metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.1C8, Dic10.1C8, M4(2).2F5, C20.3(C2×C8), C5⋊1(D4.C8), C4.3(D5⋊C8), C4○D20.1C4, C5⋊2C8.20D4, C20.C8⋊1C2, C10.5(C22⋊C8), (C5×M4(2)).2C4, (C2×C10).1M4(2), C4.39(C22⋊F5), C2.6(D10⋊C8), C20.37(C22⋊C4), C22.1(C4.F5), D20.2C4.1C2, (C2×C5⋊C16)⋊1C2, (C2×C4).68(C2×F5), (C2×C20).34(C2×C4), (C2×C5⋊2C8).186C22, SmallGroup(320,236)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, cac-1=a3, cbc-1=a17b >
2C2
20C2
10C22
10C4
2C10
4D5
2C8
5Q8
5D4
5C8
5C8
10C2×C4
10D4
2Dic5
2D10
10C16
10C2×C8
10C16
10M4(2)
2C40
Smallest permutation representation of D20.C8
►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 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 142)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)
(1 92 21 53 152 134 80 107 11 82 31 43 142 124 70 117)(2 99 30 56 153 121 69 110 12 89 40 46 143 131 79 120)(3 86 39 59 154 128 78 113 13 96 29 49 144 138 68 103)(4 93 28 42 155 135 67 116 14 83 38 52 145 125 77 106)(5 100 37 45 156 122 76 119 15 90 27 55 146 132 66 109)(6 87 26 48 157 129 65 102 16 97 36 58 147 139 75 112)(7 94 35 51 158 136 74 105 17 84 25 41 148 126 64 115)(8 81 24 54 159 123 63 108 18 91 34 44 149 133 73 118)(9 88 33 57 160 130 72 111 19 98 23 47 150 140 62 101)(10 95 22 60 141 137 61 114 20 85 32 50 151 127 71 104)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152), (1,92,21,53,152,134,80,107,11,82,31,43,142,124,70,117)(2,99,30,56,153,121,69,110,12,89,40,46,143,131,79,120)(3,86,39,59,154,128,78,113,13,96,29,49,144,138,68,103)(4,93,28,42,155,135,67,116,14,83,38,52,145,125,77,106)(5,100,37,45,156,122,76,119,15,90,27,55,146,132,66,109)(6,87,26,48,157,129,65,102,16,97,36,58,147,139,75,112)(7,94,35,51,158,136,74,105,17,84,25,41,148,126,64,115)(8,81,24,54,159,123,63,108,18,91,34,44,149,133,73,118)(9,88,33,57,160,130,72,111,19,98,23,47,150,140,62,101)(10,95,22,60,141,137,61,114,20,85,32,50,151,127,71,104)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152), (1,92,21,53,152,134,80,107,11,82,31,43,142,124,70,117)(2,99,30,56,153,121,69,110,12,89,40,46,143,131,79,120)(3,86,39,59,154,128,78,113,13,96,29,49,144,138,68,103)(4,93,28,42,155,135,67,116,14,83,38,52,145,125,77,106)(5,100,37,45,156,122,76,119,15,90,27,55,146,132,66,109)(6,87,26,48,157,129,65,102,16,97,36,58,147,139,75,112)(7,94,35,51,158,136,74,105,17,84,25,41,148,126,64,115)(8,81,24,54,159,123,63,108,18,91,34,44,149,133,73,118)(9,88,33,57,160,130,72,111,19,98,23,47,150,140,62,101)(10,95,22,60,141,137,61,114,20,85,32,50,151,127,71,104) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,107),(102,106),(103,105),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(141,142),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152)], [(1,92,21,53,152,134,80,107,11,82,31,43,142,124,70,117),(2,99,30,56,153,121,69,110,12,89,40,46,143,131,79,120),(3,86,39,59,154,128,78,113,13,96,29,49,144,138,68,103),(4,93,28,42,155,135,67,116,14,83,38,52,145,125,77,106),(5,100,37,45,156,122,76,119,15,90,27,55,146,132,66,109),(6,87,26,48,157,129,65,102,16,97,36,58,147,139,75,112),(7,94,35,51,158,136,74,105,17,84,25,41,148,126,64,115),(8,81,24,54,159,123,63,108,18,91,34,44,149,133,73,118),(9,88,33,57,160,130,72,111,19,98,23,47,150,140,62,101),(10,95,22,60,141,137,61,114,20,85,32,50,151,127,71,104)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | 20 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | D4.C8 | F5 | C2×F5 | D5⋊C8 | C22⋊F5 | C4.F5 | D20.C8 |
| kernel | D20.C8 | C2×C5⋊C16 | C20.C8 | D20.2C4 | C5×M4(2) | C4○D20 | Dic10 | D20 | C5⋊2C8 | C2×C10 | C5 | M4(2) | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 |
Matrix representation of D20.C8 ►in GL6(𝔽241)
| 64 | 239 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 64 | 239 | 0 | 0 | 0 | 0 |
| 240 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 197 | 32 | 0 | 0 | 0 | 0 |
| 76 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 132 | 0 | 211 | 211 |
| 0 | 0 | 211 | 211 | 0 | 132 |
| 0 | 0 | 30 | 162 | 30 | 0 |
| 0 | 0 | 109 | 79 | 79 | 109 |
G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,239,177,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[64,240,0,0,0,0,239,177,0,0,0,0,0,0,0,1,240,0,0,0,1,0,240,0,0,0,0,0,240,0,0,0,0,0,240,1],[197,76,0,0,0,0,32,44,0,0,0,0,0,0,132,211,30,109,0,0,0,211,162,79,0,0,211,0,30,79,0,0,211,132,0,109] >;
D20.C8 in GAP, Magma, Sage, TeX
D_{20}.C_8 % in TeX
G:=Group("D20.C8"); // GroupNames label
G:=SmallGroup(320,236);
// by ID
G=gap.SmallGroup(320,236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,100,1123,570,136,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;
// generators/relations
Export