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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C20⋊3C8, C10.Q16, C20⋊2Q8, C2×D4.D5, D4×C20, D4.2D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, D20, C5⋊D4, C22×D5, D4.D4, D4.D5, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, C2×D4.D5, D4.9D10, D4.2D20
►On 160 points
Generators in S160
(1 72 36 106)(2 73 37 107)(3 74 38 108)(4 75 39 109)(5 76 40 110)(6 77 21 111)(7 78 22 112)(8 79 23 113)(9 80 24 114)(10 61 25 115)(11 62 26 116)(12 63 27 117)(13 64 28 118)(14 65 29 119)(15 66 30 120)(16 67 31 101)(17 68 32 102)(18 69 33 103)(19 70 34 104)(20 71 35 105)(41 142 87 125)(42 143 88 126)(43 144 89 127)(44 145 90 128)(45 146 91 129)(46 147 92 130)(47 148 93 131)(48 149 94 132)(49 150 95 133)(50 151 96 134)(51 152 97 135)(52 153 98 136)(53 154 99 137)(54 155 100 138)(55 156 81 139)(56 157 82 140)(57 158 83 121)(58 159 84 122)(59 160 85 123)(60 141 86 124)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 97)(42 98)(43 99)(44 100)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(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 43 36 89)(2 42 37 88)(3 41 38 87)(4 60 39 86)(5 59 40 85)(6 58 21 84)(7 57 22 83)(8 56 23 82)(9 55 24 81)(10 54 25 100)(11 53 26 99)(12 52 27 98)(13 51 28 97)(14 50 29 96)(15 49 30 95)(16 48 31 94)(17 47 32 93)(18 46 33 92)(19 45 34 91)(20 44 35 90)(61 138 115 155)(62 137 116 154)(63 136 117 153)(64 135 118 152)(65 134 119 151)(66 133 120 150)(67 132 101 149)(68 131 102 148)(69 130 103 147)(70 129 104 146)(71 128 105 145)(72 127 106 144)(73 126 107 143)(74 125 108 142)(75 124 109 141)(76 123 110 160)(77 122 111 159)(78 121 112 158)(79 140 113 157)(80 139 114 156)
G:=sub<Sym(160)| (1,72,36,106)(2,73,37,107)(3,74,38,108)(4,75,39,109)(5,76,40,110)(6,77,21,111)(7,78,22,112)(8,79,23,113)(9,80,24,114)(10,61,25,115)(11,62,26,116)(12,63,27,117)(13,64,28,118)(14,65,29,119)(15,66,30,120)(16,67,31,101)(17,68,32,102)(18,69,33,103)(19,70,34,104)(20,71,35,105)(41,142,87,125)(42,143,88,126)(43,144,89,127)(44,145,90,128)(45,146,91,129)(46,147,92,130)(47,148,93,131)(48,149,94,132)(49,150,95,133)(50,151,96,134)(51,152,97,135)(52,153,98,136)(53,154,99,137)(54,155,100,138)(55,156,81,139)(56,157,82,140)(57,158,83,121)(58,159,84,122)(59,160,85,123)(60,141,86,124), (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,97)(42,98)(43,99)(44,100)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (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), (1,43,36,89)(2,42,37,88)(3,41,38,87)(4,60,39,86)(5,59,40,85)(6,58,21,84)(7,57,22,83)(8,56,23,82)(9,55,24,81)(10,54,25,100)(11,53,26,99)(12,52,27,98)(13,51,28,97)(14,50,29,96)(15,49,30,95)(16,48,31,94)(17,47,32,93)(18,46,33,92)(19,45,34,91)(20,44,35,90)(61,138,115,155)(62,137,116,154)(63,136,117,153)(64,135,118,152)(65,134,119,151)(66,133,120,150)(67,132,101,149)(68,131,102,148)(69,130,103,147)(70,129,104,146)(71,128,105,145)(72,127,106,144)(73,126,107,143)(74,125,108,142)(75,124,109,141)(76,123,110,160)(77,122,111,159)(78,121,112,158)(79,140,113,157)(80,139,114,156)>;
G:=Group( (1,72,36,106)(2,73,37,107)(3,74,38,108)(4,75,39,109)(5,76,40,110)(6,77,21,111)(7,78,22,112)(8,79,23,113)(9,80,24,114)(10,61,25,115)(11,62,26,116)(12,63,27,117)(13,64,28,118)(14,65,29,119)(15,66,30,120)(16,67,31,101)(17,68,32,102)(18,69,33,103)(19,70,34,104)(20,71,35,105)(41,142,87,125)(42,143,88,126)(43,144,89,127)(44,145,90,128)(45,146,91,129)(46,147,92,130)(47,148,93,131)(48,149,94,132)(49,150,95,133)(50,151,96,134)(51,152,97,135)(52,153,98,136)(53,154,99,137)(54,155,100,138)(55,156,81,139)(56,157,82,140)(57,158,83,121)(58,159,84,122)(59,160,85,123)(60,141,86,124), (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,97)(42,98)(43,99)(44,100)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (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), (1,43,36,89)(2,42,37,88)(3,41,38,87)(4,60,39,86)(5,59,40,85)(6,58,21,84)(7,57,22,83)(8,56,23,82)(9,55,24,81)(10,54,25,100)(11,53,26,99)(12,52,27,98)(13,51,28,97)(14,50,29,96)(15,49,30,95)(16,48,31,94)(17,47,32,93)(18,46,33,92)(19,45,34,91)(20,44,35,90)(61,138,115,155)(62,137,116,154)(63,136,117,153)(64,135,118,152)(65,134,119,151)(66,133,120,150)(67,132,101,149)(68,131,102,148)(69,130,103,147)(70,129,104,146)(71,128,105,145)(72,127,106,144)(73,126,107,143)(74,125,108,142)(75,124,109,141)(76,123,110,160)(77,122,111,159)(78,121,112,158)(79,140,113,157)(80,139,114,156) );
G=PermutationGroup([[(1,72,36,106),(2,73,37,107),(3,74,38,108),(4,75,39,109),(5,76,40,110),(6,77,21,111),(7,78,22,112),(8,79,23,113),(9,80,24,114),(10,61,25,115),(11,62,26,116),(12,63,27,117),(13,64,28,118),(14,65,29,119),(15,66,30,120),(16,67,31,101),(17,68,32,102),(18,69,33,103),(19,70,34,104),(20,71,35,105),(41,142,87,125),(42,143,88,126),(43,144,89,127),(44,145,90,128),(45,146,91,129),(46,147,92,130),(47,148,93,131),(48,149,94,132),(49,150,95,133),(50,151,96,134),(51,152,97,135),(52,153,98,136),(53,154,99,137),(54,155,100,138),(55,156,81,139),(56,157,82,140),(57,158,83,121),(58,159,84,122),(59,160,85,123),(60,141,86,124)], [(1,116),(2,117),(3,118),(4,119),(5,120),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,97),(42,98),(43,99),(44,100),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(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)], [(1,43,36,89),(2,42,37,88),(3,41,38,87),(4,60,39,86),(5,59,40,85),(6,58,21,84),(7,57,22,83),(8,56,23,82),(9,55,24,81),(10,54,25,100),(11,53,26,99),(12,52,27,98),(13,51,28,97),(14,50,29,96),(15,49,30,95),(16,48,31,94),(17,47,32,93),(18,46,33,92),(19,45,34,91),(20,44,35,90),(61,138,115,155),(62,137,116,154),(63,136,117,153),(64,135,118,152),(65,134,119,151),(66,133,120,150),(67,132,101,149),(68,131,102,148),(69,130,103,147),(70,129,104,146),(71,128,105,145),(72,127,106,144),(73,126,107,143),(74,125,108,142),(75,124,109,141),(76,123,110,160),(77,122,111,159),(78,121,112,158),(79,140,113,157),(80,139,114,156)]])
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