metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.12D4, C4.97(D4×D5), Q8⋊C4⋊8D5, (C2×D40).3C2, (C2×C8).20D10, D20⋊8C4⋊5C2, C4⋊C4.157D10, C4.8(C4○D20), C20.129(C2×D4), C5⋊2(D4.2D4), (C2×Q8).24D10, C20.8Q8⋊8C2, D20⋊6C4⋊13C2, C10.72(C4○D8), C20.24(C4○D4), C20.23D4⋊2C2, (C2×C40).20C22, (C2×Dic5).44D4, C22.208(D4×D5), C2.19(D40⋊C2), C10.28(C4⋊D4), C10.66(C8⋊C22), (C2×C20).259C23, (C2×D20).73C22, (Q8×C10).42C22, C2.11(Q8.D10), C2.31(D10⋊D4), (C4×Dic5).32C22, (C2×Q8⋊D5)⋊6C2, (C5×Q8⋊C4)⋊8C2, (C2×C10).272(C2×D4), (C5×C4⋊C4).60C22, (C2×C5⋊2C8).49C22, (C2×C4).366(C22×D5), SmallGroup(320,446)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.12D4
G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 630 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4.2D4, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, Q8×C10, D20⋊6C4, C20.8Q8, C5×Q8⋊C4, D20⋊8C4, C2×D40, C2×Q8⋊D5, C20.23D4, D20.12D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C22×D5, D4.2D4, C4○D20, D4×D5, D10⋊D4, D40⋊C2, Q8.D10, D20.12D4
►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)
(1 124)(2 123)(3 122)(4 121)(5 140)(6 139)(7 138)(8 137)(9 136)(10 135)(11 134)(12 133)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 126)(20 125)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 99)(42 98)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 100)(101 159)(102 158)(103 157)(104 156)(105 155)(106 154)(107 153)(108 152)(109 151)(110 150)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 160)
(1 104 65 100)(2 115 66 91)(3 106 67 82)(4 117 68 93)(5 108 69 84)(6 119 70 95)(7 110 71 86)(8 101 72 97)(9 112 73 88)(10 103 74 99)(11 114 75 90)(12 105 76 81)(13 116 77 92)(14 107 78 83)(15 118 79 94)(16 109 80 85)(17 120 61 96)(18 111 62 87)(19 102 63 98)(20 113 64 89)(21 58 122 154)(22 49 123 145)(23 60 124 156)(24 51 125 147)(25 42 126 158)(26 53 127 149)(27 44 128 160)(28 55 129 151)(29 46 130 142)(30 57 131 153)(31 48 132 144)(32 59 133 155)(33 50 134 146)(34 41 135 157)(35 52 136 148)(36 43 137 159)(37 54 138 150)(38 45 139 141)(39 56 140 152)(40 47 121 143)
(1 34 16 29 11 24 6 39)(2 35 17 30 12 25 7 40)(3 36 18 31 13 26 8 21)(4 37 19 32 14 27 9 22)(5 38 20 33 15 28 10 23)(41 90 56 85 51 100 46 95)(42 91 57 86 52 81 47 96)(43 92 58 87 53 82 48 97)(44 93 59 88 54 83 49 98)(45 94 60 89 55 84 50 99)(61 131 76 126 71 121 66 136)(62 132 77 127 72 122 67 137)(63 133 78 128 73 123 68 138)(64 134 79 129 74 124 69 139)(65 135 80 130 75 125 70 140)(101 159 116 154 111 149 106 144)(102 160 117 155 112 150 107 145)(103 141 118 156 113 151 108 146)(104 142 119 157 114 152 109 147)(105 143 120 158 115 153 110 148)
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), (1,124)(2,123)(3,122)(4,121)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,100)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,160), (1,104,65,100)(2,115,66,91)(3,106,67,82)(4,117,68,93)(5,108,69,84)(6,119,70,95)(7,110,71,86)(8,101,72,97)(9,112,73,88)(10,103,74,99)(11,114,75,90)(12,105,76,81)(13,116,77,92)(14,107,78,83)(15,118,79,94)(16,109,80,85)(17,120,61,96)(18,111,62,87)(19,102,63,98)(20,113,64,89)(21,58,122,154)(22,49,123,145)(23,60,124,156)(24,51,125,147)(25,42,126,158)(26,53,127,149)(27,44,128,160)(28,55,129,151)(29,46,130,142)(30,57,131,153)(31,48,132,144)(32,59,133,155)(33,50,134,146)(34,41,135,157)(35,52,136,148)(36,43,137,159)(37,54,138,150)(38,45,139,141)(39,56,140,152)(40,47,121,143), (1,34,16,29,11,24,6,39)(2,35,17,30,12,25,7,40)(3,36,18,31,13,26,8,21)(4,37,19,32,14,27,9,22)(5,38,20,33,15,28,10,23)(41,90,56,85,51,100,46,95)(42,91,57,86,52,81,47,96)(43,92,58,87,53,82,48,97)(44,93,59,88,54,83,49,98)(45,94,60,89,55,84,50,99)(61,131,76,126,71,121,66,136)(62,132,77,127,72,122,67,137)(63,133,78,128,73,123,68,138)(64,134,79,129,74,124,69,139)(65,135,80,130,75,125,70,140)(101,159,116,154,111,149,106,144)(102,160,117,155,112,150,107,145)(103,141,118,156,113,151,108,146)(104,142,119,157,114,152,109,147)(105,143,120,158,115,153,110,148)>;
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), (1,124)(2,123)(3,122)(4,121)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,100)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,160), (1,104,65,100)(2,115,66,91)(3,106,67,82)(4,117,68,93)(5,108,69,84)(6,119,70,95)(7,110,71,86)(8,101,72,97)(9,112,73,88)(10,103,74,99)(11,114,75,90)(12,105,76,81)(13,116,77,92)(14,107,78,83)(15,118,79,94)(16,109,80,85)(17,120,61,96)(18,111,62,87)(19,102,63,98)(20,113,64,89)(21,58,122,154)(22,49,123,145)(23,60,124,156)(24,51,125,147)(25,42,126,158)(26,53,127,149)(27,44,128,160)(28,55,129,151)(29,46,130,142)(30,57,131,153)(31,48,132,144)(32,59,133,155)(33,50,134,146)(34,41,135,157)(35,52,136,148)(36,43,137,159)(37,54,138,150)(38,45,139,141)(39,56,140,152)(40,47,121,143), (1,34,16,29,11,24,6,39)(2,35,17,30,12,25,7,40)(3,36,18,31,13,26,8,21)(4,37,19,32,14,27,9,22)(5,38,20,33,15,28,10,23)(41,90,56,85,51,100,46,95)(42,91,57,86,52,81,47,96)(43,92,58,87,53,82,48,97)(44,93,59,88,54,83,49,98)(45,94,60,89,55,84,50,99)(61,131,76,126,71,121,66,136)(62,132,77,127,72,122,67,137)(63,133,78,128,73,123,68,138)(64,134,79,129,74,124,69,139)(65,135,80,130,75,125,70,140)(101,159,116,154,111,149,106,144)(102,160,117,155,112,150,107,145)(103,141,118,156,113,151,108,146)(104,142,119,157,114,152,109,147)(105,143,120,158,115,153,110,148) );
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)], [(1,124),(2,123),(3,122),(4,121),(5,140),(6,139),(7,138),(8,137),(9,136),(10,135),(11,134),(12,133),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,126),(20,125),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,99),(42,98),(43,97),(44,96),(45,95),(46,94),(47,93),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,100),(101,159),(102,158),(103,157),(104,156),(105,155),(106,154),(107,153),(108,152),(109,151),(110,150),(111,149),(112,148),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,160)], [(1,104,65,100),(2,115,66,91),(3,106,67,82),(4,117,68,93),(5,108,69,84),(6,119,70,95),(7,110,71,86),(8,101,72,97),(9,112,73,88),(10,103,74,99),(11,114,75,90),(12,105,76,81),(13,116,77,92),(14,107,78,83),(15,118,79,94),(16,109,80,85),(17,120,61,96),(18,111,62,87),(19,102,63,98),(20,113,64,89),(21,58,122,154),(22,49,123,145),(23,60,124,156),(24,51,125,147),(25,42,126,158),(26,53,127,149),(27,44,128,160),(28,55,129,151),(29,46,130,142),(30,57,131,153),(31,48,132,144),(32,59,133,155),(33,50,134,146),(34,41,135,157),(35,52,136,148),(36,43,137,159),(37,54,138,150),(38,45,139,141),(39,56,140,152),(40,47,121,143)], [(1,34,16,29,11,24,6,39),(2,35,17,30,12,25,7,40),(3,36,18,31,13,26,8,21),(4,37,19,32,14,27,9,22),(5,38,20,33,15,28,10,23),(41,90,56,85,51,100,46,95),(42,91,57,86,52,81,47,96),(43,92,58,87,53,82,48,97),(44,93,59,88,54,83,49,98),(45,94,60,89,55,84,50,99),(61,131,76,126,71,121,66,136),(62,132,77,127,72,122,67,137),(63,133,78,128,73,123,68,138),(64,134,79,129,74,124,69,139),(65,135,80,130,75,125,70,140),(101,159,116,154,111,149,106,144),(102,160,117,155,112,150,107,145),(103,141,118,156,113,151,108,146),(104,142,119,157,114,152,109,147),(105,143,120,158,115,153,110,148)]])
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 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 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 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 | Q8.D10 |
| kernel | D20.12D4 | D20⋊6C4 | C20.8Q8 | C5×Q8⋊C4 | D20⋊8C4 | C2×D40 | C2×Q8⋊D5 | C20.23D4 | D20 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.12D4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 40 | 1 |
| 32 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 29 | 24 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 30 | 11 |
| 0 | 0 | 15 | 11 |
| 30 | 32 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 24 | 17 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,40,40,0,0,2,1],[32,11,0,0,30,9,0,0,0,0,17,29,0,0,24,24],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11],[30,9,0,0,32,11,0,0,0,0,24,12,0,0,17,0] >;
D20.12D4 in GAP, Magma, Sage, TeX
D_{20}._{12}D_4 % in TeX
G:=Group("D20.12D4"); // GroupNames label
G:=SmallGroup(320,446);
// by ID
G=gap.SmallGroup(320,446);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations