metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊21D4, C10.762- (1+4), C5⋊3(D4×Q8), C5⋊D4⋊1Q8, C20⋊Q8⋊25C2, D10⋊6(C2×Q8), C22⋊Q8⋊9D5, C22⋊1(Q8×D5), Dic5⋊4(C2×Q8), C4.113(D4×D5), C4⋊C4.190D10, C20.236(C2×D4), D10⋊2Q8⋊26C2, D10⋊Q8⋊19C2, (C2×C20).55C23, (C2×Q8).127D10, C22⋊C4.58D10, Dic5.48(C2×D4), C10.78(C22×D4), Dic5⋊3Q8⋊25C2, Dic5⋊Q8⋊15C2, C10.35(C22×Q8), (C2×C10).176C24, Dic5⋊4D4.1C2, (C22×C4).238D10, (C22×Dic10)⋊17C2, C4⋊Dic5.216C22, (Q8×C10).108C22, C22.197(C23×D5), C23.190(C22×D5), Dic5.14D4⋊23C2, (C22×C20).256C22, (C22×C10).204C23, (C4×Dic5).114C22, (C2×Dic5).245C23, C10.D4.28C22, (C22×D5).208C23, C2.36(D4.10D10), C23.D5.117C22, D10⋊C4.107C22, (C2×Dic10).256C22, (C22×Dic5).118C22, (C2×Q8×D5)⋊7C2, C2.51(C2×D4×D5), C2.18(C2×Q8×D5), (C2×C10)⋊3(C2×Q8), (C4×C5⋊D4).7C2, (C5×C22⋊Q8)⋊12C2, (C2×C4×D5).105C22, (C2×C4).49(C22×D5), (C5×C4⋊C4).159C22, (C2×C5⋊D4).132C22, (C5×C22⋊C4).31C22, SmallGroup(320,1304)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 934 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×15], C22, C22 [×2], C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×4], Q8 [×16], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×14], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C4⋊Q8 [×3], C22×Q8 [×2], Dic10 [×4], Dic10 [×10], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×C10, D4×Q8, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4 [×2], Dic5⋊4D4 [×2], Dic5⋊3Q8, C20⋊Q8 [×2], D10⋊Q8 [×2], D10⋊2Q8, C4×C5⋊D4, Dic5⋊Q8, C5×C22⋊Q8, C22×Dic10, C2×Q8×D5, Dic10⋊21D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- (1+4), C22×D5 [×7], D4×Q8, D4×D5 [×2], Q8×D5 [×2], C23×D5, C2×D4×D5, C2×Q8×D5, D4.10D10, Dic10⋊21D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a9, ad=da, bc=cb, bd=db, 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 71 11 61)(2 70 12 80)(3 69 13 79)(4 68 14 78)(5 67 15 77)(6 66 16 76)(7 65 17 75)(8 64 18 74)(9 63 19 73)(10 62 20 72)(21 116 31 106)(22 115 32 105)(23 114 33 104)(24 113 34 103)(25 112 35 102)(26 111 36 101)(27 110 37 120)(28 109 38 119)(29 108 39 118)(30 107 40 117)(41 160 51 150)(42 159 52 149)(43 158 53 148)(44 157 54 147)(45 156 55 146)(46 155 56 145)(47 154 57 144)(48 153 58 143)(49 152 59 142)(50 151 60 141)(81 136 91 126)(82 135 92 125)(83 134 93 124)(84 133 94 123)(85 132 95 122)(86 131 96 121)(87 130 97 140)(88 129 98 139)(89 128 99 138)(90 127 100 137)
(1 28 145 85)(2 37 146 94)(3 26 147 83)(4 35 148 92)(5 24 149 81)(6 33 150 90)(7 22 151 99)(8 31 152 88)(9 40 153 97)(10 29 154 86)(11 38 155 95)(12 27 156 84)(13 36 157 93)(14 25 158 82)(15 34 159 91)(16 23 160 100)(17 32 141 89)(18 21 142 98)(19 30 143 87)(20 39 144 96)(41 127 66 104)(42 136 67 113)(43 125 68 102)(44 134 69 111)(45 123 70 120)(46 132 71 109)(47 121 72 118)(48 130 73 107)(49 139 74 116)(50 128 75 105)(51 137 76 114)(52 126 77 103)(53 135 78 112)(54 124 79 101)(55 133 80 110)(56 122 61 119)(57 131 62 108)(58 140 63 117)(59 129 64 106)(60 138 65 115)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 81)(18 82)(19 83)(20 84)(21 158)(22 159)(23 160)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 101)(59 102)(60 103)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 133)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 121)
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), (1,71,11,61)(2,70,12,80)(3,69,13,79)(4,68,14,78)(5,67,15,77)(6,66,16,76)(7,65,17,75)(8,64,18,74)(9,63,19,73)(10,62,20,72)(21,116,31,106)(22,115,32,105)(23,114,33,104)(24,113,34,103)(25,112,35,102)(26,111,36,101)(27,110,37,120)(28,109,38,119)(29,108,39,118)(30,107,40,117)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(81,136,91,126)(82,135,92,125)(83,134,93,124)(84,133,94,123)(85,132,95,122)(86,131,96,121)(87,130,97,140)(88,129,98,139)(89,128,99,138)(90,127,100,137), (1,28,145,85)(2,37,146,94)(3,26,147,83)(4,35,148,92)(5,24,149,81)(6,33,150,90)(7,22,151,99)(8,31,152,88)(9,40,153,97)(10,29,154,86)(11,38,155,95)(12,27,156,84)(13,36,157,93)(14,25,158,82)(15,34,159,91)(16,23,160,100)(17,32,141,89)(18,21,142,98)(19,30,143,87)(20,39,144,96)(41,127,66,104)(42,136,67,113)(43,125,68,102)(44,134,69,111)(45,123,70,120)(46,132,71,109)(47,121,72,118)(48,130,73,107)(49,139,74,116)(50,128,75,105)(51,137,76,114)(52,126,77,103)(53,135,78,112)(54,124,79,101)(55,133,80,110)(56,122,61,119)(57,131,62,108)(58,140,63,117)(59,129,64,106)(60,138,65,115), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,81)(18,82)(19,83)(20,84)(21,158)(22,159)(23,160)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,101)(59,102)(60,103)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,121)>;
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), (1,71,11,61)(2,70,12,80)(3,69,13,79)(4,68,14,78)(5,67,15,77)(6,66,16,76)(7,65,17,75)(8,64,18,74)(9,63,19,73)(10,62,20,72)(21,116,31,106)(22,115,32,105)(23,114,33,104)(24,113,34,103)(25,112,35,102)(26,111,36,101)(27,110,37,120)(28,109,38,119)(29,108,39,118)(30,107,40,117)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(81,136,91,126)(82,135,92,125)(83,134,93,124)(84,133,94,123)(85,132,95,122)(86,131,96,121)(87,130,97,140)(88,129,98,139)(89,128,99,138)(90,127,100,137), (1,28,145,85)(2,37,146,94)(3,26,147,83)(4,35,148,92)(5,24,149,81)(6,33,150,90)(7,22,151,99)(8,31,152,88)(9,40,153,97)(10,29,154,86)(11,38,155,95)(12,27,156,84)(13,36,157,93)(14,25,158,82)(15,34,159,91)(16,23,160,100)(17,32,141,89)(18,21,142,98)(19,30,143,87)(20,39,144,96)(41,127,66,104)(42,136,67,113)(43,125,68,102)(44,134,69,111)(45,123,70,120)(46,132,71,109)(47,121,72,118)(48,130,73,107)(49,139,74,116)(50,128,75,105)(51,137,76,114)(52,126,77,103)(53,135,78,112)(54,124,79,101)(55,133,80,110)(56,122,61,119)(57,131,62,108)(58,140,63,117)(59,129,64,106)(60,138,65,115), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,81)(18,82)(19,83)(20,84)(21,158)(22,159)(23,160)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,101)(59,102)(60,103)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,121) );
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)], [(1,71,11,61),(2,70,12,80),(3,69,13,79),(4,68,14,78),(5,67,15,77),(6,66,16,76),(7,65,17,75),(8,64,18,74),(9,63,19,73),(10,62,20,72),(21,116,31,106),(22,115,32,105),(23,114,33,104),(24,113,34,103),(25,112,35,102),(26,111,36,101),(27,110,37,120),(28,109,38,119),(29,108,39,118),(30,107,40,117),(41,160,51,150),(42,159,52,149),(43,158,53,148),(44,157,54,147),(45,156,55,146),(46,155,56,145),(47,154,57,144),(48,153,58,143),(49,152,59,142),(50,151,60,141),(81,136,91,126),(82,135,92,125),(83,134,93,124),(84,133,94,123),(85,132,95,122),(86,131,96,121),(87,130,97,140),(88,129,98,139),(89,128,99,138),(90,127,100,137)], [(1,28,145,85),(2,37,146,94),(3,26,147,83),(4,35,148,92),(5,24,149,81),(6,33,150,90),(7,22,151,99),(8,31,152,88),(9,40,153,97),(10,29,154,86),(11,38,155,95),(12,27,156,84),(13,36,157,93),(14,25,158,82),(15,34,159,91),(16,23,160,100),(17,32,141,89),(18,21,142,98),(19,30,143,87),(20,39,144,96),(41,127,66,104),(42,136,67,113),(43,125,68,102),(44,134,69,111),(45,123,70,120),(46,132,71,109),(47,121,72,118),(48,130,73,107),(49,139,74,116),(50,128,75,105),(51,137,76,114),(52,126,77,103),(53,135,78,112),(54,124,79,101),(55,133,80,110),(56,122,61,119),(57,131,62,108),(58,140,63,117),(59,129,64,106),(60,138,65,115)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,81),(18,82),(19,83),(20,84),(21,158),(22,159),(23,160),(24,141),(25,142),(26,143),(27,144),(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,157),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,101),(59,102),(60,103),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,133),(73,134),(74,135),(75,136),(76,137),(77,138),(78,139),(79,140),(80,121)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 30 |
| 0 | 0 | 0 | 0 | 30 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,30,0,0,0,0,30,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[9,0,0,0,0,0,5,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 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 | C2 | D4 | Q8 | D5 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8×D5 | D4.10D10 |
| kernel | Dic10⋊21D4 | Dic5.14D4 | Dic5⋊4D4 | Dic5⋊3Q8 | C20⋊Q8 | D10⋊Q8 | D10⋊2Q8 | C4×C5⋊D4 | Dic5⋊Q8 | C5×C22⋊Q8 | C22×Dic10 | C2×Q8×D5 | Dic10 | C5⋊D4 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{21}D_4 % in TeX
G:=Group("Dic10:21D4"); // GroupNames label
G:=SmallGroup(320,1304);
// by ID
G=gap.SmallGroup(320,1304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,185,80,12550]);
// 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^9,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀