127,81,73)(40,128,82,74), (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,141)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,120)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,33,56,95)(2,32,57,94)(3,31,58,93)(4,30,59,92)(5,29,60,91)(6,28,41,90)(7,27,42,89)(8,26,43,88)(9,25,44,87)(10,24,45,86)(11,23,46,85)(12,22,47,84)(13,21,48,83)(14,40,49,82)(15,39,50,81)(16,38,51,100)(17,37,52,99)(18,36,53,98)(19,35,54,97)(20,34,55,96)(61,158,135,111)(62,157,136,110)(63,156,137,109)(64,155,138,108)(65,154,139,107)(66,153,140,106)(67,152,121,105)(68,151,122,104)(69,150,123,103)(70,149,124,102)(71,148,125,101)(72,147,126,120)(73,146,127,119)(74,145,128,118)(75,144,129,117)(76,143,130,116)(77,142,131,115)(78,141,132,114)(79,160,133,113)(80,159,134,112) );
G=PermutationGroup([(1,105,56,152),(2,106,57,153),(3,107,58,154),(4,108,59,155),(5,109,60,156),(6,110,41,157),(7,111,42,158),(8,112,43,159),(9,113,44,160),(10,114,45,141),(11,115,46,142),(12,116,47,143),(13,117,48,144),(14,118,49,145),(15,119,50,146),(16,120,51,147),(17,101,52,148),(18,102,53,149),(19,103,54,150),(20,104,55,151),(21,129,83,75),(22,130,84,76),(23,131,85,77),(24,132,86,78),(25,133,87,79),(26,134,88,80),(27,135,89,61),(28,136,90,62),(29,137,91,63),(30,138,92,64),(31,139,93,65),(32,140,94,66),(33,121,95,67),(34,122,96,68),(35,123,97,69),(36,124,98,70),(37,125,99,71),(38,126,100,72),(39,127,81,73),(40,128,82,74)], [(1,142),(2,143),(3,144),(4,145),(5,146),(6,147),(7,148),(8,149),(9,150),(10,151),(11,152),(12,153),(13,154),(14,155),(15,156),(16,157),(17,158),(18,159),(19,160),(20,141),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,120),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140)], [(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)], [(1,33,56,95),(2,32,57,94),(3,31,58,93),(4,30,59,92),(5,29,60,91),(6,28,41,90),(7,27,42,89),(8,26,43,88),(9,25,44,87),(10,24,45,86),(11,23,46,85),(12,22,47,84),(13,21,48,83),(14,40,49,82),(15,39,50,81),(16,38,51,100),(17,37,52,99),(18,36,53,98),(19,35,54,97),(20,34,55,96),(61,158,135,111),(62,157,136,110),(63,156,137,109),(64,155,138,108),(65,154,139,107),(66,153,140,106),(67,152,121,105),(68,151,122,104),(69,150,123,103),(70,149,124,102),(71,148,125,101),(72,147,126,120),(73,146,127,119),(74,145,128,118),(75,144,129,117),(76,143,130,116),(77,142,131,115),(78,141,132,114),(79,160,133,113),(80,159,134,112)])
59 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 | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | C8.C22 | D4.D5 | D4.9D10 |
| kernel | D4.2D20 | C20⋊3C8 | C10.Q16 | C20⋊2Q8 | C2×D4.D5 | D4×C20 | C2×C20 | C5×D4 | C4×D4 | C20 | C20 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of D4.2D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 22 |
| 0 | 0 | 0 | 0 | 24 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 39 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 22 |
| 0 | 0 | 0 | 0 | 39 | 39 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 39 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 23 | 25 | 0 | 0 | 0 | 0 |
| 33 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 13 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 |
| 0 | 0 | 0 | 0 | 10 | 26 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,24,0,0,0,0,22,39],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,0,0,0,0,2,39,0,0,0,0,22,39],[32,10,0,0,0,0,0,9,0,0,0,0,0,0,16,39,0,0,0,0,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[23,33,0,0,0,0,25,18,0,0,0,0,0,0,28,6,0,0,0,0,13,13,0,0,0,0,0,0,15,10,0,0,0,0,2,26] >;
D4.2D20 in GAP, Magma, Sage, TeX
D_4._2D_{20} % in TeX
G:=Group("D4.2D20"); // GroupNames label
G:=SmallGroup(320,646);
// by ID
G=gap.SmallGroup(320,646);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations