metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.9D4, C4.89(D4×D5), D4⋊C4⋊14D5, D20⋊8C4⋊4C2, C4⋊C4.141D10, (C2×D4).32D10, C4.4(C4○D20), C10.Q16⋊8C2, C20.113(C2×D4), (C2×C8).118D10, C5⋊1(D4.2D4), C10.43(C4○D8), C20.12(C4○D4), C20.17D4⋊2C2, C20.8Q8⋊12C2, (C2×Dic5).33D4, C22.184(D4×D5), C2.19(D8⋊D5), C10.20(C4⋊D4), C10.37(C8⋊C22), (C2×C20).227C23, (C2×C40).129C22, (C2×D20).59C22, (D4×C10).48C22, C2.23(D10⋊D4), (C4×Dic5).20C22, C2.13(SD16⋊3D5), (C2×Dic10).65C22, (C2×D4⋊D5).3C2, (C2×C40⋊C2)⋊16C2, (C5×D4⋊C4)⋊14C2, (C2×C10).240(C2×D4), (C5×C4⋊C4).28C22, (C2×C5⋊2C8).25C22, (C2×C4).334(C22×D5), SmallGroup(320,414)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.D4
G = < a,b,c,d | a20=b2=c4=1, d2=a5, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 566 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D4.2D4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×C10, C10.Q16, C20.8Q8, C5×D4⋊C4, D20⋊8C4, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, D20.D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C22×D5, D4.2D4, C4○D20, D4×D5, D10⋊D4, D8⋊D5, SD16⋊3D5, D20.D4
►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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 126)(7 125)(8 124)(9 123)(10 122)(11 121)(12 140)(13 139)(14 138)(15 137)(16 136)(17 135)(18 134)(19 133)(20 132)(21 153)(22 152)(23 151)(24 150)(25 149)(26 148)(27 147)(28 146)(29 145)(30 144)(31 143)(32 142)(33 141)(34 160)(35 159)(36 158)(37 157)(38 156)(39 155)(40 154)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 120)(69 119)(70 118)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 109)(80 108)
(1 156 127 34)(2 147 128 25)(3 158 129 36)(4 149 130 27)(5 160 131 38)(6 151 132 29)(7 142 133 40)(8 153 134 31)(9 144 135 22)(10 155 136 33)(11 146 137 24)(12 157 138 35)(13 148 139 26)(14 159 140 37)(15 150 121 28)(16 141 122 39)(17 152 123 30)(18 143 124 21)(19 154 125 32)(20 145 126 23)(41 105 87 80)(42 116 88 71)(43 107 89 62)(44 118 90 73)(45 109 91 64)(46 120 92 75)(47 111 93 66)(48 102 94 77)(49 113 95 68)(50 104 96 79)(51 115 97 70)(52 106 98 61)(53 117 99 72)(54 108 100 63)(55 119 81 74)(56 110 82 65)(57 101 83 76)(58 112 84 67)(59 103 85 78)(60 114 86 69)
(1 99 6 84 11 89 16 94)(2 100 7 85 12 90 17 95)(3 81 8 86 13 91 18 96)(4 82 9 87 14 92 19 97)(5 83 10 88 15 93 20 98)(21 64 26 69 31 74 36 79)(22 65 27 70 32 75 37 80)(23 66 28 71 33 76 38 61)(24 67 29 72 34 77 39 62)(25 68 30 73 35 78 40 63)(41 140 46 125 51 130 56 135)(42 121 47 126 52 131 57 136)(43 122 48 127 53 132 58 137)(44 123 49 128 54 133 59 138)(45 124 50 129 55 134 60 139)(101 160 106 145 111 150 116 155)(102 141 107 146 112 151 117 156)(103 142 108 147 113 152 118 157)(104 143 109 148 114 153 119 158)(105 144 110 149 115 154 120 159)
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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108), (1,156,127,34)(2,147,128,25)(3,158,129,36)(4,149,130,27)(5,160,131,38)(6,151,132,29)(7,142,133,40)(8,153,134,31)(9,144,135,22)(10,155,136,33)(11,146,137,24)(12,157,138,35)(13,148,139,26)(14,159,140,37)(15,150,121,28)(16,141,122,39)(17,152,123,30)(18,143,124,21)(19,154,125,32)(20,145,126,23)(41,105,87,80)(42,116,88,71)(43,107,89,62)(44,118,90,73)(45,109,91,64)(46,120,92,75)(47,111,93,66)(48,102,94,77)(49,113,95,68)(50,104,96,79)(51,115,97,70)(52,106,98,61)(53,117,99,72)(54,108,100,63)(55,119,81,74)(56,110,82,65)(57,101,83,76)(58,112,84,67)(59,103,85,78)(60,114,86,69), (1,99,6,84,11,89,16,94)(2,100,7,85,12,90,17,95)(3,81,8,86,13,91,18,96)(4,82,9,87,14,92,19,97)(5,83,10,88,15,93,20,98)(21,64,26,69,31,74,36,79)(22,65,27,70,32,75,37,80)(23,66,28,71,33,76,38,61)(24,67,29,72,34,77,39,62)(25,68,30,73,35,78,40,63)(41,140,46,125,51,130,56,135)(42,121,47,126,52,131,57,136)(43,122,48,127,53,132,58,137)(44,123,49,128,54,133,59,138)(45,124,50,129,55,134,60,139)(101,160,106,145,111,150,116,155)(102,141,107,146,112,151,117,156)(103,142,108,147,113,152,118,157)(104,143,109,148,114,153,119,158)(105,144,110,149,115,154,120,159)>;
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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,160)(35,159)(36,158)(37,157)(38,156)(39,155)(40,154)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108), (1,156,127,34)(2,147,128,25)(3,158,129,36)(4,149,130,27)(5,160,131,38)(6,151,132,29)(7,142,133,40)(8,153,134,31)(9,144,135,22)(10,155,136,33)(11,146,137,24)(12,157,138,35)(13,148,139,26)(14,159,140,37)(15,150,121,28)(16,141,122,39)(17,152,123,30)(18,143,124,21)(19,154,125,32)(20,145,126,23)(41,105,87,80)(42,116,88,71)(43,107,89,62)(44,118,90,73)(45,109,91,64)(46,120,92,75)(47,111,93,66)(48,102,94,77)(49,113,95,68)(50,104,96,79)(51,115,97,70)(52,106,98,61)(53,117,99,72)(54,108,100,63)(55,119,81,74)(56,110,82,65)(57,101,83,76)(58,112,84,67)(59,103,85,78)(60,114,86,69), (1,99,6,84,11,89,16,94)(2,100,7,85,12,90,17,95)(3,81,8,86,13,91,18,96)(4,82,9,87,14,92,19,97)(5,83,10,88,15,93,20,98)(21,64,26,69,31,74,36,79)(22,65,27,70,32,75,37,80)(23,66,28,71,33,76,38,61)(24,67,29,72,34,77,39,62)(25,68,30,73,35,78,40,63)(41,140,46,125,51,130,56,135)(42,121,47,126,52,131,57,136)(43,122,48,127,53,132,58,137)(44,123,49,128,54,133,59,138)(45,124,50,129,55,134,60,139)(101,160,106,145,111,150,116,155)(102,141,107,146,112,151,117,156)(103,142,108,147,113,152,118,157)(104,143,109,148,114,153,119,158)(105,144,110,149,115,154,120,159) );
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)], [(1,131),(2,130),(3,129),(4,128),(5,127),(6,126),(7,125),(8,124),(9,123),(10,122),(11,121),(12,140),(13,139),(14,138),(15,137),(16,136),(17,135),(18,134),(19,133),(20,132),(21,153),(22,152),(23,151),(24,150),(25,149),(26,148),(27,147),(28,146),(29,145),(30,144),(31,143),(32,142),(33,141),(34,160),(35,159),(36,158),(37,157),(38,156),(39,155),(40,154),(41,100),(42,99),(43,98),(44,97),(45,96),(46,95),(47,94),(48,93),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(61,107),(62,106),(63,105),(64,104),(65,103),(66,102),(67,101),(68,120),(69,119),(70,118),(71,117),(72,116),(73,115),(74,114),(75,113),(76,112),(77,111),(78,110),(79,109),(80,108)], [(1,156,127,34),(2,147,128,25),(3,158,129,36),(4,149,130,27),(5,160,131,38),(6,151,132,29),(7,142,133,40),(8,153,134,31),(9,144,135,22),(10,155,136,33),(11,146,137,24),(12,157,138,35),(13,148,139,26),(14,159,140,37),(15,150,121,28),(16,141,122,39),(17,152,123,30),(18,143,124,21),(19,154,125,32),(20,145,126,23),(41,105,87,80),(42,116,88,71),(43,107,89,62),(44,118,90,73),(45,109,91,64),(46,120,92,75),(47,111,93,66),(48,102,94,77),(49,113,95,68),(50,104,96,79),(51,115,97,70),(52,106,98,61),(53,117,99,72),(54,108,100,63),(55,119,81,74),(56,110,82,65),(57,101,83,76),(58,112,84,67),(59,103,85,78),(60,114,86,69)], [(1,99,6,84,11,89,16,94),(2,100,7,85,12,90,17,95),(3,81,8,86,13,91,18,96),(4,82,9,87,14,92,19,97),(5,83,10,88,15,93,20,98),(21,64,26,69,31,74,36,79),(22,65,27,70,32,75,37,80),(23,66,28,71,33,76,38,61),(24,67,29,72,34,77,39,62),(25,68,30,73,35,78,40,63),(41,140,46,125,51,130,56,135),(42,121,47,126,52,131,57,136),(43,122,48,127,53,132,58,137),(44,123,49,128,54,133,59,138),(45,124,50,129,55,134,60,139),(101,160,106,145,111,150,116,155),(102,141,107,146,112,151,117,156),(103,142,108,147,113,152,118,157),(104,143,109,148,114,153,119,158),(105,144,110,149,115,154,120,159)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 4 | 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 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | SD16⋊3D5 |
| kernel | D20.D4 | C10.Q16 | C20.8Q8 | C5×D4⋊C4 | D20⋊8C4 | C2×C40⋊C2 | C2×D4⋊D5 | C20.17D4 | D20 | C2×Dic5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.D4 ►in GL4(𝔽41) generated by
| 1 | 9 | 0 | 0 |
| 18 | 40 | 0 | 0 |
| 0 | 0 | 6 | 5 |
| 0 | 0 | 1 | 1 |
| 40 | 32 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 40 |
| 9 | 40 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 29 | 0 | 0 |
| 17 | 30 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 39 |
G:=sub<GL(4,GF(41))| [1,18,0,0,9,40,0,0,0,0,6,1,0,0,5,1],[40,0,0,0,32,1,0,0,0,0,1,0,0,0,36,40],[9,0,0,0,40,32,0,0,0,0,9,0,0,0,0,9],[0,17,0,0,29,30,0,0,0,0,2,9,0,0,4,39] >;
D20.D4 in GAP, Magma, Sage, TeX
D_{20}.D_4 % in TeX
G:=Group("D20.D4"); // GroupNames label
G:=SmallGroup(320,414);
// by ID
G=gap.SmallGroup(320,414);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations