metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊2C8, C5⋊C8⋊9D4, C5⋊2(C8×D4), C20⋊1(C2×C8), C2.3(D4×F5), C4⋊1(D5⋊C8), D10⋊2(C2×C8), C20⋊C8⋊3C2, C10.8(C4×D4), C4⋊C4.12F5, (C2×D20).11C4, C10.6(C22×C8), D10⋊C8⋊13C2, C2.2(Q8.F5), D10⋊C4.8C4, C10.19(C8○D4), Dic5.69(C2×D4), D20⋊8C4.16C2, Dic5.54(C4○D4), C22.37(C22×F5), (C2×Dic5).329C23, (C4×Dic5).190C22, (C4×C5⋊C8)⋊3C2, (C5×C4⋊C4).6C4, C2.8(C2×D5⋊C8), (C2×D5⋊C8)⋊10C2, (C2×C4).60(C2×F5), (C2×C20).42(C2×C4), (C2×C5⋊C8).26C22, (C2×C4×D5).289C22, (C2×C10).40(C22×C4), (C2×Dic5).55(C2×C4), (C22×D5).47(C2×C4), SmallGroup(320,1040)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊2C8
G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a12b >
Subgroups: 474 in 134 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊C8, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8×D4, C4×Dic5, D10⋊C4, C5×C4⋊C4, D5⋊C8, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8, D20⋊8C4, C2×D5⋊C8, D20⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C2×F5, C8×D4, D5⋊C8, C22×F5, C2×D5⋊C8, D4×F5, Q8.F5, D20⋊2C8
►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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 83)(22 82)(23 81)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 141)(42 160)(43 159)(44 158)(45 157)(46 156)(47 155)(48 154)(49 153)(50 152)(51 151)(52 150)(53 149)(54 148)(55 147)(56 146)(57 145)(58 144)(59 143)(60 142)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 140)(119 139)(120 138)
(1 83 160 133 73 27 48 111)(2 100 149 126 74 24 57 104)(3 97 158 139 75 21 46 117)(4 94 147 132 76 38 55 110)(5 91 156 125 77 35 44 103)(6 88 145 138 78 32 53 116)(7 85 154 131 79 29 42 109)(8 82 143 124 80 26 51 102)(9 99 152 137 61 23 60 115)(10 96 141 130 62 40 49 108)(11 93 150 123 63 37 58 101)(12 90 159 136 64 34 47 114)(13 87 148 129 65 31 56 107)(14 84 157 122 66 28 45 120)(15 81 146 135 67 25 54 113)(16 98 155 128 68 22 43 106)(17 95 144 121 69 39 52 119)(18 92 153 134 70 36 41 112)(19 89 142 127 71 33 50 105)(20 86 151 140 72 30 59 118)
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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,141)(42,160)(43,159)(44,158)(45,157)(46,156)(47,155)(48,154)(49,153)(50,152)(51,151)(52,150)(53,149)(54,148)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,140)(119,139)(120,138), (1,83,160,133,73,27,48,111)(2,100,149,126,74,24,57,104)(3,97,158,139,75,21,46,117)(4,94,147,132,76,38,55,110)(5,91,156,125,77,35,44,103)(6,88,145,138,78,32,53,116)(7,85,154,131,79,29,42,109)(8,82,143,124,80,26,51,102)(9,99,152,137,61,23,60,115)(10,96,141,130,62,40,49,108)(11,93,150,123,63,37,58,101)(12,90,159,136,64,34,47,114)(13,87,148,129,65,31,56,107)(14,84,157,122,66,28,45,120)(15,81,146,135,67,25,54,113)(16,98,155,128,68,22,43,106)(17,95,144,121,69,39,52,119)(18,92,153,134,70,36,41,112)(19,89,142,127,71,33,50,105)(20,86,151,140,72,30,59,118)>;
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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,83)(22,82)(23,81)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,141)(42,160)(43,159)(44,158)(45,157)(46,156)(47,155)(48,154)(49,153)(50,152)(51,151)(52,150)(53,149)(54,148)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,140)(119,139)(120,138), (1,83,160,133,73,27,48,111)(2,100,149,126,74,24,57,104)(3,97,158,139,75,21,46,117)(4,94,147,132,76,38,55,110)(5,91,156,125,77,35,44,103)(6,88,145,138,78,32,53,116)(7,85,154,131,79,29,42,109)(8,82,143,124,80,26,51,102)(9,99,152,137,61,23,60,115)(10,96,141,130,62,40,49,108)(11,93,150,123,63,37,58,101)(12,90,159,136,64,34,47,114)(13,87,148,129,65,31,56,107)(14,84,157,122,66,28,45,120)(15,81,146,135,67,25,54,113)(16,98,155,128,68,22,43,106)(17,95,144,121,69,39,52,119)(18,92,153,134,70,36,41,112)(19,89,142,127,71,33,50,105)(20,86,151,140,72,30,59,118) );
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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,70),(19,69),(20,68),(21,83),(22,82),(23,81),(24,100),(25,99),(26,98),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,141),(42,160),(43,159),(44,158),(45,157),(46,156),(47,155),(48,154),(49,153),(50,152),(51,151),(52,150),(53,149),(54,148),(55,147),(56,146),(57,145),(58,144),(59,143),(60,142),(101,137),(102,136),(103,135),(104,134),(105,133),(106,132),(107,131),(108,130),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,140),(119,139),(120,138)], [(1,83,160,133,73,27,48,111),(2,100,149,126,74,24,57,104),(3,97,158,139,75,21,46,117),(4,94,147,132,76,38,55,110),(5,91,156,125,77,35,44,103),(6,88,145,138,78,32,53,116),(7,85,154,131,79,29,42,109),(8,82,143,124,80,26,51,102),(9,99,152,137,61,23,60,115),(10,96,141,130,62,40,49,108),(11,93,150,123,63,37,58,101),(12,90,159,136,64,34,47,114),(13,87,148,129,65,31,56,107),(14,84,157,122,66,28,45,120),(15,81,146,135,67,25,54,113),(16,98,155,128,68,22,43,106),(17,95,144,121,69,39,52,119),(18,92,153,134,70,36,41,112),(19,89,142,127,71,33,50,105),(20,86,151,140,72,30,59,118)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | F5 | C2×F5 | D5⋊C8 | D4×F5 | Q8.F5 |
| kernel | D20⋊2C8 | C4×C5⋊C8 | C20⋊C8 | D10⋊C8 | D20⋊8C4 | C2×D5⋊C8 | D10⋊C4 | C5×C4⋊C4 | C2×D20 | D20 | C5⋊C8 | Dic5 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
Matrix representation of D20⋊2C8 ►in GL6(𝔽41)
| 28 | 5 | 0 | 0 | 0 | 0 |
| 7 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 1 | 20 |
| 0 | 0 | 22 | 0 | 21 | 21 |
| 0 | 0 | 21 | 21 | 0 | 22 |
| 0 | 0 | 20 | 1 | 20 | 0 |
G:=sub<GL(6,GF(41))| [28,7,0,0,0,0,5,13,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1],[40,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0,0,1,0,40,0,0,0,1,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,22,21,20,0,0,20,0,21,1,0,0,1,21,0,20,0,0,20,21,22,0] >;
D20⋊2C8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_2C_8 % in TeX
G:=Group("D20:2C8"); // GroupNames label
G:=SmallGroup(320,1040);
// by ID
G=gap.SmallGroup(320,1040);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations