metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5Q8, C20.18SD16, C42.79D10, C4⋊Q8⋊2D5, C4.11(Q8×D5), C4⋊C4.82D10, C5⋊5(D4⋊2Q8), C20⋊3C8⋊34C2, C20.38(C2×Q8), (C4×D20).18C2, (C2×C20).155D4, C4.10(Q8⋊D5), C20.81(C4○D4), C20.Q8⋊42C2, C10.76(C2×SD16), D20⋊6C4.14C2, C10.98(C8⋊C22), (C4×C20).131C22, (C2×C20).402C23, C4.34(Q8⋊2D5), C10.75(C22⋊Q8), C2.12(D10⋊3Q8), (C2×D20).256C22, C4⋊Dic5.347C22, C2.19(D4.D10), (C5×C4⋊Q8)⋊2C2, C2.14(C2×Q8⋊D5), (C2×C10).533(C2×D4), (C2×C4).188(C5⋊D4), (C5×C4⋊C4).129C22, (C2×C4).499(C22×D5), C22.205(C2×C5⋊D4), (C2×C5⋊2C8).136C22, SmallGroup(320,711)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 438 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C5⋊2C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C22×D5, D4⋊2Q8, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C20⋊3C8, C20.Q8 [×2], D20⋊6C4 [×2], C4×D20, C5×C4⋊Q8, D20⋊5Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊2Q8, Q8⋊D5 [×2], Q8×D5, Q8⋊2D5, C2×C5⋊D4, D4.D10, C2×Q8⋊D5, D10⋊3Q8, D20⋊5Q8
►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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 141)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)
(1 34 102 134)(2 25 103 125)(3 36 104 136)(4 27 105 127)(5 38 106 138)(6 29 107 129)(7 40 108 140)(8 31 109 131)(9 22 110 122)(10 33 111 133)(11 24 112 124)(12 35 113 135)(13 26 114 126)(14 37 115 137)(15 28 116 128)(16 39 117 139)(17 30 118 130)(18 21 119 121)(19 32 120 132)(20 23 101 123)(41 155 87 63)(42 146 88 74)(43 157 89 65)(44 148 90 76)(45 159 91 67)(46 150 92 78)(47 141 93 69)(48 152 94 80)(49 143 95 71)(50 154 96 62)(51 145 97 73)(52 156 98 64)(53 147 99 75)(54 158 100 66)(55 149 81 77)(56 160 82 68)(57 151 83 79)(58 142 84 70)(59 153 85 61)(60 144 86 72)
(1 87 102 41)(2 88 103 42)(3 89 104 43)(4 90 105 44)(5 91 106 45)(6 92 107 46)(7 93 108 47)(8 94 109 48)(9 95 110 49)(10 96 111 50)(11 97 112 51)(12 98 113 52)(13 99 114 53)(14 100 115 54)(15 81 116 55)(16 82 117 56)(17 83 118 57)(18 84 119 58)(19 85 120 59)(20 86 101 60)(21 142 121 70)(22 143 122 71)(23 144 123 72)(24 145 124 73)(25 146 125 74)(26 147 126 75)(27 148 127 76)(28 149 128 77)(29 150 129 78)(30 151 130 79)(31 152 131 80)(32 153 132 61)(33 154 133 62)(34 155 134 63)(35 156 135 64)(36 157 136 65)(37 158 137 66)(38 159 138 67)(39 160 139 68)(40 141 140 69)
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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,141)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142), (1,34,102,134)(2,25,103,125)(3,36,104,136)(4,27,105,127)(5,38,106,138)(6,29,107,129)(7,40,108,140)(8,31,109,131)(9,22,110,122)(10,33,111,133)(11,24,112,124)(12,35,113,135)(13,26,114,126)(14,37,115,137)(15,28,116,128)(16,39,117,139)(17,30,118,130)(18,21,119,121)(19,32,120,132)(20,23,101,123)(41,155,87,63)(42,146,88,74)(43,157,89,65)(44,148,90,76)(45,159,91,67)(46,150,92,78)(47,141,93,69)(48,152,94,80)(49,143,95,71)(50,154,96,62)(51,145,97,73)(52,156,98,64)(53,147,99,75)(54,158,100,66)(55,149,81,77)(56,160,82,68)(57,151,83,79)(58,142,84,70)(59,153,85,61)(60,144,86,72), (1,87,102,41)(2,88,103,42)(3,89,104,43)(4,90,105,44)(5,91,106,45)(6,92,107,46)(7,93,108,47)(8,94,109,48)(9,95,110,49)(10,96,111,50)(11,97,112,51)(12,98,113,52)(13,99,114,53)(14,100,115,54)(15,81,116,55)(16,82,117,56)(17,83,118,57)(18,84,119,58)(19,85,120,59)(20,86,101,60)(21,142,121,70)(22,143,122,71)(23,144,123,72)(24,145,124,73)(25,146,125,74)(26,147,126,75)(27,148,127,76)(28,149,128,77)(29,150,129,78)(30,151,130,79)(31,152,131,80)(32,153,132,61)(33,154,133,62)(34,155,134,63)(35,156,135,64)(36,157,136,65)(37,158,137,66)(38,159,138,67)(39,160,139,68)(40,141,140,69)>;
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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,141)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142), (1,34,102,134)(2,25,103,125)(3,36,104,136)(4,27,105,127)(5,38,106,138)(6,29,107,129)(7,40,108,140)(8,31,109,131)(9,22,110,122)(10,33,111,133)(11,24,112,124)(12,35,113,135)(13,26,114,126)(14,37,115,137)(15,28,116,128)(16,39,117,139)(17,30,118,130)(18,21,119,121)(19,32,120,132)(20,23,101,123)(41,155,87,63)(42,146,88,74)(43,157,89,65)(44,148,90,76)(45,159,91,67)(46,150,92,78)(47,141,93,69)(48,152,94,80)(49,143,95,71)(50,154,96,62)(51,145,97,73)(52,156,98,64)(53,147,99,75)(54,158,100,66)(55,149,81,77)(56,160,82,68)(57,151,83,79)(58,142,84,70)(59,153,85,61)(60,144,86,72), (1,87,102,41)(2,88,103,42)(3,89,104,43)(4,90,105,44)(5,91,106,45)(6,92,107,46)(7,93,108,47)(8,94,109,48)(9,95,110,49)(10,96,111,50)(11,97,112,51)(12,98,113,52)(13,99,114,53)(14,100,115,54)(15,81,116,55)(16,82,117,56)(17,83,118,57)(18,84,119,58)(19,85,120,59)(20,86,101,60)(21,142,121,70)(22,143,122,71)(23,144,123,72)(24,145,124,73)(25,146,125,74)(26,147,126,75)(27,148,127,76)(28,149,128,77)(29,150,129,78)(30,151,130,79)(31,152,131,80)(32,153,132,61)(33,154,133,62)(34,155,134,63)(35,156,135,64)(36,157,136,65)(37,158,137,66)(38,159,138,67)(39,160,139,68)(40,141,140,69) );
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)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,120),(13,119),(14,118),(15,117),(16,116),(17,115),(18,114),(19,113),(20,112),(21,121),(22,140),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,100),(48,99),(49,98),(50,97),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,141),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142)], [(1,34,102,134),(2,25,103,125),(3,36,104,136),(4,27,105,127),(5,38,106,138),(6,29,107,129),(7,40,108,140),(8,31,109,131),(9,22,110,122),(10,33,111,133),(11,24,112,124),(12,35,113,135),(13,26,114,126),(14,37,115,137),(15,28,116,128),(16,39,117,139),(17,30,118,130),(18,21,119,121),(19,32,120,132),(20,23,101,123),(41,155,87,63),(42,146,88,74),(43,157,89,65),(44,148,90,76),(45,159,91,67),(46,150,92,78),(47,141,93,69),(48,152,94,80),(49,143,95,71),(50,154,96,62),(51,145,97,73),(52,156,98,64),(53,147,99,75),(54,158,100,66),(55,149,81,77),(56,160,82,68),(57,151,83,79),(58,142,84,70),(59,153,85,61),(60,144,86,72)], [(1,87,102,41),(2,88,103,42),(3,89,104,43),(4,90,105,44),(5,91,106,45),(6,92,107,46),(7,93,108,47),(8,94,109,48),(9,95,110,49),(10,96,111,50),(11,97,112,51),(12,98,113,52),(13,99,114,53),(14,100,115,54),(15,81,116,55),(16,82,117,56),(17,83,118,57),(18,84,119,58),(19,85,120,59),(20,86,101,60),(21,142,121,70),(22,143,122,71),(23,144,123,72),(24,145,124,73),(25,146,125,74),(26,147,126,75),(27,148,127,76),(28,149,128,77),(29,150,129,78),(30,151,130,79),(31,152,131,80),(32,153,132,61),(33,154,133,62),(34,155,134,63),(35,156,135,64),(36,157,136,65),(37,158,137,66),(38,159,138,67),(39,160,139,68),(40,141,140,69)])
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 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | Q8⋊D5 | Q8×D5 | Q8⋊2D5 | D4.D10 |
| kernel | D20⋊5Q8 | C20⋊3C8 | C20.Q8 | 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⋊5Q8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 4 |
| 0 | 0 | 0 | 0 | 20 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 19 |
| 0 | 0 | 0 | 0 | 13 | 30 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 37 |
| 0 | 0 | 0 | 0 | 21 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,20,0,0,0,0,4,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[2,40,0,0,0,0,5,39,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,13,0,0,0,0,19,30],[32,40,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,37,40] >;
D20⋊5Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5Q_8 % in TeX
G:=Group("D20:5Q8"); // GroupNames label
G:=SmallGroup(320,711);
// by ID
G=gap.SmallGroup(320,711);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,268,1123,297,136,12550]);
// 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,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations