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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D4.Q8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C20⋊3C8, C10.D8, C20.Q8, D20⋊6C4, C4×D20, C5×C42.C2, D20.4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, C5⋊D4, 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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 142)(22 141)(23 160)(24 159)(25 158)(26 157)(27 156)(28 155)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 122)(82 121)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)
(1 63 47 116)(2 74 48 107)(3 65 49 118)(4 76 50 109)(5 67 51 120)(6 78 52 111)(7 69 53 102)(8 80 54 113)(9 71 55 104)(10 62 56 115)(11 73 57 106)(12 64 58 117)(13 75 59 108)(14 66 60 119)(15 77 41 110)(16 68 42 101)(17 79 43 112)(18 70 44 103)(19 61 45 114)(20 72 46 105)(21 85 158 139)(22 96 159 130)(23 87 160 121)(24 98 141 132)(25 89 142 123)(26 100 143 134)(27 91 144 125)(28 82 145 136)(29 93 146 127)(30 84 147 138)(31 95 148 129)(32 86 149 140)(33 97 150 131)(34 88 151 122)(35 99 152 133)(36 90 153 124)(37 81 154 135)(38 92 155 126)(39 83 156 137)(40 94 157 128)
(1 124 57 100)(2 125 58 81)(3 126 59 82)(4 127 60 83)(5 128 41 84)(6 129 42 85)(7 130 43 86)(8 131 44 87)(9 132 45 88)(10 133 46 89)(11 134 47 90)(12 135 48 91)(13 136 49 92)(14 137 50 93)(15 138 51 94)(16 139 52 95)(17 140 53 96)(18 121 54 97)(19 122 55 98)(20 123 56 99)(21 78 148 101)(22 79 149 102)(23 80 150 103)(24 61 151 104)(25 62 152 105)(26 63 153 106)(27 64 154 107)(28 65 155 108)(29 66 156 109)(30 67 157 110)(31 68 158 111)(32 69 159 112)(33 70 160 113)(34 71 141 114)(35 72 142 115)(36 73 143 116)(37 74 144 117)(38 75 145 118)(39 76 146 119)(40 77 147 120)
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,142)(22,141)(23,160)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123), (1,63,47,116)(2,74,48,107)(3,65,49,118)(4,76,50,109)(5,67,51,120)(6,78,52,111)(7,69,53,102)(8,80,54,113)(9,71,55,104)(10,62,56,115)(11,73,57,106)(12,64,58,117)(13,75,59,108)(14,66,60,119)(15,77,41,110)(16,68,42,101)(17,79,43,112)(18,70,44,103)(19,61,45,114)(20,72,46,105)(21,85,158,139)(22,96,159,130)(23,87,160,121)(24,98,141,132)(25,89,142,123)(26,100,143,134)(27,91,144,125)(28,82,145,136)(29,93,146,127)(30,84,147,138)(31,95,148,129)(32,86,149,140)(33,97,150,131)(34,88,151,122)(35,99,152,133)(36,90,153,124)(37,81,154,135)(38,92,155,126)(39,83,156,137)(40,94,157,128), (1,124,57,100)(2,125,58,81)(3,126,59,82)(4,127,60,83)(5,128,41,84)(6,129,42,85)(7,130,43,86)(8,131,44,87)(9,132,45,88)(10,133,46,89)(11,134,47,90)(12,135,48,91)(13,136,49,92)(14,137,50,93)(15,138,51,94)(16,139,52,95)(17,140,53,96)(18,121,54,97)(19,122,55,98)(20,123,56,99)(21,78,148,101)(22,79,149,102)(23,80,150,103)(24,61,151,104)(25,62,152,105)(26,63,153,106)(27,64,154,107)(28,65,155,108)(29,66,156,109)(30,67,157,110)(31,68,158,111)(32,69,159,112)(33,70,160,113)(34,71,141,114)(35,72,142,115)(36,73,143,116)(37,74,144,117)(38,75,145,118)(39,76,146,119)(40,77,147,120)>;
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,142)(22,141)(23,160)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123), (1,63,47,116)(2,74,48,107)(3,65,49,118)(4,76,50,109)(5,67,51,120)(6,78,52,111)(7,69,53,102)(8,80,54,113)(9,71,55,104)(10,62,56,115)(11,73,57,106)(12,64,58,117)(13,75,59,108)(14,66,60,119)(15,77,41,110)(16,68,42,101)(17,79,43,112)(18,70,44,103)(19,61,45,114)(20,72,46,105)(21,85,158,139)(22,96,159,130)(23,87,160,121)(24,98,141,132)(25,89,142,123)(26,100,143,134)(27,91,144,125)(28,82,145,136)(29,93,146,127)(30,84,147,138)(31,95,148,129)(32,86,149,140)(33,97,150,131)(34,88,151,122)(35,99,152,133)(36,90,153,124)(37,81,154,135)(38,92,155,126)(39,83,156,137)(40,94,157,128), (1,124,57,100)(2,125,58,81)(3,126,59,82)(4,127,60,83)(5,128,41,84)(6,129,42,85)(7,130,43,86)(8,131,44,87)(9,132,45,88)(10,133,46,89)(11,134,47,90)(12,135,48,91)(13,136,49,92)(14,137,50,93)(15,138,51,94)(16,139,52,95)(17,140,53,96)(18,121,54,97)(19,122,55,98)(20,123,56,99)(21,78,148,101)(22,79,149,102)(23,80,150,103)(24,61,151,104)(25,62,152,105)(26,63,153,106)(27,64,154,107)(28,65,155,108)(29,66,156,109)(30,67,157,110)(31,68,158,111)(32,69,159,112)(33,70,160,113)(34,71,141,114)(35,72,142,115)(36,73,143,116)(37,74,144,117)(38,75,145,118)(39,76,146,119)(40,77,147,120) );
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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,60),(18,59),(19,58),(20,57),(21,142),(22,141),(23,160),(24,159),(25,158),(26,157),(27,156),(28,155),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,143),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113),(81,122),(82,121),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123)], [(1,63,47,116),(2,74,48,107),(3,65,49,118),(4,76,50,109),(5,67,51,120),(6,78,52,111),(7,69,53,102),(8,80,54,113),(9,71,55,104),(10,62,56,115),(11,73,57,106),(12,64,58,117),(13,75,59,108),(14,66,60,119),(15,77,41,110),(16,68,42,101),(17,79,43,112),(18,70,44,103),(19,61,45,114),(20,72,46,105),(21,85,158,139),(22,96,159,130),(23,87,160,121),(24,98,141,132),(25,89,142,123),(26,100,143,134),(27,91,144,125),(28,82,145,136),(29,93,146,127),(30,84,147,138),(31,95,148,129),(32,86,149,140),(33,97,150,131),(34,88,151,122),(35,99,152,133),(36,90,153,124),(37,81,154,135),(38,92,155,126),(39,83,156,137),(40,94,157,128)], [(1,124,57,100),(2,125,58,81),(3,126,59,82),(4,127,60,83),(5,128,41,84),(6,129,42,85),(7,130,43,86),(8,131,44,87),(9,132,45,88),(10,133,46,89),(11,134,47,90),(12,135,48,91),(13,136,49,92),(14,137,50,93),(15,138,51,94),(16,139,52,95),(17,140,53,96),(18,121,54,97),(19,122,55,98),(20,123,56,99),(21,78,148,101),(22,79,149,102),(23,80,150,103),(24,61,151,104),(25,62,152,105),(26,63,153,106),(27,64,154,107),(28,65,155,108),(29,66,156,109),(30,67,157,110),(31,68,158,111),(32,69,159,112),(33,70,160,113),(34,71,141,114),(35,72,142,115),(36,73,143,116),(37,74,144,117),(38,75,145,118),(39,76,146,119),(40,77,147,120)]])
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