metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊3Q8, C20.13SD16, C42.33D10, C4⋊C8⋊9D5, C4.43(Q8×D5), C5⋊2(D4⋊2Q8), C40⋊6C4⋊16C2, C20⋊2Q8⋊12C2, (C4×D20).11C2, (C2×C8).129D10, (C2×C20).121D4, (C2×C4).132D20, C20.102(C2×Q8), D20⋊5C4.5C2, C4.13(C40⋊C2), (C4×C20).68C22, C10.11(C2×SD16), C20.286(C4○D4), C2.16(C8⋊D10), C10.13(C8⋊C22), (C2×C40).139C22, (C2×C20).752C23, C22.115(C2×D20), C10.30(C22⋊Q8), C4⋊Dic5.18C22, C4.110(D4⋊2D5), C2.11(D10⋊2Q8), (C2×D20).201C22, (C5×C4⋊C8)⋊11C2, C2.14(C2×C40⋊C2), (C2×C10).135(C2×D4), (C2×C4).697(C22×D5), SmallGroup(320,469)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊3Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 518 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, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4⋊2Q8, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C40⋊6C4, D20⋊5C4, C5×C4⋊C8, C20⋊2Q8, C4×D20, D20⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, D20, C22×D5, D4⋊2Q8, C40⋊C2, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8, C2×C40⋊C2, C8⋊D10, D20⋊3Q8
►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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(41 125)(42 124)(43 123)(44 122)(45 121)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 83)(62 82)(63 81)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(101 158)(102 157)(103 156)(104 155)(105 154)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 144)(116 143)(117 142)(118 141)(119 160)(120 159)
(1 77 37 93)(2 76 38 92)(3 75 39 91)(4 74 40 90)(5 73 21 89)(6 72 22 88)(7 71 23 87)(8 70 24 86)(9 69 25 85)(10 68 26 84)(11 67 27 83)(12 66 28 82)(13 65 29 81)(14 64 30 100)(15 63 31 99)(16 62 32 98)(17 61 33 97)(18 80 34 96)(19 79 35 95)(20 78 36 94)(41 109 130 142)(42 108 131 141)(43 107 132 160)(44 106 133 159)(45 105 134 158)(46 104 135 157)(47 103 136 156)(48 102 137 155)(49 101 138 154)(50 120 139 153)(51 119 140 152)(52 118 121 151)(53 117 122 150)(54 116 123 149)(55 115 124 148)(56 114 125 147)(57 113 126 146)(58 112 127 145)(59 111 128 144)(60 110 129 143)
(1 49 37 138)(2 50 38 139)(3 51 39 140)(4 52 40 121)(5 53 21 122)(6 54 22 123)(7 55 23 124)(8 56 24 125)(9 57 25 126)(10 58 26 127)(11 59 27 128)(12 60 28 129)(13 41 29 130)(14 42 30 131)(15 43 31 132)(16 44 32 133)(17 45 33 134)(18 46 34 135)(19 47 35 136)(20 48 36 137)(61 158 97 105)(62 159 98 106)(63 160 99 107)(64 141 100 108)(65 142 81 109)(66 143 82 110)(67 144 83 111)(68 145 84 112)(69 146 85 113)(70 147 86 114)(71 148 87 115)(72 149 88 116)(73 150 89 117)(74 151 90 118)(75 152 91 119)(76 153 92 120)(77 154 93 101)(78 155 94 102)(79 156 95 103)(80 157 96 104)
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,125)(42,124)(43,123)(44,122)(45,121)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,83)(62,82)(63,81)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,160)(120,159), (1,77,37,93)(2,76,38,92)(3,75,39,91)(4,74,40,90)(5,73,21,89)(6,72,22,88)(7,71,23,87)(8,70,24,86)(9,69,25,85)(10,68,26,84)(11,67,27,83)(12,66,28,82)(13,65,29,81)(14,64,30,100)(15,63,31,99)(16,62,32,98)(17,61,33,97)(18,80,34,96)(19,79,35,95)(20,78,36,94)(41,109,130,142)(42,108,131,141)(43,107,132,160)(44,106,133,159)(45,105,134,158)(46,104,135,157)(47,103,136,156)(48,102,137,155)(49,101,138,154)(50,120,139,153)(51,119,140,152)(52,118,121,151)(53,117,122,150)(54,116,123,149)(55,115,124,148)(56,114,125,147)(57,113,126,146)(58,112,127,145)(59,111,128,144)(60,110,129,143), (1,49,37,138)(2,50,38,139)(3,51,39,140)(4,52,40,121)(5,53,21,122)(6,54,22,123)(7,55,23,124)(8,56,24,125)(9,57,25,126)(10,58,26,127)(11,59,27,128)(12,60,28,129)(13,41,29,130)(14,42,30,131)(15,43,31,132)(16,44,32,133)(17,45,33,134)(18,46,34,135)(19,47,35,136)(20,48,36,137)(61,158,97,105)(62,159,98,106)(63,160,99,107)(64,141,100,108)(65,142,81,109)(66,143,82,110)(67,144,83,111)(68,145,84,112)(69,146,85,113)(70,147,86,114)(71,148,87,115)(72,149,88,116)(73,150,89,117)(74,151,90,118)(75,152,91,119)(76,153,92,120)(77,154,93,101)(78,155,94,102)(79,156,95,103)(80,157,96,104)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,125)(42,124)(43,123)(44,122)(45,121)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,83)(62,82)(63,81)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,160)(120,159), (1,77,37,93)(2,76,38,92)(3,75,39,91)(4,74,40,90)(5,73,21,89)(6,72,22,88)(7,71,23,87)(8,70,24,86)(9,69,25,85)(10,68,26,84)(11,67,27,83)(12,66,28,82)(13,65,29,81)(14,64,30,100)(15,63,31,99)(16,62,32,98)(17,61,33,97)(18,80,34,96)(19,79,35,95)(20,78,36,94)(41,109,130,142)(42,108,131,141)(43,107,132,160)(44,106,133,159)(45,105,134,158)(46,104,135,157)(47,103,136,156)(48,102,137,155)(49,101,138,154)(50,120,139,153)(51,119,140,152)(52,118,121,151)(53,117,122,150)(54,116,123,149)(55,115,124,148)(56,114,125,147)(57,113,126,146)(58,112,127,145)(59,111,128,144)(60,110,129,143), (1,49,37,138)(2,50,38,139)(3,51,39,140)(4,52,40,121)(5,53,21,122)(6,54,22,123)(7,55,23,124)(8,56,24,125)(9,57,25,126)(10,58,26,127)(11,59,27,128)(12,60,28,129)(13,41,29,130)(14,42,30,131)(15,43,31,132)(16,44,32,133)(17,45,33,134)(18,46,34,135)(19,47,35,136)(20,48,36,137)(61,158,97,105)(62,159,98,106)(63,160,99,107)(64,141,100,108)(65,142,81,109)(66,143,82,110)(67,144,83,111)(68,145,84,112)(69,146,85,113)(70,147,86,114)(71,148,87,115)(72,149,88,116)(73,150,89,117)(74,151,90,118)(75,152,91,119)(76,153,92,120)(77,154,93,101)(78,155,94,102)(79,156,95,103)(80,157,96,104) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(41,125),(42,124),(43,123),(44,122),(45,121),(46,140),(47,139),(48,138),(49,137),(50,136),(51,135),(52,134),(53,133),(54,132),(55,131),(56,130),(57,129),(58,128),(59,127),(60,126),(61,83),(62,82),(63,81),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,84),(101,158),(102,157),(103,156),(104,155),(105,154),(106,153),(107,152),(108,151),(109,150),(110,149),(111,148),(112,147),(113,146),(114,145),(115,144),(116,143),(117,142),(118,141),(119,160),(120,159)], [(1,77,37,93),(2,76,38,92),(3,75,39,91),(4,74,40,90),(5,73,21,89),(6,72,22,88),(7,71,23,87),(8,70,24,86),(9,69,25,85),(10,68,26,84),(11,67,27,83),(12,66,28,82),(13,65,29,81),(14,64,30,100),(15,63,31,99),(16,62,32,98),(17,61,33,97),(18,80,34,96),(19,79,35,95),(20,78,36,94),(41,109,130,142),(42,108,131,141),(43,107,132,160),(44,106,133,159),(45,105,134,158),(46,104,135,157),(47,103,136,156),(48,102,137,155),(49,101,138,154),(50,120,139,153),(51,119,140,152),(52,118,121,151),(53,117,122,150),(54,116,123,149),(55,115,124,148),(56,114,125,147),(57,113,126,146),(58,112,127,145),(59,111,128,144),(60,110,129,143)], [(1,49,37,138),(2,50,38,139),(3,51,39,140),(4,52,40,121),(5,53,21,122),(6,54,22,123),(7,55,23,124),(8,56,24,125),(9,57,25,126),(10,58,26,127),(11,59,27,128),(12,60,28,129),(13,41,29,130),(14,42,30,131),(15,43,31,132),(16,44,32,133),(17,45,33,134),(18,46,34,135),(19,47,35,136),(20,48,36,137),(61,158,97,105),(62,159,98,106),(63,160,99,107),(64,141,100,108),(65,142,81,109),(66,143,82,110),(67,144,83,111),(68,145,84,112),(69,146,85,113),(70,147,86,114),(71,148,87,115),(72,149,88,116),(73,150,89,117),(74,151,90,118),(75,152,91,119),(76,153,92,120),(77,154,93,101),(78,155,94,102),(79,156,95,103),(80,157,96,104)]])
59 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 | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| 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 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D20 | C40⋊C2 | C8⋊C22 | D4⋊2D5 | Q8×D5 | C8⋊D10 |
| kernel | D20⋊3Q8 | C40⋊6C4 | D20⋊5C4 | C5×C4⋊C8 | C20⋊2Q8 | C4×D20 | D20 | C2×C20 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of D20⋊3Q8 ►in GL4(𝔽41) generated by
| 28 | 39 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 28 | 39 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 40 |
| 37 | 10 | 0 | 0 |
| 27 | 4 | 0 | 0 |
| 0 | 0 | 30 | 2 |
| 0 | 0 | 21 | 11 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 24 | 9 |
G:=sub<GL(4,GF(41))| [28,2,0,0,39,16,0,0,0,0,40,0,0,0,0,40],[28,2,0,0,39,13,0,0,0,0,1,11,0,0,0,40],[37,27,0,0,10,4,0,0,0,0,30,21,0,0,2,11],[39,13,0,0,28,2,0,0,0,0,32,24,0,0,0,9] >;
D20⋊3Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_3Q_8 % in TeX
G:=Group("D20:3Q8"); // GroupNames label
G:=SmallGroup(320,469);
// by ID
G=gap.SmallGroup(320,469);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations