metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.30D4, C20.13C24, C40.42C23, Q16.11D10, D20.8C23, Dic10.30D4, D40.13C22, Dic10.8C23, Dic20.15C22, C5⋊2(Q8○D8), (D5×Q16)⋊6C2, C4.77(D4×D5), Q8⋊D5.C22, (C2×Q16)⋊12D5, (C10×Q16)⋊3C2, C20.88(C2×D4), C5⋊D4.10D4, Q8.D10⋊6C2, D40⋊7C2⋊4C2, Q16⋊D5⋊5C2, D10.51(C2×D4), (C2×C8).105D10, C5⋊2C8.5C23, (C2×Q8).91D10, (C4×D5).7C23, (C8×D5).7C22, C8.14(C22×D5), C4.13(C23×D5), C22.22(D4×D5), D20.3C4⋊3C2, Q8.7(C22×D5), (Q8×D5).1C22, (C5×Q8).7C23, C20.C23⋊8C2, (C2×C40).35C22, Dic5.57(C2×D4), C8⋊D5.3C22, C40⋊C2.3C22, C5⋊Q16.1C22, (C2×C20).530C23, Q8.10D10⋊4C2, C4○D20.53C22, C10.114(C22×D4), (C5×Q16).11C22, Q8⋊2D5.1C22, (Q8×C10).152C22, C4.Dic5.48C22, C2.87(C2×D4×D5), (C2×C10).403(C2×D4), (C2×C4).231(C22×D5), SmallGroup(320,1438)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 886 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×11], Q8 [×4], Q8 [×9], D5 [×4], C10, C10, C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×6], Q16 [×4], Q16 [×5], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×13], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C8○D4, C2×Q16, C2×Q16 [×2], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×4], C4×D5 [×2], C4×D5 [×10], D20, D20 [×2], D20 [×4], C5⋊D4 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, Q8⋊D5 [×4], C5⋊Q16 [×4], C2×C40, C5×Q16 [×4], C4○D20, C4○D20 [×2], C4○D20 [×4], Q8×D5 [×4], Q8×D5 [×2], Q8⋊2D5 [×4], Q8⋊2D5 [×2], Q8×C10 [×2], D20.3C4, D40⋊7C2, D5×Q16 [×2], Q16⋊D5 [×4], Q8.D10 [×2], C20.C23 [×2], C10×Q16, Q8.10D10 [×2], D20.30D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.30D4
Generators and relations
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c3 >
►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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 140)(17 139)(18 138)(19 137)(20 136)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 84)(42 83)(43 82)(44 81)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(101 154)(102 153)(103 152)(104 151)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)(113 142)(114 141)(115 160)(116 159)(117 158)(118 157)(119 156)(120 155)
(1 62 106 82 11 72 116 92)(2 63 107 83 12 73 117 93)(3 64 108 84 13 74 118 94)(4 65 109 85 14 75 119 95)(5 66 110 86 15 76 120 96)(6 67 111 87 16 77 101 97)(7 68 112 88 17 78 102 98)(8 69 113 89 18 79 103 99)(9 70 114 90 19 80 104 100)(10 71 115 91 20 61 105 81)(21 160 54 136 31 150 44 126)(22 141 55 137 32 151 45 127)(23 142 56 138 33 152 46 128)(24 143 57 139 34 153 47 129)(25 144 58 140 35 154 48 130)(26 145 59 121 36 155 49 131)(27 146 60 122 37 156 50 132)(28 147 41 123 38 157 51 133)(29 148 42 124 39 158 52 134)(30 149 43 125 40 159 53 135)
(1 145 11 155)(2 156 12 146)(3 147 13 157)(4 158 14 148)(5 149 15 159)(6 160 16 150)(7 151 17 141)(8 142 18 152)(9 153 19 143)(10 144 20 154)(21 77 31 67)(22 68 32 78)(23 79 33 69)(24 70 34 80)(25 61 35 71)(26 72 36 62)(27 63 37 73)(28 74 38 64)(29 65 39 75)(30 76 40 66)(41 84 51 94)(42 95 52 85)(43 86 53 96)(44 97 54 87)(45 88 55 98)(46 99 56 89)(47 90 57 100)(48 81 58 91)(49 92 59 82)(50 83 60 93)(101 136 111 126)(102 127 112 137)(103 138 113 128)(104 129 114 139)(105 140 115 130)(106 131 116 121)(107 122 117 132)(108 133 118 123)(109 124 119 134)(110 135 120 125)
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,140)(17,139)(18,138)(19,137)(20,136)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,84)(42,83)(43,82)(44,81)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,160)(116,159)(117,158)(118,157)(119,156)(120,155), (1,62,106,82,11,72,116,92)(2,63,107,83,12,73,117,93)(3,64,108,84,13,74,118,94)(4,65,109,85,14,75,119,95)(5,66,110,86,15,76,120,96)(6,67,111,87,16,77,101,97)(7,68,112,88,17,78,102,98)(8,69,113,89,18,79,103,99)(9,70,114,90,19,80,104,100)(10,71,115,91,20,61,105,81)(21,160,54,136,31,150,44,126)(22,141,55,137,32,151,45,127)(23,142,56,138,33,152,46,128)(24,143,57,139,34,153,47,129)(25,144,58,140,35,154,48,130)(26,145,59,121,36,155,49,131)(27,146,60,122,37,156,50,132)(28,147,41,123,38,157,51,133)(29,148,42,124,39,158,52,134)(30,149,43,125,40,159,53,135), (1,145,11,155)(2,156,12,146)(3,147,13,157)(4,158,14,148)(5,149,15,159)(6,160,16,150)(7,151,17,141)(8,142,18,152)(9,153,19,143)(10,144,20,154)(21,77,31,67)(22,68,32,78)(23,79,33,69)(24,70,34,80)(25,61,35,71)(26,72,36,62)(27,63,37,73)(28,74,38,64)(29,65,39,75)(30,76,40,66)(41,84,51,94)(42,95,52,85)(43,86,53,96)(44,97,54,87)(45,88,55,98)(46,99,56,89)(47,90,57,100)(48,81,58,91)(49,92,59,82)(50,83,60,93)(101,136,111,126)(102,127,112,137)(103,138,113,128)(104,129,114,139)(105,140,115,130)(106,131,116,121)(107,122,117,132)(108,133,118,123)(109,124,119,134)(110,135,120,125)>;
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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,140)(17,139)(18,138)(19,137)(20,136)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,84)(42,83)(43,82)(44,81)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,160)(116,159)(117,158)(118,157)(119,156)(120,155), (1,62,106,82,11,72,116,92)(2,63,107,83,12,73,117,93)(3,64,108,84,13,74,118,94)(4,65,109,85,14,75,119,95)(5,66,110,86,15,76,120,96)(6,67,111,87,16,77,101,97)(7,68,112,88,17,78,102,98)(8,69,113,89,18,79,103,99)(9,70,114,90,19,80,104,100)(10,71,115,91,20,61,105,81)(21,160,54,136,31,150,44,126)(22,141,55,137,32,151,45,127)(23,142,56,138,33,152,46,128)(24,143,57,139,34,153,47,129)(25,144,58,140,35,154,48,130)(26,145,59,121,36,155,49,131)(27,146,60,122,37,156,50,132)(28,147,41,123,38,157,51,133)(29,148,42,124,39,158,52,134)(30,149,43,125,40,159,53,135), (1,145,11,155)(2,156,12,146)(3,147,13,157)(4,158,14,148)(5,149,15,159)(6,160,16,150)(7,151,17,141)(8,142,18,152)(9,153,19,143)(10,144,20,154)(21,77,31,67)(22,68,32,78)(23,79,33,69)(24,70,34,80)(25,61,35,71)(26,72,36,62)(27,63,37,73)(28,74,38,64)(29,65,39,75)(30,76,40,66)(41,84,51,94)(42,95,52,85)(43,86,53,96)(44,97,54,87)(45,88,55,98)(46,99,56,89)(47,90,57,100)(48,81,58,91)(49,92,59,82)(50,83,60,93)(101,136,111,126)(102,127,112,137)(103,138,113,128)(104,129,114,139)(105,140,115,130)(106,131,116,121)(107,122,117,132)(108,133,118,123)(109,124,119,134)(110,135,120,125) );
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)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,140),(17,139),(18,138),(19,137),(20,136),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,84),(42,83),(43,82),(44,81),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(101,154),(102,153),(103,152),(104,151),(105,150),(106,149),(107,148),(108,147),(109,146),(110,145),(111,144),(112,143),(113,142),(114,141),(115,160),(116,159),(117,158),(118,157),(119,156),(120,155)], [(1,62,106,82,11,72,116,92),(2,63,107,83,12,73,117,93),(3,64,108,84,13,74,118,94),(4,65,109,85,14,75,119,95),(5,66,110,86,15,76,120,96),(6,67,111,87,16,77,101,97),(7,68,112,88,17,78,102,98),(8,69,113,89,18,79,103,99),(9,70,114,90,19,80,104,100),(10,71,115,91,20,61,105,81),(21,160,54,136,31,150,44,126),(22,141,55,137,32,151,45,127),(23,142,56,138,33,152,46,128),(24,143,57,139,34,153,47,129),(25,144,58,140,35,154,48,130),(26,145,59,121,36,155,49,131),(27,146,60,122,37,156,50,132),(28,147,41,123,38,157,51,133),(29,148,42,124,39,158,52,134),(30,149,43,125,40,159,53,135)], [(1,145,11,155),(2,156,12,146),(3,147,13,157),(4,158,14,148),(5,149,15,159),(6,160,16,150),(7,151,17,141),(8,142,18,152),(9,153,19,143),(10,144,20,154),(21,77,31,67),(22,68,32,78),(23,79,33,69),(24,70,34,80),(25,61,35,71),(26,72,36,62),(27,63,37,73),(28,74,38,64),(29,65,39,75),(30,76,40,66),(41,84,51,94),(42,95,52,85),(43,86,53,96),(44,97,54,87),(45,88,55,98),(46,99,56,89),(47,90,57,100),(48,81,58,91),(49,92,59,82),(50,83,60,93),(101,136,111,126),(102,127,112,137),(103,138,113,128),(104,129,114,139),(105,140,115,130),(106,131,116,121),(107,122,117,132),(108,133,118,123),(109,124,119,134),(110,135,120,125)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 6 | 23 |
| 0 | 0 | 18 | 21 |
| 35 | 18 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 8 | 17 | 0 | 15 |
| 24 | 33 | 15 | 0 |
| 0 | 26 | 8 | 17 |
| 26 | 0 | 24 | 33 |
| 12 | 0 | 29 | 0 |
| 0 | 12 | 0 | 29 |
| 12 | 0 | 12 | 0 |
| 0 | 12 | 0 | 12 |
| 15 | 0 | 17 | 8 |
| 0 | 15 | 33 | 24 |
| 17 | 8 | 26 | 0 |
| 33 | 24 | 0 | 26 |
G:=sub<GL(4,GF(41))| [0,0,35,23,0,0,18,20,6,18,0,0,23,21,0,0],[8,24,0,26,17,33,26,0,0,15,8,24,15,0,17,33],[12,0,12,0,0,12,0,12,29,0,12,0,0,29,0,12],[15,0,17,33,0,15,8,24,17,33,26,0,8,24,0,26] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 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 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | Q8○D8 | D4×D5 | D4×D5 | D20.30D4 |
| kernel | D20.30D4 | D20.3C4 | D40⋊7C2 | D5×Q16 | Q16⋊D5 | Q8.D10 | C20.C23 | C10×Q16 | Q8.10D10 | Dic10 | D20 | C5⋊D4 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_{20}._{30}D_4 % in TeX
G:=Group("D20.30D4"); // GroupNames label
G:=SmallGroup(320,1438);
// by ID
G=gap.SmallGroup(320,1438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀