metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊22D4, C10.192- (1+4), C5⋊4(Q8⋊5D4), C4.114(D4×D5), C22⋊Q8⋊10D5, C4⋊2D20⋊26C2, C4⋊C4.191D10, C20.237(C2×D4), D10⋊14(C4○D4), D10⋊D4⋊26C2, D20⋊8C4⋊27C2, D10⋊Q8⋊20C2, (C2×C20).56C23, (C2×Q8).128D10, C22⋊C4.17D10, Dic5.49(C2×D4), C10.79(C22×D4), Dic5⋊3Q8⋊26C2, C20.23D4⋊13C2, (C2×C10).177C24, (C22×C4).239D10, Dic5.5D4⋊25C2, (C2×D20).273C22, (Q8×C10).109C22, C23.120(C22×D5), C22.198(C23×D5), (C22×C20).257C22, (C22×C10).205C23, (C4×Dic5).115C22, (C2×Dic5).246C23, C10.D4.29C22, (C22×D5).209C23, C23.D5.118C22, D10⋊C4.128C22, C2.20(Q8.10D10), (C2×Dic10).303C22, (C2×Q8×D5)⋊8C2, C2.52(C2×D4×D5), (C4×C5⋊D4)⋊24C2, C2.50(D5×C4○D4), (C2×C4○D20)⋊25C2, (C5×C22⋊Q8)⋊13C2, C10.162(C2×C4○D4), (C2×C4×D5).106C22, (C5×C4⋊C4).160C22, (C2×C4).592(C22×D5), (C2×C5⋊D4).133C22, (C5×C22⋊C4).32C22, SmallGroup(320,1305)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1078 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×12], Q8 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×3], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8, C22⋊Q8 [×2], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10, Q8⋊5D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×6], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C4○D20 [×4], Q8×D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10⋊D4 [×2], Dic5.5D4 [×2], Dic5⋊3Q8, D20⋊8C4 [×2], C4⋊2D20, D10⋊Q8 [×2], C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, Dic10⋊22D4
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⋊5D4, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10, D5×C4○D4, Dic10⋊22D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=dad=a9, cbc-1=a10b, 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 98 11 88)(2 97 12 87)(3 96 13 86)(4 95 14 85)(5 94 15 84)(6 93 16 83)(7 92 17 82)(8 91 18 81)(9 90 19 100)(10 89 20 99)(21 77 31 67)(22 76 32 66)(23 75 33 65)(24 74 34 64)(25 73 35 63)(26 72 36 62)(27 71 37 61)(28 70 38 80)(29 69 39 79)(30 68 40 78)(41 125 51 135)(42 124 52 134)(43 123 53 133)(44 122 54 132)(45 121 55 131)(46 140 56 130)(47 139 57 129)(48 138 58 128)(49 137 59 127)(50 136 60 126)(101 149 111 159)(102 148 112 158)(103 147 113 157)(104 146 114 156)(105 145 115 155)(106 144 116 154)(107 143 117 153)(108 142 118 152)(109 141 119 151)(110 160 120 150)
(1 63 128 115)(2 72 129 104)(3 61 130 113)(4 70 131 102)(5 79 132 111)(6 68 133 120)(7 77 134 109)(8 66 135 118)(9 75 136 107)(10 64 137 116)(11 73 138 105)(12 62 139 114)(13 71 140 103)(14 80 121 112)(15 69 122 101)(16 78 123 110)(17 67 124 119)(18 76 125 108)(19 65 126 117)(20 74 127 106)(21 42 151 92)(22 51 152 81)(23 60 153 90)(24 49 154 99)(25 58 155 88)(26 47 156 97)(27 56 157 86)(28 45 158 95)(29 54 159 84)(30 43 160 93)(31 52 141 82)(32 41 142 91)(33 50 143 100)(34 59 144 89)(35 48 145 98)(36 57 146 87)(37 46 147 96)(38 55 148 85)(39 44 149 94)(40 53 150 83)
(1 145)(2 154)(3 143)(4 152)(5 141)(6 150)(7 159)(8 148)(9 157)(10 146)(11 155)(12 144)(13 153)(14 142)(15 151)(16 160)(17 149)(18 158)(19 147)(20 156)(21 122)(22 131)(23 140)(24 129)(25 138)(26 127)(27 136)(28 125)(29 134)(30 123)(31 132)(32 121)(33 130)(34 139)(35 128)(36 137)(37 126)(38 135)(39 124)(40 133)(41 80)(42 69)(43 78)(44 67)(45 76)(46 65)(47 74)(48 63)(49 72)(50 61)(51 70)(52 79)(53 68)(54 77)(55 66)(56 75)(57 64)(58 73)(59 62)(60 71)(81 102)(82 111)(83 120)(84 109)(85 118)(86 107)(87 116)(88 105)(89 114)(90 103)(91 112)(92 101)(93 110)(94 119)(95 108)(96 117)(97 106)(98 115)(99 104)(100 113)
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), (1,98,11,88)(2,97,12,87)(3,96,13,86)(4,95,14,85)(5,94,15,84)(6,93,16,83)(7,92,17,82)(8,91,18,81)(9,90,19,100)(10,89,20,99)(21,77,31,67)(22,76,32,66)(23,75,33,65)(24,74,34,64)(25,73,35,63)(26,72,36,62)(27,71,37,61)(28,70,38,80)(29,69,39,79)(30,68,40,78)(41,125,51,135)(42,124,52,134)(43,123,53,133)(44,122,54,132)(45,121,55,131)(46,140,56,130)(47,139,57,129)(48,138,58,128)(49,137,59,127)(50,136,60,126)(101,149,111,159)(102,148,112,158)(103,147,113,157)(104,146,114,156)(105,145,115,155)(106,144,116,154)(107,143,117,153)(108,142,118,152)(109,141,119,151)(110,160,120,150), (1,63,128,115)(2,72,129,104)(3,61,130,113)(4,70,131,102)(5,79,132,111)(6,68,133,120)(7,77,134,109)(8,66,135,118)(9,75,136,107)(10,64,137,116)(11,73,138,105)(12,62,139,114)(13,71,140,103)(14,80,121,112)(15,69,122,101)(16,78,123,110)(17,67,124,119)(18,76,125,108)(19,65,126,117)(20,74,127,106)(21,42,151,92)(22,51,152,81)(23,60,153,90)(24,49,154,99)(25,58,155,88)(26,47,156,97)(27,56,157,86)(28,45,158,95)(29,54,159,84)(30,43,160,93)(31,52,141,82)(32,41,142,91)(33,50,143,100)(34,59,144,89)(35,48,145,98)(36,57,146,87)(37,46,147,96)(38,55,148,85)(39,44,149,94)(40,53,150,83), (1,145)(2,154)(3,143)(4,152)(5,141)(6,150)(7,159)(8,148)(9,157)(10,146)(11,155)(12,144)(13,153)(14,142)(15,151)(16,160)(17,149)(18,158)(19,147)(20,156)(21,122)(22,131)(23,140)(24,129)(25,138)(26,127)(27,136)(28,125)(29,134)(30,123)(31,132)(32,121)(33,130)(34,139)(35,128)(36,137)(37,126)(38,135)(39,124)(40,133)(41,80)(42,69)(43,78)(44,67)(45,76)(46,65)(47,74)(48,63)(49,72)(50,61)(51,70)(52,79)(53,68)(54,77)(55,66)(56,75)(57,64)(58,73)(59,62)(60,71)(81,102)(82,111)(83,120)(84,109)(85,118)(86,107)(87,116)(88,105)(89,114)(90,103)(91,112)(92,101)(93,110)(94,119)(95,108)(96,117)(97,106)(98,115)(99,104)(100,113)>;
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), (1,98,11,88)(2,97,12,87)(3,96,13,86)(4,95,14,85)(5,94,15,84)(6,93,16,83)(7,92,17,82)(8,91,18,81)(9,90,19,100)(10,89,20,99)(21,77,31,67)(22,76,32,66)(23,75,33,65)(24,74,34,64)(25,73,35,63)(26,72,36,62)(27,71,37,61)(28,70,38,80)(29,69,39,79)(30,68,40,78)(41,125,51,135)(42,124,52,134)(43,123,53,133)(44,122,54,132)(45,121,55,131)(46,140,56,130)(47,139,57,129)(48,138,58,128)(49,137,59,127)(50,136,60,126)(101,149,111,159)(102,148,112,158)(103,147,113,157)(104,146,114,156)(105,145,115,155)(106,144,116,154)(107,143,117,153)(108,142,118,152)(109,141,119,151)(110,160,120,150), (1,63,128,115)(2,72,129,104)(3,61,130,113)(4,70,131,102)(5,79,132,111)(6,68,133,120)(7,77,134,109)(8,66,135,118)(9,75,136,107)(10,64,137,116)(11,73,138,105)(12,62,139,114)(13,71,140,103)(14,80,121,112)(15,69,122,101)(16,78,123,110)(17,67,124,119)(18,76,125,108)(19,65,126,117)(20,74,127,106)(21,42,151,92)(22,51,152,81)(23,60,153,90)(24,49,154,99)(25,58,155,88)(26,47,156,97)(27,56,157,86)(28,45,158,95)(29,54,159,84)(30,43,160,93)(31,52,141,82)(32,41,142,91)(33,50,143,100)(34,59,144,89)(35,48,145,98)(36,57,146,87)(37,46,147,96)(38,55,148,85)(39,44,149,94)(40,53,150,83), (1,145)(2,154)(3,143)(4,152)(5,141)(6,150)(7,159)(8,148)(9,157)(10,146)(11,155)(12,144)(13,153)(14,142)(15,151)(16,160)(17,149)(18,158)(19,147)(20,156)(21,122)(22,131)(23,140)(24,129)(25,138)(26,127)(27,136)(28,125)(29,134)(30,123)(31,132)(32,121)(33,130)(34,139)(35,128)(36,137)(37,126)(38,135)(39,124)(40,133)(41,80)(42,69)(43,78)(44,67)(45,76)(46,65)(47,74)(48,63)(49,72)(50,61)(51,70)(52,79)(53,68)(54,77)(55,66)(56,75)(57,64)(58,73)(59,62)(60,71)(81,102)(82,111)(83,120)(84,109)(85,118)(86,107)(87,116)(88,105)(89,114)(90,103)(91,112)(92,101)(93,110)(94,119)(95,108)(96,117)(97,106)(98,115)(99,104)(100,113) );
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)], [(1,98,11,88),(2,97,12,87),(3,96,13,86),(4,95,14,85),(5,94,15,84),(6,93,16,83),(7,92,17,82),(8,91,18,81),(9,90,19,100),(10,89,20,99),(21,77,31,67),(22,76,32,66),(23,75,33,65),(24,74,34,64),(25,73,35,63),(26,72,36,62),(27,71,37,61),(28,70,38,80),(29,69,39,79),(30,68,40,78),(41,125,51,135),(42,124,52,134),(43,123,53,133),(44,122,54,132),(45,121,55,131),(46,140,56,130),(47,139,57,129),(48,138,58,128),(49,137,59,127),(50,136,60,126),(101,149,111,159),(102,148,112,158),(103,147,113,157),(104,146,114,156),(105,145,115,155),(106,144,116,154),(107,143,117,153),(108,142,118,152),(109,141,119,151),(110,160,120,150)], [(1,63,128,115),(2,72,129,104),(3,61,130,113),(4,70,131,102),(5,79,132,111),(6,68,133,120),(7,77,134,109),(8,66,135,118),(9,75,136,107),(10,64,137,116),(11,73,138,105),(12,62,139,114),(13,71,140,103),(14,80,121,112),(15,69,122,101),(16,78,123,110),(17,67,124,119),(18,76,125,108),(19,65,126,117),(20,74,127,106),(21,42,151,92),(22,51,152,81),(23,60,153,90),(24,49,154,99),(25,58,155,88),(26,47,156,97),(27,56,157,86),(28,45,158,95),(29,54,159,84),(30,43,160,93),(31,52,141,82),(32,41,142,91),(33,50,143,100),(34,59,144,89),(35,48,145,98),(36,57,146,87),(37,46,147,96),(38,55,148,85),(39,44,149,94),(40,53,150,83)], [(1,145),(2,154),(3,143),(4,152),(5,141),(6,150),(7,159),(8,148),(9,157),(10,146),(11,155),(12,144),(13,153),(14,142),(15,151),(16,160),(17,149),(18,158),(19,147),(20,156),(21,122),(22,131),(23,140),(24,129),(25,138),(26,127),(27,136),(28,125),(29,134),(30,123),(31,132),(32,121),(33,130),(34,139),(35,128),(36,137),(37,126),(38,135),(39,124),(40,133),(41,80),(42,69),(43,78),(44,67),(45,76),(46,65),(47,74),(48,63),(49,72),(50,61),(51,70),(52,79),(53,68),(54,77),(55,66),(56,75),(57,64),(58,73),(59,62),(60,71),(81,102),(82,111),(83,120),(84,109),(85,118),(86,107),(87,116),(88,105),(89,114),(90,103),(91,112),(92,101),(93,110),(94,119),(95,108),(96,117),(97,106),(98,115),(99,104),(100,113)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 32 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,1,32,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,18,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 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 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8.10D10 | D5×C4○D4 |
| kernel | Dic10⋊22D4 | D10⋊D4 | Dic5.5D4 | Dic5⋊3Q8 | D20⋊8C4 | C4⋊2D20 | D10⋊Q8 | C4×C5⋊D4 | C20.23D4 | C5×C22⋊Q8 | C2×C4○D20 | C2×Q8×D5 | Dic10 | C22⋊Q8 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{22}D_4 % in TeX
G:=Group("Dic10:22D4"); // GroupNames label
G:=SmallGroup(320,1305);
// by ID
G=gap.SmallGroup(320,1305);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,12550]);
// 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=d*a*d=a^9,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀