metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.17D4, (C2×Q16)⋊4D5, C4.68(D4×D5), C20.53(C2×D4), (C2×C8).39D10, (C5×Q8).10D4, D10⋊3Q8⋊7C2, (C10×Q16)⋊14C2, C5⋊6(D4.7D4), (C2×Q8).63D10, Q8.9(C5⋊D4), D20⋊5C4⋊30C2, D10⋊1C8⋊29C2, C10.62C22≀C2, C10.80(C4○D8), Q8⋊Dic5⋊34C2, (C22×D5).48D4, C22.278(D4×D5), (C2×C20).461C23, (C2×C40).253C22, (C2×Dic5).244D4, (Q8×C10).90C22, C2.30(C23⋊D10), C2.17(Q8.D10), C2.29(Q16⋊D5), (C2×D20).129C22, C10.79(C8.C22), C4⋊Dic5.184C22, (C2×Q8⋊D5)⋊20C2, C4.49(C2×C5⋊D4), (C2×C4×D5).57C22, (C2×C10).372(C2×D4), (C2×Q8⋊2D5).5C2, (C2×C4).549(C22×D5), (C2×C5⋊2C8).166C22, SmallGroup(320,814)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.17D4
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=cac-1=a-1, dad-1=a11, cbc-1=a3b, dbd-1=a5b, dcd-1=a10c-1 >
Subgroups: 670 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, D4.7D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5, C2×C40, C5×Q16, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8×C10, D10⋊1C8, D20⋊5C4, Q8⋊Dic5, C2×Q8⋊D5, D10⋊3Q8, C10×Q16, C2×Q8⋊2D5, D20.17D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, C5⋊D4, C22×D5, D4.7D4, D4×D5, C2×C5⋊D4, Q16⋊D5, Q8.D10, C23⋊D10, D20.17D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)
(1 42 80 40)(2 41 61 39)(3 60 62 38)(4 59 63 37)(5 58 64 36)(6 57 65 35)(7 56 66 34)(8 55 67 33)(9 54 68 32)(10 53 69 31)(11 52 70 30)(12 51 71 29)(13 50 72 28)(14 49 73 27)(15 48 74 26)(16 47 75 25)(17 46 76 24)(18 45 77 23)(19 44 78 22)(20 43 79 21)(81 116 145 121)(82 115 146 140)(83 114 147 139)(84 113 148 138)(85 112 149 137)(86 111 150 136)(87 110 151 135)(88 109 152 134)(89 108 153 133)(90 107 154 132)(91 106 155 131)(92 105 156 130)(93 104 157 129)(94 103 158 128)(95 102 159 127)(96 101 160 126)(97 120 141 125)(98 119 142 124)(99 118 143 123)(100 117 144 122)
(1 158 11 148)(2 149 12 159)(3 160 13 150)(4 151 14 141)(5 142 15 152)(6 153 16 143)(7 144 17 154)(8 155 18 145)(9 146 19 156)(10 157 20 147)(21 129 31 139)(22 140 32 130)(23 131 33 121)(24 122 34 132)(25 133 35 123)(26 124 36 134)(27 135 37 125)(28 126 38 136)(29 137 39 127)(30 128 40 138)(41 102 51 112)(42 113 52 103)(43 104 53 114)(44 115 54 105)(45 106 55 116)(46 117 56 107)(47 108 57 118)(48 119 58 109)(49 110 59 120)(50 101 60 111)(61 85 71 95)(62 96 72 86)(63 87 73 97)(64 98 74 88)(65 89 75 99)(66 100 76 90)(67 91 77 81)(68 82 78 92)(69 93 79 83)(70 84 80 94)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,42,80,40)(2,41,61,39)(3,60,62,38)(4,59,63,37)(5,58,64,36)(6,57,65,35)(7,56,66,34)(8,55,67,33)(9,54,68,32)(10,53,69,31)(11,52,70,30)(12,51,71,29)(13,50,72,28)(14,49,73,27)(15,48,74,26)(16,47,75,25)(17,46,76,24)(18,45,77,23)(19,44,78,22)(20,43,79,21)(81,116,145,121)(82,115,146,140)(83,114,147,139)(84,113,148,138)(85,112,149,137)(86,111,150,136)(87,110,151,135)(88,109,152,134)(89,108,153,133)(90,107,154,132)(91,106,155,131)(92,105,156,130)(93,104,157,129)(94,103,158,128)(95,102,159,127)(96,101,160,126)(97,120,141,125)(98,119,142,124)(99,118,143,123)(100,117,144,122), (1,158,11,148)(2,149,12,159)(3,160,13,150)(4,151,14,141)(5,142,15,152)(6,153,16,143)(7,144,17,154)(8,155,18,145)(9,146,19,156)(10,157,20,147)(21,129,31,139)(22,140,32,130)(23,131,33,121)(24,122,34,132)(25,133,35,123)(26,124,36,134)(27,135,37,125)(28,126,38,136)(29,137,39,127)(30,128,40,138)(41,102,51,112)(42,113,52,103)(43,104,53,114)(44,115,54,105)(45,106,55,116)(46,117,56,107)(47,108,57,118)(48,119,58,109)(49,110,59,120)(50,101,60,111)(61,85,71,95)(62,96,72,86)(63,87,73,97)(64,98,74,88)(65,89,75,99)(66,100,76,90)(67,91,77,81)(68,82,78,92)(69,93,79,83)(70,84,80,94)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,42,80,40)(2,41,61,39)(3,60,62,38)(4,59,63,37)(5,58,64,36)(6,57,65,35)(7,56,66,34)(8,55,67,33)(9,54,68,32)(10,53,69,31)(11,52,70,30)(12,51,71,29)(13,50,72,28)(14,49,73,27)(15,48,74,26)(16,47,75,25)(17,46,76,24)(18,45,77,23)(19,44,78,22)(20,43,79,21)(81,116,145,121)(82,115,146,140)(83,114,147,139)(84,113,148,138)(85,112,149,137)(86,111,150,136)(87,110,151,135)(88,109,152,134)(89,108,153,133)(90,107,154,132)(91,106,155,131)(92,105,156,130)(93,104,157,129)(94,103,158,128)(95,102,159,127)(96,101,160,126)(97,120,141,125)(98,119,142,124)(99,118,143,123)(100,117,144,122), (1,158,11,148)(2,149,12,159)(3,160,13,150)(4,151,14,141)(5,142,15,152)(6,153,16,143)(7,144,17,154)(8,155,18,145)(9,146,19,156)(10,157,20,147)(21,129,31,139)(22,140,32,130)(23,131,33,121)(24,122,34,132)(25,133,35,123)(26,124,36,134)(27,135,37,125)(28,126,38,136)(29,137,39,127)(30,128,40,138)(41,102,51,112)(42,113,52,103)(43,104,53,114)(44,115,54,105)(45,106,55,116)(46,117,56,107)(47,108,57,118)(48,119,58,109)(49,110,59,120)(50,101,60,111)(61,85,71,95)(62,96,72,86)(63,87,73,97)(64,98,74,88)(65,89,75,99)(66,100,76,90)(67,91,77,81)(68,82,78,92)(69,93,79,83)(70,84,80,94) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,23),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(121,124),(122,123),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(141,149),(142,148),(143,147),(144,146),(150,160),(151,159),(152,158),(153,157),(154,156)], [(1,42,80,40),(2,41,61,39),(3,60,62,38),(4,59,63,37),(5,58,64,36),(6,57,65,35),(7,56,66,34),(8,55,67,33),(9,54,68,32),(10,53,69,31),(11,52,70,30),(12,51,71,29),(13,50,72,28),(14,49,73,27),(15,48,74,26),(16,47,75,25),(17,46,76,24),(18,45,77,23),(19,44,78,22),(20,43,79,21),(81,116,145,121),(82,115,146,140),(83,114,147,139),(84,113,148,138),(85,112,149,137),(86,111,150,136),(87,110,151,135),(88,109,152,134),(89,108,153,133),(90,107,154,132),(91,106,155,131),(92,105,156,130),(93,104,157,129),(94,103,158,128),(95,102,159,127),(96,101,160,126),(97,120,141,125),(98,119,142,124),(99,118,143,123),(100,117,144,122)], [(1,158,11,148),(2,149,12,159),(3,160,13,150),(4,151,14,141),(5,142,15,152),(6,153,16,143),(7,144,17,154),(8,155,18,145),(9,146,19,156),(10,157,20,147),(21,129,31,139),(22,140,32,130),(23,131,33,121),(24,122,34,132),(25,133,35,123),(26,124,36,134),(27,135,37,125),(28,126,38,136),(29,137,39,127),(30,128,40,138),(41,102,51,112),(42,113,52,103),(43,104,53,114),(44,115,54,105),(45,106,55,116),(46,117,56,107),(47,108,57,118),(48,119,58,109),(49,110,59,120),(50,101,60,111),(61,85,71,95),(62,96,72,86),(63,87,73,97),(64,98,74,88),(65,89,75,99),(66,100,76,90),(67,91,77,81),(68,82,78,92),(69,93,79,83),(70,84,80,94)]])
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 | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | Q16⋊D5 | Q8.D10 |
| kernel | D20.17D4 | D10⋊1C8 | D20⋊5C4 | Q8⋊Dic5 | C2×Q8⋊D5 | D10⋊3Q8 | C10×Q16 | C2×Q8⋊2D5 | D20 | C2×Dic5 | C5×Q8 | C22×D5 | C2×Q16 | C2×C8 | C2×Q8 | C10 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.17D4 ►in GL4(𝔽41) generated by
| 7 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 39 | 40 |
| 40 | 7 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 2 | 1 |
| 3 | 3 | 0 | 0 |
| 24 | 38 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 24 | 0 |
| 17 | 1 | 0 | 0 |
| 40 | 24 | 0 | 0 |
| 0 | 0 | 11 | 26 |
| 0 | 0 | 30 | 30 |
G:=sub<GL(4,GF(41))| [7,1,0,0,40,0,0,0,0,0,1,39,0,0,1,40],[40,0,0,0,7,1,0,0,0,0,40,2,0,0,0,1],[3,24,0,0,3,38,0,0,0,0,0,24,0,0,12,0],[17,40,0,0,1,24,0,0,0,0,11,30,0,0,26,30] >;
D20.17D4 in GAP, Magma, Sage, TeX
D_{20}._{17}D_4 % in TeX
G:=Group("D20.17D4"); // GroupNames label
G:=SmallGroup(320,814);
// by ID
G=gap.SmallGroup(320,814);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,758,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^11,c*b*c^-1=a^3*b,d*b*d^-1=a^5*b,d*c*d^-1=a^10*c^-1>;
// generators/relations