metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.4Q8, C42.69D10, C5⋊7(D4.Q8), C4.10(Q8×D5), C4⋊C4.75D10, C20⋊3C8⋊30C2, C20.34(C2×Q8), C42.C2⋊1D5, (C4×D20).16C2, (C2×C20).275D4, C20.70(C4○D4), C10.D8⋊42C2, C20.Q8⋊41C2, D20⋊6C4.12C2, C10.109(C4○D8), (C4×C20).114C22, (C2×C20).384C23, C4.33(Q8⋊2D5), C10.74(C22⋊Q8), C2.21(D4⋊D10), C10.122(C8⋊C22), C2.11(D10⋊3Q8), (C2×D20).253C22, C4⋊Dic5.343C22, C2.28(D4.8D10), (C5×C42.C2)⋊1C2, (C2×C10).515(C2×D4), (C2×C4).66(C5⋊D4), (C5×C4⋊C4).122C22, (C2×C4).482(C22×D5), C22.188(C2×C5⋊D4), (C2×C5⋊2C8).126C22, SmallGroup(320,693)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.4Q8
G = < a,b,c,d | a20=b2=c4=1, d2=a10c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 422 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, Dic5, C20 [×2], C20 [×4], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C22×D5, D4.Q8, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×D20, C20⋊3C8, C10.D8, C20.Q8, D20⋊6C4 [×2], C4×D20, C5×C42.C2, D20.4Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4.Q8, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D10⋊3Q8, D4⋊D10, D4.8D10, D20.4Q8
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(81 149)(82 148)(83 147)(84 146)(85 145)(86 144)(87 143)(88 142)(89 141)(90 160)(91 159)(92 158)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)(99 151)(100 150)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)
(1 46 68 26)(2 57 69 37)(3 48 70 28)(4 59 71 39)(5 50 72 30)(6 41 73 21)(7 52 74 32)(8 43 75 23)(9 54 76 34)(10 45 77 25)(11 56 78 36)(12 47 79 27)(13 58 80 38)(14 49 61 29)(15 60 62 40)(16 51 63 31)(17 42 64 22)(18 53 65 33)(19 44 66 24)(20 55 67 35)(81 140 158 111)(82 131 159 102)(83 122 160 113)(84 133 141 104)(85 124 142 115)(86 135 143 106)(87 126 144 117)(88 137 145 108)(89 128 146 119)(90 139 147 110)(91 130 148 101)(92 121 149 112)(93 132 150 103)(94 123 151 114)(95 134 152 105)(96 125 153 116)(97 136 154 107)(98 127 155 118)(99 138 156 109)(100 129 157 120)
(1 92 78 159)(2 93 79 160)(3 94 80 141)(4 95 61 142)(5 96 62 143)(6 97 63 144)(7 98 64 145)(8 99 65 146)(9 100 66 147)(10 81 67 148)(11 82 68 149)(12 83 69 150)(13 84 70 151)(14 85 71 152)(15 86 72 153)(16 87 73 154)(17 88 74 155)(18 89 75 156)(19 90 76 157)(20 91 77 158)(21 136 51 117)(22 137 52 118)(23 138 53 119)(24 139 54 120)(25 140 55 101)(26 121 56 102)(27 122 57 103)(28 123 58 104)(29 124 59 105)(30 125 60 106)(31 126 41 107)(32 127 42 108)(33 128 43 109)(34 129 44 110)(35 130 45 111)(36 131 46 112)(37 132 47 113)(38 133 48 114)(39 134 49 115)(40 135 50 116)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142)(89,141)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127), (1,46,68,26)(2,57,69,37)(3,48,70,28)(4,59,71,39)(5,50,72,30)(6,41,73,21)(7,52,74,32)(8,43,75,23)(9,54,76,34)(10,45,77,25)(11,56,78,36)(12,47,79,27)(13,58,80,38)(14,49,61,29)(15,60,62,40)(16,51,63,31)(17,42,64,22)(18,53,65,33)(19,44,66,24)(20,55,67,35)(81,140,158,111)(82,131,159,102)(83,122,160,113)(84,133,141,104)(85,124,142,115)(86,135,143,106)(87,126,144,117)(88,137,145,108)(89,128,146,119)(90,139,147,110)(91,130,148,101)(92,121,149,112)(93,132,150,103)(94,123,151,114)(95,134,152,105)(96,125,153,116)(97,136,154,107)(98,127,155,118)(99,138,156,109)(100,129,157,120), (1,92,78,159)(2,93,79,160)(3,94,80,141)(4,95,61,142)(5,96,62,143)(6,97,63,144)(7,98,64,145)(8,99,65,146)(9,100,66,147)(10,81,67,148)(11,82,68,149)(12,83,69,150)(13,84,70,151)(14,85,71,152)(15,86,72,153)(16,87,73,154)(17,88,74,155)(18,89,75,156)(19,90,76,157)(20,91,77,158)(21,136,51,117)(22,137,52,118)(23,138,53,119)(24,139,54,120)(25,140,55,101)(26,121,56,102)(27,122,57,103)(28,123,58,104)(29,124,59,105)(30,125,60,106)(31,126,41,107)(32,127,42,108)(33,128,43,109)(34,129,44,110)(35,130,45,111)(36,131,46,112)(37,132,47,113)(38,133,48,114)(39,134,49,115)(40,135,50,116)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142)(89,141)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127), (1,46,68,26)(2,57,69,37)(3,48,70,28)(4,59,71,39)(5,50,72,30)(6,41,73,21)(7,52,74,32)(8,43,75,23)(9,54,76,34)(10,45,77,25)(11,56,78,36)(12,47,79,27)(13,58,80,38)(14,49,61,29)(15,60,62,40)(16,51,63,31)(17,42,64,22)(18,53,65,33)(19,44,66,24)(20,55,67,35)(81,140,158,111)(82,131,159,102)(83,122,160,113)(84,133,141,104)(85,124,142,115)(86,135,143,106)(87,126,144,117)(88,137,145,108)(89,128,146,119)(90,139,147,110)(91,130,148,101)(92,121,149,112)(93,132,150,103)(94,123,151,114)(95,134,152,105)(96,125,153,116)(97,136,154,107)(98,127,155,118)(99,138,156,109)(100,129,157,120), (1,92,78,159)(2,93,79,160)(3,94,80,141)(4,95,61,142)(5,96,62,143)(6,97,63,144)(7,98,64,145)(8,99,65,146)(9,100,66,147)(10,81,67,148)(11,82,68,149)(12,83,69,150)(13,84,70,151)(14,85,71,152)(15,86,72,153)(16,87,73,154)(17,88,74,155)(18,89,75,156)(19,90,76,157)(20,91,77,158)(21,136,51,117)(22,137,52,118)(23,138,53,119)(24,139,54,120)(25,140,55,101)(26,121,56,102)(27,122,57,103)(28,123,58,104)(29,124,59,105)(30,125,60,106)(31,126,41,107)(32,127,42,108)(33,128,43,109)(34,129,44,110)(35,130,45,111)(36,131,46,112)(37,132,47,113)(38,133,48,114)(39,134,49,115)(40,135,50,116) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,80),(19,79),(20,78),(21,45),(22,44),(23,43),(24,42),(25,41),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(81,149),(82,148),(83,147),(84,146),(85,145),(86,144),(87,143),(88,142),(89,141),(90,160),(91,159),(92,158),(93,157),(94,156),(95,155),(96,154),(97,153),(98,152),(99,151),(100,150),(101,126),(102,125),(103,124),(104,123),(105,122),(106,121),(107,140),(108,139),(109,138),(110,137),(111,136),(112,135),(113,134),(114,133),(115,132),(116,131),(117,130),(118,129),(119,128),(120,127)], [(1,46,68,26),(2,57,69,37),(3,48,70,28),(4,59,71,39),(5,50,72,30),(6,41,73,21),(7,52,74,32),(8,43,75,23),(9,54,76,34),(10,45,77,25),(11,56,78,36),(12,47,79,27),(13,58,80,38),(14,49,61,29),(15,60,62,40),(16,51,63,31),(17,42,64,22),(18,53,65,33),(19,44,66,24),(20,55,67,35),(81,140,158,111),(82,131,159,102),(83,122,160,113),(84,133,141,104),(85,124,142,115),(86,135,143,106),(87,126,144,117),(88,137,145,108),(89,128,146,119),(90,139,147,110),(91,130,148,101),(92,121,149,112),(93,132,150,103),(94,123,151,114),(95,134,152,105),(96,125,153,116),(97,136,154,107),(98,127,155,118),(99,138,156,109),(100,129,157,120)], [(1,92,78,159),(2,93,79,160),(3,94,80,141),(4,95,61,142),(5,96,62,143),(6,97,63,144),(7,98,64,145),(8,99,65,146),(9,100,66,147),(10,81,67,148),(11,82,68,149),(12,83,69,150),(13,84,70,151),(14,85,71,152),(15,86,72,153),(16,87,73,154),(17,88,74,155),(18,89,75,156),(19,90,76,157),(20,91,77,158),(21,136,51,117),(22,137,52,118),(23,138,53,119),(24,139,54,120),(25,140,55,101),(26,121,56,102),(27,122,57,103),(28,123,58,104),(29,124,59,105),(30,125,60,106),(31,126,41,107),(32,127,42,108),(33,128,43,109),(34,129,44,110),(35,130,45,111),(36,131,46,112),(37,132,47,113),(38,133,48,114),(39,134,49,115),(40,135,50,116)])
47 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 | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C5⋊D4 | C8⋊C22 | Q8×D5 | Q8⋊2D5 | D4⋊D10 | D4.8D10 |
| kernel | D20.4Q8 | C20⋊3C8 | C10.D8 | C20.Q8 | D20⋊6C4 | C4×D20 | C5×C42.C2 | D20 | C2×C20 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C2×C4 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.4Q8 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 17 |
| 0 | 0 | 0 | 0 | 12 | 17 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 39 | 0 | 0 |
| 0 | 0 | 39 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,12,0,0,0,0,17,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,39,0,0,0,0,39,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
D20.4Q8 in GAP, Magma, Sage, TeX
D_{20}._4Q_8 % in TeX
G:=Group("D20.4Q8"); // GroupNames label
G:=SmallGroup(320,693);
// by ID
G=gap.SmallGroup(320,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10*c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations