metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊7D4, (C5×Q8)⋊6D4, C4.64(D4×D5), C20⋊2D4⋊7C2, C5⋊6(D4⋊D4), Q8⋊3(C5⋊D4), C20.49(C2×D4), (C2×SD16)⋊13D5, (C2×D4).74D10, (C2×C8).148D10, D20⋊5C4⋊36C2, D10⋊1C8⋊34C2, C10.59C22≀C2, (C10×SD16)⋊22C2, C10.64(C4○D8), Q8⋊Dic5⋊30C2, (C2×Q8).117D10, (C22×D5).45D4, C22.269(D4×D5), C2.29(D40⋊C2), C10.79(C8⋊C22), (C2×C40).295C22, (C2×C20).449C23, (C2×Dic5).240D4, (D4×C10).98C22, (Q8×C10).78C22, C2.27(C23⋊D10), (C2×D20).125C22, C4⋊Dic5.176C22, C2.30(SD16⋊3D5), (C2×D4⋊D5)⋊20C2, C4.44(C2×C5⋊D4), (C2×Q8⋊2D5)⋊2C2, (C2×C4×D5).53C22, (C2×C10).361(C2×D4), (C2×C4).538(C22×D5), (C2×C5⋊2C8).158C22, SmallGroup(320,799)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊7D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, dad=a11, cbc-1=a3b, dbd=a5b, dcd=c-1 >
Subgroups: 750 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 [×3], C10 [×3], C10, 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 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, C4×D5 [×6], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C22×C10, D4⋊D4, C2×C5⋊2C8, C4⋊Dic5, D4⋊D5 [×2], C23.D5, C2×C40, C5×SD16 [×2], C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5 [×4], C2×C5⋊D4, D4×C10, Q8×C10, D10⋊1C8, D20⋊5C4, Q8⋊Dic5, C2×D4⋊D5, C20⋊2D4, C10×SD16, C2×Q8⋊2D5, D20⋊7D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊D4, D4×D5 [×2], C2×C5⋊D4, D40⋊C2, SD16⋊3D5, C23⋊D10, D20⋊7D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)
(1 157 52 93)(2 156 53 92)(3 155 54 91)(4 154 55 90)(5 153 56 89)(6 152 57 88)(7 151 58 87)(8 150 59 86)(9 149 60 85)(10 148 41 84)(11 147 42 83)(12 146 43 82)(13 145 44 81)(14 144 45 100)(15 143 46 99)(16 142 47 98)(17 141 48 97)(18 160 49 96)(19 159 50 95)(20 158 51 94)(21 105 69 122)(22 104 70 121)(23 103 71 140)(24 102 72 139)(25 101 73 138)(26 120 74 137)(27 119 75 136)(28 118 76 135)(29 117 77 134)(30 116 78 133)(31 115 79 132)(32 114 80 131)(33 113 61 130)(34 112 62 129)(35 111 63 128)(36 110 64 127)(37 109 65 126)(38 108 66 125)(39 107 67 124)(40 106 68 123)
(1 24)(2 35)(3 26)(4 37)(5 28)(6 39)(7 30)(8 21)(9 32)(10 23)(11 34)(12 25)(13 36)(14 27)(15 38)(16 29)(17 40)(18 31)(19 22)(20 33)(41 71)(42 62)(43 73)(44 64)(45 75)(46 66)(47 77)(48 68)(49 79)(50 70)(51 61)(52 72)(53 63)(54 74)(55 65)(56 76)(57 67)(58 78)(59 69)(60 80)(81 110)(82 101)(83 112)(84 103)(85 114)(86 105)(87 116)(88 107)(89 118)(90 109)(91 120)(92 111)(93 102)(94 113)(95 104)(96 115)(97 106)(98 117)(99 108)(100 119)(121 159)(122 150)(123 141)(124 152)(125 143)(126 154)(127 145)(128 156)(129 147)(130 158)(131 149)(132 160)(133 151)(134 142)(135 153)(136 144)(137 155)(138 146)(139 157)(140 148)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,160), (1,157,52,93)(2,156,53,92)(3,155,54,91)(4,154,55,90)(5,153,56,89)(6,152,57,88)(7,151,58,87)(8,150,59,86)(9,149,60,85)(10,148,41,84)(11,147,42,83)(12,146,43,82)(13,145,44,81)(14,144,45,100)(15,143,46,99)(16,142,47,98)(17,141,48,97)(18,160,49,96)(19,159,50,95)(20,158,51,94)(21,105,69,122)(22,104,70,121)(23,103,71,140)(24,102,72,139)(25,101,73,138)(26,120,74,137)(27,119,75,136)(28,118,76,135)(29,117,77,134)(30,116,78,133)(31,115,79,132)(32,114,80,131)(33,113,61,130)(34,112,62,129)(35,111,63,128)(36,110,64,127)(37,109,65,126)(38,108,66,125)(39,107,67,124)(40,106,68,123), (1,24)(2,35)(3,26)(4,37)(5,28)(6,39)(7,30)(8,21)(9,32)(10,23)(11,34)(12,25)(13,36)(14,27)(15,38)(16,29)(17,40)(18,31)(19,22)(20,33)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,61)(52,72)(53,63)(54,74)(55,65)(56,76)(57,67)(58,78)(59,69)(60,80)(81,110)(82,101)(83,112)(84,103)(85,114)(86,105)(87,116)(88,107)(89,118)(90,109)(91,120)(92,111)(93,102)(94,113)(95,104)(96,115)(97,106)(98,117)(99,108)(100,119)(121,159)(122,150)(123,141)(124,152)(125,143)(126,154)(127,145)(128,156)(129,147)(130,158)(131,149)(132,160)(133,151)(134,142)(135,153)(136,144)(137,155)(138,146)(139,157)(140,148)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,160), (1,157,52,93)(2,156,53,92)(3,155,54,91)(4,154,55,90)(5,153,56,89)(6,152,57,88)(7,151,58,87)(8,150,59,86)(9,149,60,85)(10,148,41,84)(11,147,42,83)(12,146,43,82)(13,145,44,81)(14,144,45,100)(15,143,46,99)(16,142,47,98)(17,141,48,97)(18,160,49,96)(19,159,50,95)(20,158,51,94)(21,105,69,122)(22,104,70,121)(23,103,71,140)(24,102,72,139)(25,101,73,138)(26,120,74,137)(27,119,75,136)(28,118,76,135)(29,117,77,134)(30,116,78,133)(31,115,79,132)(32,114,80,131)(33,113,61,130)(34,112,62,129)(35,111,63,128)(36,110,64,127)(37,109,65,126)(38,108,66,125)(39,107,67,124)(40,106,68,123), (1,24)(2,35)(3,26)(4,37)(5,28)(6,39)(7,30)(8,21)(9,32)(10,23)(11,34)(12,25)(13,36)(14,27)(15,38)(16,29)(17,40)(18,31)(19,22)(20,33)(41,71)(42,62)(43,73)(44,64)(45,75)(46,66)(47,77)(48,68)(49,79)(50,70)(51,61)(52,72)(53,63)(54,74)(55,65)(56,76)(57,67)(58,78)(59,69)(60,80)(81,110)(82,101)(83,112)(84,103)(85,114)(86,105)(87,116)(88,107)(89,118)(90,109)(91,120)(92,111)(93,102)(94,113)(95,104)(96,115)(97,106)(98,117)(99,108)(100,119)(121,159)(122,150)(123,141)(124,152)(125,143)(126,154)(127,145)(128,156)(129,147)(130,158)(131,149)(132,160)(133,151)(134,142)(135,153)(136,144)(137,155)(138,146)(139,157)(140,148) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80),(81,89),(82,88),(83,87),(84,86),(90,100),(91,99),(92,98),(93,97),(94,96),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117),(121,126),(122,125),(123,124),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150),(158,160)], [(1,157,52,93),(2,156,53,92),(3,155,54,91),(4,154,55,90),(5,153,56,89),(6,152,57,88),(7,151,58,87),(8,150,59,86),(9,149,60,85),(10,148,41,84),(11,147,42,83),(12,146,43,82),(13,145,44,81),(14,144,45,100),(15,143,46,99),(16,142,47,98),(17,141,48,97),(18,160,49,96),(19,159,50,95),(20,158,51,94),(21,105,69,122),(22,104,70,121),(23,103,71,140),(24,102,72,139),(25,101,73,138),(26,120,74,137),(27,119,75,136),(28,118,76,135),(29,117,77,134),(30,116,78,133),(31,115,79,132),(32,114,80,131),(33,113,61,130),(34,112,62,129),(35,111,63,128),(36,110,64,127),(37,109,65,126),(38,108,66,125),(39,107,67,124),(40,106,68,123)], [(1,24),(2,35),(3,26),(4,37),(5,28),(6,39),(7,30),(8,21),(9,32),(10,23),(11,34),(12,25),(13,36),(14,27),(15,38),(16,29),(17,40),(18,31),(19,22),(20,33),(41,71),(42,62),(43,73),(44,64),(45,75),(46,66),(47,77),(48,68),(49,79),(50,70),(51,61),(52,72),(53,63),(54,74),(55,65),(56,76),(57,67),(58,78),(59,69),(60,80),(81,110),(82,101),(83,112),(84,103),(85,114),(86,105),(87,116),(88,107),(89,118),(90,109),(91,120),(92,111),(93,102),(94,113),(95,104),(96,115),(97,106),(98,117),(99,108),(100,119),(121,159),(122,150),(123,141),(124,152),(125,143),(126,154),(127,145),(128,156),(129,147),(130,158),(131,149),(132,160),(133,151),(134,142),(135,153),(136,144),(137,155),(138,146),(139,157),(140,148)])
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 | 8 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 40 | 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 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 | SD16⋊3D5 |
| kernel | D20⋊7D4 | D10⋊1C8 | D20⋊5C4 | Q8⋊Dic5 | C2×D4⋊D5 | C20⋊2D4 | C10×SD16 | C2×Q8⋊2D5 | D20 | C2×Dic5 | C5×Q8 | C22×D5 | C2×SD16 | C2×C8 | C2×D4 | 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⋊7D4 ►in GL4(𝔽41) generated by
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 8 | 1 |
| 1 | 2 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 40 |
| 30 | 30 | 0 | 0 |
| 26 | 11 | 0 | 0 |
| 0 | 0 | 38 | 20 |
| 0 | 0 | 20 | 3 |
| 0 | 17 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 17 | 35 |
| 0 | 0 | 7 | 24 |
G:=sub<GL(4,GF(41))| [1,40,0,0,2,40,0,0,0,0,34,8,0,0,40,1],[1,0,0,0,2,40,0,0,0,0,1,0,0,0,1,40],[30,26,0,0,30,11,0,0,0,0,38,20,0,0,20,3],[0,29,0,0,17,0,0,0,0,0,17,7,0,0,35,24] >;
D20⋊7D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_7D_4 % in TeX
G:=Group("D20:7D4"); // GroupNames label
G:=SmallGroup(320,799);
// by ID
G=gap.SmallGroup(320,799);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^11,c*b*c^-1=a^3*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations