metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊20D4, C10.332+ (1+4), C4⋊D4⋊7D5, C5⋊3(Q8⋊6D4), C4.109(D4×D5), C20⋊D4⋊15C2, C4⋊2D20⋊19C2, C4⋊C4.177D10, (C2×D4).90D10, C20.225(C2×D4), D10⋊D4⋊17C2, Dic5⋊8(C4○D4), Dic5⋊4D4⋊6C2, (C2×C20).35C23, C22⋊C4.46D10, Dic5.45(C2×D4), C10.62(C22×D4), Dic5⋊D4⋊10C2, Dic5⋊3Q8⋊20C2, (C2×C10).143C24, (C22×C4).219D10, C2.35(D4⋊6D10), C23.10(C22×D5), (C2×D20).148C22, (D4×C10).117C22, (C22×C10).14C23, (C4×Dic5).98C22, (C22×D5).62C23, C22.164(C23×D5), D10⋊C4.12C22, (C22×C20).237C22, (C2×Dic5).235C23, C10.D4.14C22, C23.D5.110C22, (C2×Dic10).301C22, (C22×Dic5).104C22, C2.35(C2×D4×D5), (C5×C4⋊D4)⋊8C2, (C4×C5⋊D4)⋊15C2, C2.34(D5×C4○D4), (C2×C4○D20)⋊19C2, (C2×D4⋊2D5)⋊11C2, (C2×C4×D5).91C22, C10.148(C2×C4○D4), (C5×C4⋊C4).139C22, (C2×C4).585(C22×D5), (C2×C5⋊D4).25C22, (C5×C22⋊C4).8C22, SmallGroup(320,1271)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1222 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×18], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×24], Q8 [×4], C23, C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×12], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×2], C20 [×2], C20 [×3], D10 [×9], C2×C10, C2×C10 [×9], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×5], C4⋊1D4 [×3], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×6], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×C10, C22×C10 [×2], Q8⋊6D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C4○D20 [×4], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×8], C22×C20, D4×C10, D4×C10 [×2], Dic5⋊4D4 [×2], D10⋊D4 [×2], Dic5⋊3Q8, C4⋊2D20, C4×C5⋊D4, Dic5⋊D4 [×2], C20⋊D4, C20⋊D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D4⋊2D5, Dic10⋊20D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], Q8⋊6D4, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10, D5×C4○D4, Dic10⋊20D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=dbd=a10b, dcd=c-1 >
►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 95 11 85)(2 94 12 84)(3 93 13 83)(4 92 14 82)(5 91 15 81)(6 90 16 100)(7 89 17 99)(8 88 18 98)(9 87 19 97)(10 86 20 96)(21 61 31 71)(22 80 32 70)(23 79 33 69)(24 78 34 68)(25 77 35 67)(26 76 36 66)(27 75 37 65)(28 74 38 64)(29 73 39 63)(30 72 40 62)(41 151 51 141)(42 150 52 160)(43 149 53 159)(44 148 54 158)(45 147 55 157)(46 146 56 156)(47 145 57 155)(48 144 58 154)(49 143 59 153)(50 142 60 152)(101 123 111 133)(102 122 112 132)(103 121 113 131)(104 140 114 130)(105 139 115 129)(106 138 116 128)(107 137 117 127)(108 136 118 126)(109 135 119 125)(110 134 120 124)
(1 63 155 129)(2 74 156 140)(3 65 157 131)(4 76 158 122)(5 67 159 133)(6 78 160 124)(7 69 141 135)(8 80 142 126)(9 71 143 137)(10 62 144 128)(11 73 145 139)(12 64 146 130)(13 75 147 121)(14 66 148 132)(15 77 149 123)(16 68 150 134)(17 79 151 125)(18 70 152 136)(19 61 153 127)(20 72 154 138)(21 49 117 97)(22 60 118 88)(23 51 119 99)(24 42 120 90)(25 53 101 81)(26 44 102 92)(27 55 103 83)(28 46 104 94)(29 57 105 85)(30 48 106 96)(31 59 107 87)(32 50 108 98)(33 41 109 89)(34 52 110 100)(35 43 111 91)(36 54 112 82)(37 45 113 93)(38 56 114 84)(39 47 115 95)(40 58 116 86)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 121)(37 122)(38 123)(39 124)(40 125)(41 154)(42 155)(43 156)(44 157)(45 158)(46 159)(47 160)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 118)(62 119)(63 120)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)
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), (1,95,11,85)(2,94,12,84)(3,93,13,83)(4,92,14,82)(5,91,15,81)(6,90,16,100)(7,89,17,99)(8,88,18,98)(9,87,19,97)(10,86,20,96)(21,61,31,71)(22,80,32,70)(23,79,33,69)(24,78,34,68)(25,77,35,67)(26,76,36,66)(27,75,37,65)(28,74,38,64)(29,73,39,63)(30,72,40,62)(41,151,51,141)(42,150,52,160)(43,149,53,159)(44,148,54,158)(45,147,55,157)(46,146,56,156)(47,145,57,155)(48,144,58,154)(49,143,59,153)(50,142,60,152)(101,123,111,133)(102,122,112,132)(103,121,113,131)(104,140,114,130)(105,139,115,129)(106,138,116,128)(107,137,117,127)(108,136,118,126)(109,135,119,125)(110,134,120,124), (1,63,155,129)(2,74,156,140)(3,65,157,131)(4,76,158,122)(5,67,159,133)(6,78,160,124)(7,69,141,135)(8,80,142,126)(9,71,143,137)(10,62,144,128)(11,73,145,139)(12,64,146,130)(13,75,147,121)(14,66,148,132)(15,77,149,123)(16,68,150,134)(17,79,151,125)(18,70,152,136)(19,61,153,127)(20,72,154,138)(21,49,117,97)(22,60,118,88)(23,51,119,99)(24,42,120,90)(25,53,101,81)(26,44,102,92)(27,55,103,83)(28,46,104,94)(29,57,105,85)(30,48,106,96)(31,59,107,87)(32,50,108,98)(33,41,109,89)(34,52,110,100)(35,43,111,91)(36,54,112,82)(37,45,113,93)(38,56,114,84)(39,47,115,95)(40,58,116,86), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,121)(37,122)(38,123)(39,124)(40,125)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,118)(62,119)(63,120)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)>;
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), (1,95,11,85)(2,94,12,84)(3,93,13,83)(4,92,14,82)(5,91,15,81)(6,90,16,100)(7,89,17,99)(8,88,18,98)(9,87,19,97)(10,86,20,96)(21,61,31,71)(22,80,32,70)(23,79,33,69)(24,78,34,68)(25,77,35,67)(26,76,36,66)(27,75,37,65)(28,74,38,64)(29,73,39,63)(30,72,40,62)(41,151,51,141)(42,150,52,160)(43,149,53,159)(44,148,54,158)(45,147,55,157)(46,146,56,156)(47,145,57,155)(48,144,58,154)(49,143,59,153)(50,142,60,152)(101,123,111,133)(102,122,112,132)(103,121,113,131)(104,140,114,130)(105,139,115,129)(106,138,116,128)(107,137,117,127)(108,136,118,126)(109,135,119,125)(110,134,120,124), (1,63,155,129)(2,74,156,140)(3,65,157,131)(4,76,158,122)(5,67,159,133)(6,78,160,124)(7,69,141,135)(8,80,142,126)(9,71,143,137)(10,62,144,128)(11,73,145,139)(12,64,146,130)(13,75,147,121)(14,66,148,132)(15,77,149,123)(16,68,150,134)(17,79,151,125)(18,70,152,136)(19,61,153,127)(20,72,154,138)(21,49,117,97)(22,60,118,88)(23,51,119,99)(24,42,120,90)(25,53,101,81)(26,44,102,92)(27,55,103,83)(28,46,104,94)(29,57,105,85)(30,48,106,96)(31,59,107,87)(32,50,108,98)(33,41,109,89)(34,52,110,100)(35,43,111,91)(36,54,112,82)(37,45,113,93)(38,56,114,84)(39,47,115,95)(40,58,116,86), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,121)(37,122)(38,123)(39,124)(40,125)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,118)(62,119)(63,120)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117) );
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)], [(1,95,11,85),(2,94,12,84),(3,93,13,83),(4,92,14,82),(5,91,15,81),(6,90,16,100),(7,89,17,99),(8,88,18,98),(9,87,19,97),(10,86,20,96),(21,61,31,71),(22,80,32,70),(23,79,33,69),(24,78,34,68),(25,77,35,67),(26,76,36,66),(27,75,37,65),(28,74,38,64),(29,73,39,63),(30,72,40,62),(41,151,51,141),(42,150,52,160),(43,149,53,159),(44,148,54,158),(45,147,55,157),(46,146,56,156),(47,145,57,155),(48,144,58,154),(49,143,59,153),(50,142,60,152),(101,123,111,133),(102,122,112,132),(103,121,113,131),(104,140,114,130),(105,139,115,129),(106,138,116,128),(107,137,117,127),(108,136,118,126),(109,135,119,125),(110,134,120,124)], [(1,63,155,129),(2,74,156,140),(3,65,157,131),(4,76,158,122),(5,67,159,133),(6,78,160,124),(7,69,141,135),(8,80,142,126),(9,71,143,137),(10,62,144,128),(11,73,145,139),(12,64,146,130),(13,75,147,121),(14,66,148,132),(15,77,149,123),(16,68,150,134),(17,79,151,125),(18,70,152,136),(19,61,153,127),(20,72,154,138),(21,49,117,97),(22,60,118,88),(23,51,119,99),(24,42,120,90),(25,53,101,81),(26,44,102,92),(27,55,103,83),(28,46,104,94),(29,57,105,85),(30,48,106,96),(31,59,107,87),(32,50,108,98),(33,41,109,89),(34,52,110,100),(35,43,111,91),(36,54,112,82),(37,45,113,93),(38,56,114,84),(39,47,115,95),(40,58,116,86)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,126),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,121),(37,122),(38,123),(39,124),(40,125),(41,154),(42,155),(43,156),(44,157),(45,158),(46,159),(47,160),(48,141),(49,142),(50,143),(51,144),(52,145),(53,146),(54,147),(55,148),(56,149),(57,150),(58,151),(59,152),(60,153),(61,118),(62,119),(63,120),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 2 | 0 | 0 |
| 0 | 0 | 24 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 7 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 23 | 0 | 0 |
| 0 | 0 | 21 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 23 | 0 | 0 |
| 0 | 0 | 30 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,24,0,0,0,0,2,19,0,0,0,0,0,0,7,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,7,0,0,0,0,0,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,7,21,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,30,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | Dic10⋊20D4 | Dic5⋊4D4 | D10⋊D4 | Dic5⋊3Q8 | C4⋊2D20 | C4×C5⋊D4 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4⋊2D5 | Dic10 | C4⋊D4 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{20}D_4 % in TeX
G:=Group("Dic10:20D4"); // GroupNames label
G:=SmallGroup(320,1271);
// by ID
G=gap.SmallGroup(320,1271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,184,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀