metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊3D20, Dic10⋊3D4, (C5×D4)⋊2D4, C20.2(C2×D4), C4.86(D4×D5), C4.3(C2×D20), C4⋊D20⋊3C2, C5⋊2(D4⋊D4), C4⋊C4.12D10, D4⋊C4⋊13D5, C10.Q16⋊6C2, (C2×C8).117D10, D10⋊1C8⋊12C2, C10.21C22≀C2, (C2×D4).136D10, C10.42(C4○D8), (C22×D5).21D4, C22.179(D4×D5), C2.17(D8⋊D5), C10.35(C8⋊C22), (C2×C40).128C22, (C2×C20).221C23, (C2×Dic5).198D4, (D4×C10).42C22, (C2×D20).55C22, C2.24(C22⋊D20), C2.12(SD16⋊3D5), (C2×Dic10).63C22, (C2×D4⋊D5)⋊4C2, (C2×D4⋊2D5)⋊1C2, (C2×C40⋊C2)⋊15C2, (C5×D4⋊C4)⋊13C2, (C2×C4×D5).17C22, (C2×C10).234(C2×D4), (C5×C4⋊C4).22C22, (C2×C5⋊2C8).19C22, (C2×C4).328(C22×D5), SmallGroup(320,408)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊3D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 782 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 [×2], D4 [×9], Q8 [×3], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×3], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, Dic10 [×2], Dic10, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×D5, C22×C10, D4⋊D4, C40⋊C2 [×2], C2×C5⋊2C8, D10⋊C4, D4⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D4⋊2D5 [×4], C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D10⋊1C8, C5×D4⋊C4, C4⋊D20, C2×C40⋊C2, C2×D4⋊D5, C2×D4⋊2D5, D4⋊3D20
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, D8⋊D5, SD16⋊3D5, D4⋊3D20
►On 160 points
Generators in S160
(1 109 151 35)(2 36 152 110)(3 111 153 37)(4 38 154 112)(5 113 155 39)(6 40 156 114)(7 115 157 21)(8 22 158 116)(9 117 159 23)(10 24 160 118)(11 119 141 25)(12 26 142 120)(13 101 143 27)(14 28 144 102)(15 103 145 29)(16 30 146 104)(17 105 147 31)(18 32 148 106)(19 107 149 33)(20 34 150 108)(41 99 131 79)(42 80 132 100)(43 81 133 61)(44 62 134 82)(45 83 135 63)(46 64 136 84)(47 85 137 65)(48 66 138 86)(49 87 139 67)(50 68 140 88)(51 89 121 69)(52 70 122 90)(53 91 123 71)(54 72 124 92)(55 93 125 73)(56 74 126 94)(57 95 127 75)(58 76 128 96)(59 97 129 77)(60 78 130 98)
(1 91)(2 124)(3 93)(4 126)(5 95)(6 128)(7 97)(8 130)(9 99)(10 132)(11 81)(12 134)(13 83)(14 136)(15 85)(16 138)(17 87)(18 140)(19 89)(20 122)(21 129)(22 78)(23 131)(24 80)(25 133)(26 62)(27 135)(28 64)(29 137)(30 66)(31 139)(32 68)(33 121)(34 70)(35 123)(36 72)(37 125)(38 74)(39 127)(40 76)(41 117)(42 160)(43 119)(44 142)(45 101)(46 144)(47 103)(48 146)(49 105)(50 148)(51 107)(52 150)(53 109)(54 152)(55 111)(56 154)(57 113)(58 156)(59 115)(60 158)(61 141)(63 143)(65 145)(67 147)(69 149)(71 151)(73 153)(75 155)(77 157)(79 159)(82 120)(84 102)(86 104)(88 106)(90 108)(92 110)(94 112)(96 114)(98 116)(100 118)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 120)(39 119)(40 118)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 100)(59 99)(60 98)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 140)(69 139)(70 138)(71 137)(72 136)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
G:=sub<Sym(160)| (1,109,151,35)(2,36,152,110)(3,111,153,37)(4,38,154,112)(5,113,155,39)(6,40,156,114)(7,115,157,21)(8,22,158,116)(9,117,159,23)(10,24,160,118)(11,119,141,25)(12,26,142,120)(13,101,143,27)(14,28,144,102)(15,103,145,29)(16,30,146,104)(17,105,147,31)(18,32,148,106)(19,107,149,33)(20,34,150,108)(41,99,131,79)(42,80,132,100)(43,81,133,61)(44,62,134,82)(45,83,135,63)(46,64,136,84)(47,85,137,65)(48,66,138,86)(49,87,139,67)(50,68,140,88)(51,89,121,69)(52,70,122,90)(53,91,123,71)(54,72,124,92)(55,93,125,73)(56,74,126,94)(57,95,127,75)(58,76,128,96)(59,97,129,77)(60,78,130,98), (1,91)(2,124)(3,93)(4,126)(5,95)(6,128)(7,97)(8,130)(9,99)(10,132)(11,81)(12,134)(13,83)(14,136)(15,85)(16,138)(17,87)(18,140)(19,89)(20,122)(21,129)(22,78)(23,131)(24,80)(25,133)(26,62)(27,135)(28,64)(29,137)(30,66)(31,139)(32,68)(33,121)(34,70)(35,123)(36,72)(37,125)(38,74)(39,127)(40,76)(41,117)(42,160)(43,119)(44,142)(45,101)(46,144)(47,103)(48,146)(49,105)(50,148)(51,107)(52,150)(53,109)(54,152)(55,111)(56,154)(57,113)(58,156)(59,115)(60,158)(61,141)(63,143)(65,145)(67,147)(69,149)(71,151)(73,153)(75,155)(77,157)(79,159)(82,120)(84,102)(86,104)(88,106)(90,108)(92,110)(94,112)(96,114)(98,116)(100,118), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,120)(39,119)(40,118)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)>;
G:=Group( (1,109,151,35)(2,36,152,110)(3,111,153,37)(4,38,154,112)(5,113,155,39)(6,40,156,114)(7,115,157,21)(8,22,158,116)(9,117,159,23)(10,24,160,118)(11,119,141,25)(12,26,142,120)(13,101,143,27)(14,28,144,102)(15,103,145,29)(16,30,146,104)(17,105,147,31)(18,32,148,106)(19,107,149,33)(20,34,150,108)(41,99,131,79)(42,80,132,100)(43,81,133,61)(44,62,134,82)(45,83,135,63)(46,64,136,84)(47,85,137,65)(48,66,138,86)(49,87,139,67)(50,68,140,88)(51,89,121,69)(52,70,122,90)(53,91,123,71)(54,72,124,92)(55,93,125,73)(56,74,126,94)(57,95,127,75)(58,76,128,96)(59,97,129,77)(60,78,130,98), (1,91)(2,124)(3,93)(4,126)(5,95)(6,128)(7,97)(8,130)(9,99)(10,132)(11,81)(12,134)(13,83)(14,136)(15,85)(16,138)(17,87)(18,140)(19,89)(20,122)(21,129)(22,78)(23,131)(24,80)(25,133)(26,62)(27,135)(28,64)(29,137)(30,66)(31,139)(32,68)(33,121)(34,70)(35,123)(36,72)(37,125)(38,74)(39,127)(40,76)(41,117)(42,160)(43,119)(44,142)(45,101)(46,144)(47,103)(48,146)(49,105)(50,148)(51,107)(52,150)(53,109)(54,152)(55,111)(56,154)(57,113)(58,156)(59,115)(60,158)(61,141)(63,143)(65,145)(67,147)(69,149)(71,151)(73,153)(75,155)(77,157)(79,159)(82,120)(84,102)(86,104)(88,106)(90,108)(92,110)(94,112)(96,114)(98,116)(100,118), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,120)(39,119)(40,118)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159) );
G=PermutationGroup([(1,109,151,35),(2,36,152,110),(3,111,153,37),(4,38,154,112),(5,113,155,39),(6,40,156,114),(7,115,157,21),(8,22,158,116),(9,117,159,23),(10,24,160,118),(11,119,141,25),(12,26,142,120),(13,101,143,27),(14,28,144,102),(15,103,145,29),(16,30,146,104),(17,105,147,31),(18,32,148,106),(19,107,149,33),(20,34,150,108),(41,99,131,79),(42,80,132,100),(43,81,133,61),(44,62,134,82),(45,83,135,63),(46,64,136,84),(47,85,137,65),(48,66,138,86),(49,87,139,67),(50,68,140,88),(51,89,121,69),(52,70,122,90),(53,91,123,71),(54,72,124,92),(55,93,125,73),(56,74,126,94),(57,95,127,75),(58,76,128,96),(59,97,129,77),(60,78,130,98)], [(1,91),(2,124),(3,93),(4,126),(5,95),(6,128),(7,97),(8,130),(9,99),(10,132),(11,81),(12,134),(13,83),(14,136),(15,85),(16,138),(17,87),(18,140),(19,89),(20,122),(21,129),(22,78),(23,131),(24,80),(25,133),(26,62),(27,135),(28,64),(29,137),(30,66),(31,139),(32,68),(33,121),(34,70),(35,123),(36,72),(37,125),(38,74),(39,127),(40,76),(41,117),(42,160),(43,119),(44,142),(45,101),(46,144),(47,103),(48,146),(49,105),(50,148),(51,107),(52,150),(53,109),(54,152),(55,111),(56,154),(57,113),(58,156),(59,115),(60,158),(61,141),(63,143),(65,145),(67,147),(69,149),(71,151),(73,153),(75,155),(77,157),(79,159),(82,120),(84,102),(86,104),(88,106),(90,108),(92,110),(94,112),(96,114),(98,116),(100,118)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,117),(22,116),(23,115),(24,114),(25,113),(26,112),(27,111),(28,110),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,120),(39,119),(40,118),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,100),(59,99),(60,98),(61,127),(62,126),(63,125),(64,124),(65,123),(66,122),(67,121),(68,140),(69,139),(70,138),(71,137),(72,136),(73,135),(74,134),(75,133),(76,132),(77,131),(78,130),(79,129),(80,128),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)])
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 | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 40 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | D8⋊D5 | SD16⋊3D5 |
| kernel | D4⋊3D20 | C10.Q16 | D10⋊1C8 | C5×D4⋊C4 | C4⋊D20 | C2×C40⋊C2 | C2×D4⋊D5 | C2×D4⋊2D5 | Dic10 | C2×Dic5 | C5×D4 | C22×D5 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | 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 D4⋊3D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 37 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 34 | 0 |
| 2 | 30 | 0 | 0 |
| 27 | 16 | 0 | 0 |
| 0 | 0 | 9 | 25 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 4 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[2,27,0,0,30,16,0,0,0,0,9,0,0,0,25,32],[40,5,0,0,0,1,0,0,0,0,40,4,0,0,0,1] >;
D4⋊3D20 in GAP, Magma, Sage, TeX
D_4\rtimes_3D_{20} % in TeX
G:=Group("D4:3D20"); // GroupNames label
G:=SmallGroup(320,408);
// by ID
G=gap.SmallGroup(320,408);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations