metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊6Q8, C20.18D8, C42.81D10, C4⋊Q8⋊5D5, C4.12(Q8×D5), C4⋊C4.83D10, C5⋊5(D4⋊Q8), C20⋊3C8⋊35C2, C10.60(C2×D8), C20.39(C2×Q8), C4.16(D4⋊D5), (C4×D20).19C2, (C2×C20).157D4, C20.82(C4○D4), C10.D8⋊43C2, D20⋊6C4.15C2, (C4×C20).134C22, (C2×C20).405C23, C4.35(Q8⋊2D5), C10.76(C22⋊Q8), C2.13(D10⋊3Q8), (C2×D20).257C22, C10.97(C8.C22), C4⋊Dic5.348C22, C2.18(C20.C23), (C5×C4⋊Q8)⋊5C2, C2.15(C2×D4⋊D5), (C2×C10).536(C2×D4), (C2×C4).189(C5⋊D4), (C5×C4⋊C4).130C22, (C2×C4).502(C22×D5), C22.208(C2×C5⋊D4), (C2×C5⋊2C8).138C22, SmallGroup(320,714)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊6Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 438 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4⋊Q8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C20⋊3C8, C10.D8, D20⋊6C4, C4×D20, C5×C4⋊Q8, D20⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, C5⋊D4, C22×D5, D4⋊Q8, D4⋊D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C2×D4⋊D5, C20.C23, D10⋊3Q8, D20⋊6Q8
►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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 140)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 158)(42 157)(43 156)(44 155)(45 154)(46 153)(47 152)(48 151)(49 150)(50 149)(51 148)(52 147)(53 146)(54 145)(55 144)(56 143)(57 142)(58 141)(59 160)(60 159)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)
(1 112 140 98)(2 103 121 89)(3 114 122 100)(4 105 123 91)(5 116 124 82)(6 107 125 93)(7 118 126 84)(8 109 127 95)(9 120 128 86)(10 111 129 97)(11 102 130 88)(12 113 131 99)(13 104 132 90)(14 115 133 81)(15 106 134 92)(16 117 135 83)(17 108 136 94)(18 119 137 85)(19 110 138 96)(20 101 139 87)(21 47 68 148)(22 58 69 159)(23 49 70 150)(24 60 71 141)(25 51 72 152)(26 42 73 143)(27 53 74 154)(28 44 75 145)(29 55 76 156)(30 46 77 147)(31 57 78 158)(32 48 79 149)(33 59 80 160)(34 50 61 151)(35 41 62 142)(36 52 63 153)(37 43 64 144)(38 54 65 155)(39 45 66 146)(40 56 67 157)
(1 31 140 78)(2 32 121 79)(3 33 122 80)(4 34 123 61)(5 35 124 62)(6 36 125 63)(7 37 126 64)(8 38 127 65)(9 39 128 66)(10 40 129 67)(11 21 130 68)(12 22 131 69)(13 23 132 70)(14 24 133 71)(15 25 134 72)(16 26 135 73)(17 27 136 74)(18 28 137 75)(19 29 138 76)(20 30 139 77)(41 116 142 82)(42 117 143 83)(43 118 144 84)(44 119 145 85)(45 120 146 86)(46 101 147 87)(47 102 148 88)(48 103 149 89)(49 104 150 90)(50 105 151 91)(51 106 152 92)(52 107 153 93)(53 108 154 94)(54 109 155 95)(55 110 156 96)(56 111 157 97)(57 112 158 98)(58 113 159 99)(59 114 160 100)(60 115 141 81)
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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,160)(60,159)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114), (1,112,140,98)(2,103,121,89)(3,114,122,100)(4,105,123,91)(5,116,124,82)(6,107,125,93)(7,118,126,84)(8,109,127,95)(9,120,128,86)(10,111,129,97)(11,102,130,88)(12,113,131,99)(13,104,132,90)(14,115,133,81)(15,106,134,92)(16,117,135,83)(17,108,136,94)(18,119,137,85)(19,110,138,96)(20,101,139,87)(21,47,68,148)(22,58,69,159)(23,49,70,150)(24,60,71,141)(25,51,72,152)(26,42,73,143)(27,53,74,154)(28,44,75,145)(29,55,76,156)(30,46,77,147)(31,57,78,158)(32,48,79,149)(33,59,80,160)(34,50,61,151)(35,41,62,142)(36,52,63,153)(37,43,64,144)(38,54,65,155)(39,45,66,146)(40,56,67,157), (1,31,140,78)(2,32,121,79)(3,33,122,80)(4,34,123,61)(5,35,124,62)(6,36,125,63)(7,37,126,64)(8,38,127,65)(9,39,128,66)(10,40,129,67)(11,21,130,68)(12,22,131,69)(13,23,132,70)(14,24,133,71)(15,25,134,72)(16,26,135,73)(17,27,136,74)(18,28,137,75)(19,29,138,76)(20,30,139,77)(41,116,142,82)(42,117,143,83)(43,118,144,84)(44,119,145,85)(45,120,146,86)(46,101,147,87)(47,102,148,88)(48,103,149,89)(49,104,150,90)(50,105,151,91)(51,106,152,92)(52,107,153,93)(53,108,154,94)(54,109,155,95)(55,110,156,96)(56,111,157,97)(57,112,158,98)(58,113,159,99)(59,114,160,100)(60,115,141,81)>;
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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,160)(60,159)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114), (1,112,140,98)(2,103,121,89)(3,114,122,100)(4,105,123,91)(5,116,124,82)(6,107,125,93)(7,118,126,84)(8,109,127,95)(9,120,128,86)(10,111,129,97)(11,102,130,88)(12,113,131,99)(13,104,132,90)(14,115,133,81)(15,106,134,92)(16,117,135,83)(17,108,136,94)(18,119,137,85)(19,110,138,96)(20,101,139,87)(21,47,68,148)(22,58,69,159)(23,49,70,150)(24,60,71,141)(25,51,72,152)(26,42,73,143)(27,53,74,154)(28,44,75,145)(29,55,76,156)(30,46,77,147)(31,57,78,158)(32,48,79,149)(33,59,80,160)(34,50,61,151)(35,41,62,142)(36,52,63,153)(37,43,64,144)(38,54,65,155)(39,45,66,146)(40,56,67,157), (1,31,140,78)(2,32,121,79)(3,33,122,80)(4,34,123,61)(5,35,124,62)(6,36,125,63)(7,37,126,64)(8,38,127,65)(9,39,128,66)(10,40,129,67)(11,21,130,68)(12,22,131,69)(13,23,132,70)(14,24,133,71)(15,25,134,72)(16,26,135,73)(17,27,136,74)(18,28,137,75)(19,29,138,76)(20,30,139,77)(41,116,142,82)(42,117,143,83)(43,118,144,84)(44,119,145,85)(45,120,146,86)(46,101,147,87)(47,102,148,88)(48,103,149,89)(49,104,150,90)(50,105,151,91)(51,106,152,92)(52,107,153,93)(53,108,154,94)(54,109,155,95)(55,110,156,96)(56,111,157,97)(57,112,158,98)(58,113,159,99)(59,114,160,100)(60,115,141,81) );
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)], [(1,139),(2,138),(3,137),(4,136),(5,135),(6,134),(7,133),(8,132),(9,131),(10,130),(11,129),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,121),(20,140),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,158),(42,157),(43,156),(44,155),(45,154),(46,153),(47,152),(48,151),(49,150),(50,149),(51,148),(52,147),(53,146),(54,145),(55,144),(56,143),(57,142),(58,141),(59,160),(60,159),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114)], [(1,112,140,98),(2,103,121,89),(3,114,122,100),(4,105,123,91),(5,116,124,82),(6,107,125,93),(7,118,126,84),(8,109,127,95),(9,120,128,86),(10,111,129,97),(11,102,130,88),(12,113,131,99),(13,104,132,90),(14,115,133,81),(15,106,134,92),(16,117,135,83),(17,108,136,94),(18,119,137,85),(19,110,138,96),(20,101,139,87),(21,47,68,148),(22,58,69,159),(23,49,70,150),(24,60,71,141),(25,51,72,152),(26,42,73,143),(27,53,74,154),(28,44,75,145),(29,55,76,156),(30,46,77,147),(31,57,78,158),(32,48,79,149),(33,59,80,160),(34,50,61,151),(35,41,62,142),(36,52,63,153),(37,43,64,144),(38,54,65,155),(39,45,66,146),(40,56,67,157)], [(1,31,140,78),(2,32,121,79),(3,33,122,80),(4,34,123,61),(5,35,124,62),(6,36,125,63),(7,37,126,64),(8,38,127,65),(9,39,128,66),(10,40,129,67),(11,21,130,68),(12,22,131,69),(13,23,132,70),(14,24,133,71),(15,25,134,72),(16,26,135,73),(17,27,136,74),(18,28,137,75),(19,29,138,76),(20,30,139,77),(41,116,142,82),(42,117,143,83),(43,118,144,84),(44,119,145,85),(45,120,146,86),(46,101,147,87),(47,102,148,88),(48,103,149,89),(49,104,150,90),(50,105,151,91),(51,106,152,92),(52,107,153,93),(53,108,154,94),(54,109,155,95),(55,110,156,96),(56,111,157,97),(57,112,158,98),(58,113,159,99),(59,114,160,100),(60,115,141,81)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D8 | C4○D4 | D10 | D10 | C5⋊D4 | C8.C22 | D4⋊D5 | Q8×D5 | Q8⋊2D5 | C20.C23 |
| kernel | D20⋊6Q8 | C20⋊3C8 | C10.D8 | D20⋊6C4 | C4×D20 | C5×C4⋊Q8 | D20 | C2×C20 | C4⋊Q8 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of D20⋊6Q8 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 37 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 37 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 7 | 0 | 0 |
| 0 | 0 | 35 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,21,0,0,0,0,37,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,37,40,0,0,0,0,0,0,1,0,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,35,0,0,0,0,7,24,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,39,9] >;
D20⋊6Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_6Q_8 % in TeX
G:=Group("D20:6Q8"); // GroupNames label
G:=SmallGroup(320,714);
// by ID
G=gap.SmallGroup(320,714);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=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