metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊4D4, Q8⋊3D20, (C5×Q8)⋊2D4, (C2×D40)⋊6C2, C4.8(C2×D20), C4.95(D4×D5), C4⋊D20⋊5C2, C5⋊3(D4⋊D4), C4⋊C4.27D10, Q8⋊C4⋊6D5, (C2×C8).18D10, D10⋊1C8⋊8C2, C20.124(C2×D4), D20⋊6C4⋊12C2, C10.26C22≀C2, C10.70(C4○D8), (C2×C40).18C22, (C2×Q8).110D10, (C22×D5).26D4, C22.201(D4×D5), C2.9(Q8.D10), C2.17(D40⋊C2), C10.63(C8⋊C22), (C2×C20).251C23, (C2×Dic5).211D4, (C2×D20).69C22, (Q8×C10).34C22, C2.29(C22⋊D20), (C2×Q8⋊D5)⋊4C2, (C5×Q8⋊C4)⋊6C2, (C2×Q8⋊2D5)⋊1C2, (C2×C4×D5).28C22, (C2×C10).264(C2×D4), (C5×C4⋊C4).52C22, (C2×C5⋊2C8).42C22, (C2×C4).358(C22×D5), SmallGroup(320,438)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊4D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a11, cbc-1=a5b, dbd=a3b, dcd=c-1 >
Subgroups: 846 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, D4⋊D4, D40, C2×C5⋊2C8, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8×C10, D20⋊6C4, D10⋊1C8, C5×Q8⋊C4, C4⋊D20, C2×D40, C2×Q8⋊D5, C2×Q8⋊2D5, D20⋊4D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, D20, C22×D5, D4⋊D4, C2×D20, D4×D5, C22⋊D20, D40⋊C2, Q8.D10, D20⋊4D4
►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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 120)(21 83)(22 82)(23 81)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 129)(62 128)(63 127)(64 126)(65 125)(66 124)(67 123)(68 122)(69 121)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)
(1 84 124 155)(2 95 125 146)(3 86 126 157)(4 97 127 148)(5 88 128 159)(6 99 129 150)(7 90 130 141)(8 81 131 152)(9 92 132 143)(10 83 133 154)(11 94 134 145)(12 85 135 156)(13 96 136 147)(14 87 137 158)(15 98 138 149)(16 89 139 160)(17 100 140 151)(18 91 121 142)(19 82 122 153)(20 93 123 144)(21 72 52 105)(22 63 53 116)(23 74 54 107)(24 65 55 118)(25 76 56 109)(26 67 57 120)(27 78 58 111)(28 69 59 102)(29 80 60 113)(30 71 41 104)(31 62 42 115)(32 73 43 106)(33 64 44 117)(34 75 45 108)(35 66 46 119)(36 77 47 110)(37 68 48 101)(38 79 49 112)(39 70 50 103)(40 61 51 114)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 160)(100 159)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)
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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,120)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,129)(62,128)(63,127)(64,126)(65,125)(66,124)(67,123)(68,122)(69,121)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130), (1,84,124,155)(2,95,125,146)(3,86,126,157)(4,97,127,148)(5,88,128,159)(6,99,129,150)(7,90,130,141)(8,81,131,152)(9,92,132,143)(10,83,133,154)(11,94,134,145)(12,85,135,156)(13,96,136,147)(14,87,137,158)(15,98,138,149)(16,89,139,160)(17,100,140,151)(18,91,121,142)(19,82,122,153)(20,93,123,144)(21,72,52,105)(22,63,53,116)(23,74,54,107)(24,65,55,118)(25,76,56,109)(26,67,57,120)(27,78,58,111)(28,69,59,102)(29,80,60,113)(30,71,41,104)(31,62,42,115)(32,73,43,106)(33,64,44,117)(34,75,45,108)(35,66,46,119)(36,77,47,110)(37,68,48,101)(38,79,49,112)(39,70,50,103)(40,61,51,114), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,160)(100,159)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)>;
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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,120)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,129)(62,128)(63,127)(64,126)(65,125)(66,124)(67,123)(68,122)(69,121)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130), (1,84,124,155)(2,95,125,146)(3,86,126,157)(4,97,127,148)(5,88,128,159)(6,99,129,150)(7,90,130,141)(8,81,131,152)(9,92,132,143)(10,83,133,154)(11,94,134,145)(12,85,135,156)(13,96,136,147)(14,87,137,158)(15,98,138,149)(16,89,139,160)(17,100,140,151)(18,91,121,142)(19,82,122,153)(20,93,123,144)(21,72,52,105)(22,63,53,116)(23,74,54,107)(24,65,55,118)(25,76,56,109)(26,67,57,120)(27,78,58,111)(28,69,59,102)(29,80,60,113)(30,71,41,104)(31,62,42,115)(32,73,43,106)(33,64,44,117)(34,75,45,108)(35,66,46,119)(36,77,47,110)(37,68,48,101)(38,79,49,112)(39,70,50,103)(40,61,51,114), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,160)(100,159)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135) );
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)], [(1,119),(2,118),(3,117),(4,116),(5,115),(6,114),(7,113),(8,112),(9,111),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,120),(21,83),(22,82),(23,81),(24,100),(25,99),(26,98),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,145),(42,144),(43,143),(44,142),(45,141),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,149),(58,148),(59,147),(60,146),(61,129),(62,128),(63,127),(64,126),(65,125),(66,124),(67,123),(68,122),(69,121),(70,140),(71,139),(72,138),(73,137),(74,136),(75,135),(76,134),(77,133),(78,132),(79,131),(80,130)], [(1,84,124,155),(2,95,125,146),(3,86,126,157),(4,97,127,148),(5,88,128,159),(6,99,129,150),(7,90,130,141),(8,81,131,152),(9,92,132,143),(10,83,133,154),(11,94,134,145),(12,85,135,156),(13,96,136,147),(14,87,137,158),(15,98,138,149),(16,89,139,160),(17,100,140,151),(18,91,121,142),(19,82,122,153),(20,93,123,144),(21,72,52,105),(22,63,53,116),(23,74,54,107),(24,65,55,118),(25,76,56,109),(26,67,57,120),(27,78,58,111),(28,69,59,102),(29,80,60,113),(30,71,41,104),(31,62,42,115),(32,73,43,106),(33,64,44,117),(34,75,45,108),(35,66,46,119),(36,77,47,110),(37,68,48,101),(38,79,49,112),(39,70,50,103),(40,61,51,114)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(81,158),(82,157),(83,156),(84,155),(85,154),(86,153),(87,152),(88,151),(89,150),(90,149),(91,148),(92,147),(93,146),(94,145),(95,144),(96,143),(97,142),(98,141),(99,160),(100,159),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135)]])
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 | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | D20 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 | Q8.D10 |
| kernel | D20⋊4D4 | D20⋊6C4 | D10⋊1C8 | C5×Q8⋊C4 | C4⋊D20 | C2×D40 | C2×Q8⋊D5 | C2×Q8⋊2D5 | D20 | C2×Dic5 | C5×Q8 | C22×D5 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20⋊4D4 ►in GL4(𝔽41) generated by
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 7 |
| 0 | 17 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 9 |
| 9 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 |
| 0 | 0 | 30 | 32 |
| 0 | 0 | 9 | 11 |
| 40 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 40 |
G:=sub<GL(4,GF(41))| [1,1,0,0,39,40,0,0,0,0,0,1,0,0,40,7],[0,29,0,0,17,0,0,0,0,0,32,11,0,0,30,9],[9,9,0,0,0,32,0,0,0,0,30,9,0,0,32,11],[40,0,0,0,2,1,0,0,0,0,1,34,0,0,0,40] >;
D20⋊4D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_4D_4 % in TeX
G:=Group("D20:4D4"); // GroupNames label
G:=SmallGroup(320,438);
// by ID
G=gap.SmallGroup(320,438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations