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 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×11], Q8 [×2], Q8, C23 [×3], D5 [×4], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5, C20 [×2], C20 [×3], D10 [×10], C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, C4×D5 [×6], D20 [×2], D20 [×9], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×2], D4⋊D4, D40 [×2], C2×C5⋊2C8, D10⋊C4, Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20 [×2], C2×D20 [×2], Q8⋊2D5 [×4], Q8×C10, D20⋊6C4, D10⋊1C8, C5×Q8⋊C4, C4⋊D20, C2×D40, C2×Q8⋊D5, C2×Q8⋊2D5, D20⋊4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, D20 [×2], C22×D5, D4⋊D4, C2×D20, D4×D5 [×2], 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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 140)(9 139)(10 138)(11 137)(12 136)(13 135)(14 134)(15 133)(16 132)(17 131)(18 130)(19 129)(20 128)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)
(1 148 70 58)(2 159 71 49)(3 150 72 60)(4 141 73 51)(5 152 74 42)(6 143 75 53)(7 154 76 44)(8 145 77 55)(9 156 78 46)(10 147 79 57)(11 158 80 48)(12 149 61 59)(13 160 62 50)(14 151 63 41)(15 142 64 52)(16 153 65 43)(17 144 66 54)(18 155 67 45)(19 146 68 56)(20 157 69 47)(21 126 106 100)(22 137 107 91)(23 128 108 82)(24 139 109 93)(25 130 110 84)(26 121 111 95)(27 132 112 86)(28 123 113 97)(29 134 114 88)(30 125 115 99)(31 136 116 90)(32 127 117 81)(33 138 118 92)(34 129 119 83)(35 140 120 94)(36 131 101 85)(37 122 102 96)(38 133 103 87)(39 124 104 98)(40 135 105 89)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 120)(37 119)(38 118)(39 117)(40 116)(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 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)
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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150), (1,148,70,58)(2,159,71,49)(3,150,72,60)(4,141,73,51)(5,152,74,42)(6,143,75,53)(7,154,76,44)(8,145,77,55)(9,156,78,46)(10,147,79,57)(11,158,80,48)(12,149,61,59)(13,160,62,50)(14,151,63,41)(15,142,64,52)(16,153,65,43)(17,144,66,54)(18,155,67,45)(19,146,68,56)(20,157,69,47)(21,126,106,100)(22,137,107,91)(23,128,108,82)(24,139,109,93)(25,130,110,84)(26,121,111,95)(27,132,112,86)(28,123,113,97)(29,134,114,88)(30,125,115,99)(31,136,116,90)(32,127,117,81)(33,138,118,92)(34,129,119,83)(35,140,120,94)(36,131,101,85)(37,122,102,96)(38,133,103,87)(39,124,104,98)(40,135,105,89), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,120)(37,119)(38,118)(39,117)(40,116)(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)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)>;
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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150), (1,148,70,58)(2,159,71,49)(3,150,72,60)(4,141,73,51)(5,152,74,42)(6,143,75,53)(7,154,76,44)(8,145,77,55)(9,156,78,46)(10,147,79,57)(11,158,80,48)(12,149,61,59)(13,160,62,50)(14,151,63,41)(15,142,64,52)(16,153,65,43)(17,144,66,54)(18,155,67,45)(19,146,68,56)(20,157,69,47)(21,126,106,100)(22,137,107,91)(23,128,108,82)(24,139,109,93)(25,130,110,84)(26,121,111,95)(27,132,112,86)(28,123,113,97)(29,134,114,88)(30,125,115,99)(31,136,116,90)(32,127,117,81)(33,138,118,92)(34,129,119,83)(35,140,120,94)(36,131,101,85)(37,122,102,96)(38,133,103,87)(39,124,104,98)(40,135,105,89), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,120)(37,119)(38,118)(39,117)(40,116)(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)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136) );
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)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,140),(9,139),(10,138),(11,137),(12,136),(13,135),(14,134),(15,133),(16,132),(17,131),(18,130),(19,129),(20,128),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,160),(111,159),(112,158),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150)], [(1,148,70,58),(2,159,71,49),(3,150,72,60),(4,141,73,51),(5,152,74,42),(6,143,75,53),(7,154,76,44),(8,145,77,55),(9,156,78,46),(10,147,79,57),(11,158,80,48),(12,149,61,59),(13,160,62,50),(14,151,63,41),(15,142,64,52),(16,153,65,43),(17,144,66,54),(18,155,67,45),(19,146,68,56),(20,157,69,47),(21,126,106,100),(22,137,107,91),(23,128,108,82),(24,139,109,93),(25,130,110,84),(26,121,111,95),(27,132,112,86),(28,123,113,97),(29,134,114,88),(30,125,115,99),(31,136,116,90),(32,127,117,81),(33,138,118,92),(34,129,119,83),(35,140,120,94),(36,131,101,85),(37,122,102,96),(38,133,103,87),(39,124,104,98),(40,135,105,89)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,109),(28,108),(29,107),(30,106),(31,105),(32,104),(33,103),(34,102),(35,101),(36,120),(37,119),(38,118),(39,117),(40,116),(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),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(99,100),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136)])
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