metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊3C8, C20.35D8, C4.17D40, C20.29SD16, C42.249D10, C20.36M4(2), (C4×C8)⋊1D5, (C4×C40)⋊1C2, C5⋊3(D4⋊C8), C4.7(C8×D5), C20⋊3C8⋊1C2, C10.16C4≀C2, C20.51(C2×C8), (C4×D20).1C2, (C2×D20).16C4, (C2×C20).436D4, (C2×C4).161D20, C4.5(C8⋊D5), C4⋊Dic5.17C4, C4.16(C40⋊C2), C2.1(D20⋊5C4), C2.4(D10⋊1C8), C2.2(D20⋊4C4), C10.17(C22⋊C8), (C4×C20).320C22, C10.25(D4⋊C4), C22.31(D10⋊C4), (C2×C4).96(C4×D5), (C2×C20).389(C2×C4), (C2×C4).205(C5⋊D4), (C2×C10).100(C22⋊C4), SmallGroup(320,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊3C8
G = < a,b,c | a20=b2=c8=1, bab=a-1, ac=ca, cbc-1=a15b >
Subgroups: 374 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×C8, C4⋊C8, C4×D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4⋊C8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C20⋊3C8, C4×C40, C4×D20, D20⋊3C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D8, SD16, D10, C22⋊C8, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, D4⋊C8, C8×D5, C8⋊D5, C40⋊C2, D40, D10⋊C4, D20⋊4C4, D10⋊1C8, D20⋊5C4, D20⋊3C8
►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 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 131)(22 130)(23 129)(24 128)(25 127)(26 126)(27 125)(28 124)(29 123)(30 122)(31 121)(32 140)(33 139)(34 138)(35 137)(36 136)(37 135)(38 134)(39 133)(40 132)(41 61)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(101 148)(102 147)(103 146)(104 145)(105 144)(106 143)(107 142)(108 141)(109 160)(110 159)(111 158)(112 157)(113 156)(114 155)(115 154)(116 153)(117 152)(118 151)(119 150)(120 149)
(1 28 144 61 99 130 106 57)(2 29 145 62 100 131 107 58)(3 30 146 63 81 132 108 59)(4 31 147 64 82 133 109 60)(5 32 148 65 83 134 110 41)(6 33 149 66 84 135 111 42)(7 34 150 67 85 136 112 43)(8 35 151 68 86 137 113 44)(9 36 152 69 87 138 114 45)(10 37 153 70 88 139 115 46)(11 38 154 71 89 140 116 47)(12 39 155 72 90 121 117 48)(13 40 156 73 91 122 118 49)(14 21 157 74 92 123 119 50)(15 22 158 75 93 124 120 51)(16 23 159 76 94 125 101 52)(17 24 160 77 95 126 102 53)(18 25 141 78 96 127 103 54)(19 26 142 79 97 128 104 55)(20 27 143 80 98 129 105 56)
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), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,140)(33,139)(34,138)(35,137)(36,136)(37,135)(38,134)(39,133)(40,132)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,160)(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149), (1,28,144,61,99,130,106,57)(2,29,145,62,100,131,107,58)(3,30,146,63,81,132,108,59)(4,31,147,64,82,133,109,60)(5,32,148,65,83,134,110,41)(6,33,149,66,84,135,111,42)(7,34,150,67,85,136,112,43)(8,35,151,68,86,137,113,44)(9,36,152,69,87,138,114,45)(10,37,153,70,88,139,115,46)(11,38,154,71,89,140,116,47)(12,39,155,72,90,121,117,48)(13,40,156,73,91,122,118,49)(14,21,157,74,92,123,119,50)(15,22,158,75,93,124,120,51)(16,23,159,76,94,125,101,52)(17,24,160,77,95,126,102,53)(18,25,141,78,96,127,103,54)(19,26,142,79,97,128,104,55)(20,27,143,80,98,129,105,56)>;
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), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,140)(33,139)(34,138)(35,137)(36,136)(37,135)(38,134)(39,133)(40,132)(41,61)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,160)(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149), (1,28,144,61,99,130,106,57)(2,29,145,62,100,131,107,58)(3,30,146,63,81,132,108,59)(4,31,147,64,82,133,109,60)(5,32,148,65,83,134,110,41)(6,33,149,66,84,135,111,42)(7,34,150,67,85,136,112,43)(8,35,151,68,86,137,113,44)(9,36,152,69,87,138,114,45)(10,37,153,70,88,139,115,46)(11,38,154,71,89,140,116,47)(12,39,155,72,90,121,117,48)(13,40,156,73,91,122,118,49)(14,21,157,74,92,123,119,50)(15,22,158,75,93,124,120,51)(16,23,159,76,94,125,101,52)(17,24,160,77,95,126,102,53)(18,25,141,78,96,127,103,54)(19,26,142,79,97,128,104,55)(20,27,143,80,98,129,105,56) );
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)], [(1,88),(2,87),(3,86),(4,85),(5,84),(6,83),(7,82),(8,81),(9,100),(10,99),(11,98),(12,97),(13,96),(14,95),(15,94),(16,93),(17,92),(18,91),(19,90),(20,89),(21,131),(22,130),(23,129),(24,128),(25,127),(26,126),(27,125),(28,124),(29,123),(30,122),(31,121),(32,140),(33,139),(34,138),(35,137),(36,136),(37,135),(38,134),(39,133),(40,132),(41,61),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(101,148),(102,147),(103,146),(104,145),(105,144),(106,143),(107,142),(108,141),(109,160),(110,159),(111,158),(112,157),(113,156),(114,155),(115,154),(116,153),(117,152),(118,151),(119,150),(120,149)], [(1,28,144,61,99,130,106,57),(2,29,145,62,100,131,107,58),(3,30,146,63,81,132,108,59),(4,31,147,64,82,133,109,60),(5,32,148,65,83,134,110,41),(6,33,149,66,84,135,111,42),(7,34,150,67,85,136,112,43),(8,35,151,68,86,137,113,44),(9,36,152,69,87,138,114,45),(10,37,153,70,88,139,115,46),(11,38,154,71,89,140,116,47),(12,39,155,72,90,121,117,48),(13,40,156,73,91,122,118,49),(14,21,157,74,92,123,119,50),(15,22,158,75,93,124,120,51),(16,23,159,76,94,125,101,52),(17,24,160,77,95,126,102,53),(18,25,141,78,96,127,103,54),(19,26,142,79,97,128,104,55),(20,27,143,80,98,129,105,56)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D5 | M4(2) | D8 | SD16 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | C8×D5 | C8⋊D5 | C40⋊C2 | D40 | D20⋊4C4 |
| kernel | D20⋊3C8 | C20⋊3C8 | C4×C40 | C4×D20 | C4⋊Dic5 | C2×D20 | D20 | C2×C20 | C4×C8 | C20 | C20 | C20 | C42 | C10 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of D20⋊3C8 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 38 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 29 | 29 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,0,1,0,0,1,0],[38,0,0,0,0,38,0,0,0,0,29,29,0,0,12,29] >;
D20⋊3C8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_3C_8 % in TeX
G:=Group("D20:3C8"); // GroupNames label
G:=SmallGroup(320,17);
// by ID
G=gap.SmallGroup(320,17);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,100,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations