metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊Q8⋊18C2, C20⋊9(C4○D4), C4⋊D4⋊25D5, C20⋊7D4⋊31C2, C20⋊D4⋊14C2, C4⋊3(D4⋊2D5), C22.1(D4×D5), C4⋊C4.175D10, (C2×Dic5)⋊13D4, (D4×Dic5)⋊14C2, D20⋊8C4⋊19C2, Dic5⋊2(C4○D4), Dic5⋊D4⋊7C2, Dic5⋊4D4⋊5C2, (C2×D4).150D10, (C2×C20).34C23, C22⋊C4.45D10, Dic5.17(C2×D4), C10.59(C22×D4), (C2×C10).140C24, (C22×C4).366D10, Dic5.5D4⋊16C2, (D4×C10).114C22, (C2×D20).147C22, C4⋊Dic5.203C22, (C22×C10).11C23, C5⋊3(C22.26C24), (C22×D5).59C23, C23.178(C22×D5), C22.161(C23×D5), C23.D5.18C22, D10⋊C4.57C22, (C22×C20).235C22, (C2×Dic5).233C23, (C4×Dic5).285C22, C10.D4.12C22, (C2×Dic10).156C22, (C22×Dic5).244C22, C2.32(C2×D4×D5), (C2×C4×Dic5)⋊7C2, (C5×C4⋊D4)⋊5C2, C2.33(D5×C4○D4), (C2×C10).3(C2×D4), (C2×D4⋊2D5)⋊8C2, C10.79(C2×C4○D4), (C2×C4×D5).88C22, C2.30(C2×D4⋊2D5), (C2×C4).34(C22×D5), (C5×C4⋊C4).136C22, (C2×C5⋊D4).23C22, (C5×C22⋊C4).5C22, SmallGroup(320,1268)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1102 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×22], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], Dic5 [×3], C20 [×2], C20 [×3], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.26C24, C4×Dic5 [×4], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], Dic5⋊4D4 [×2], Dic5.5D4 [×2], C20⋊Q8, D20⋊8C4, C2×C4×Dic5, C20⋊7D4, D4×Dic5, Dic5⋊D4 [×2], C20⋊D4, C5×C4⋊D4, C2×D4⋊2D5 [×2], C20⋊(C4○D4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], D4⋊2D5 [×2], C23×D5, C2×D4×D5, C2×D4⋊2D5, D5×C4○D4, C20⋊(C4○D4)
Generators and relations
G = < a,b,c,d | a20=b4=d2=1, c2=b2, bab-1=a9, cac-1=a11, ad=da, bc=cb, bd=db, dcd=b2c >
►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)
(1 135 149 23)(2 124 150 32)(3 133 151 21)(4 122 152 30)(5 131 153 39)(6 140 154 28)(7 129 155 37)(8 138 156 26)(9 127 157 35)(10 136 158 24)(11 125 159 33)(12 134 160 22)(13 123 141 31)(14 132 142 40)(15 121 143 29)(16 130 144 38)(17 139 145 27)(18 128 146 36)(19 137 147 25)(20 126 148 34)(41 109 83 65)(42 118 84 74)(43 107 85 63)(44 116 86 72)(45 105 87 61)(46 114 88 70)(47 103 89 79)(48 112 90 68)(49 101 91 77)(50 110 92 66)(51 119 93 75)(52 108 94 64)(53 117 95 73)(54 106 96 62)(55 115 97 71)(56 104 98 80)(57 113 99 69)(58 102 100 78)(59 111 81 67)(60 120 82 76)
(1 23 149 135)(2 34 150 126)(3 25 151 137)(4 36 152 128)(5 27 153 139)(6 38 154 130)(7 29 155 121)(8 40 156 132)(9 31 157 123)(10 22 158 134)(11 33 159 125)(12 24 160 136)(13 35 141 127)(14 26 142 138)(15 37 143 129)(16 28 144 140)(17 39 145 131)(18 30 146 122)(19 21 147 133)(20 32 148 124)(41 117 83 73)(42 108 84 64)(43 119 85 75)(44 110 86 66)(45 101 87 77)(46 112 88 68)(47 103 89 79)(48 114 90 70)(49 105 91 61)(50 116 92 72)(51 107 93 63)(52 118 94 74)(53 109 95 65)(54 120 96 76)(55 111 97 67)(56 102 98 78)(57 113 99 69)(58 104 100 80)(59 115 81 71)(60 106 82 62)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 137)(82 138)(83 139)(84 140)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)
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), (1,135,149,23)(2,124,150,32)(3,133,151,21)(4,122,152,30)(5,131,153,39)(6,140,154,28)(7,129,155,37)(8,138,156,26)(9,127,157,35)(10,136,158,24)(11,125,159,33)(12,134,160,22)(13,123,141,31)(14,132,142,40)(15,121,143,29)(16,130,144,38)(17,139,145,27)(18,128,146,36)(19,137,147,25)(20,126,148,34)(41,109,83,65)(42,118,84,74)(43,107,85,63)(44,116,86,72)(45,105,87,61)(46,114,88,70)(47,103,89,79)(48,112,90,68)(49,101,91,77)(50,110,92,66)(51,119,93,75)(52,108,94,64)(53,117,95,73)(54,106,96,62)(55,115,97,71)(56,104,98,80)(57,113,99,69)(58,102,100,78)(59,111,81,67)(60,120,82,76), (1,23,149,135)(2,34,150,126)(3,25,151,137)(4,36,152,128)(5,27,153,139)(6,38,154,130)(7,29,155,121)(8,40,156,132)(9,31,157,123)(10,22,158,134)(11,33,159,125)(12,24,160,136)(13,35,141,127)(14,26,142,138)(15,37,143,129)(16,28,144,140)(17,39,145,131)(18,30,146,122)(19,21,147,133)(20,32,148,124)(41,117,83,73)(42,108,84,64)(43,119,85,75)(44,110,86,66)(45,101,87,77)(46,112,88,68)(47,103,89,79)(48,114,90,70)(49,105,91,61)(50,116,92,72)(51,107,93,63)(52,118,94,74)(53,109,95,65)(54,120,96,76)(55,111,97,67)(56,102,98,78)(57,113,99,69)(58,104,100,80)(59,115,81,71)(60,106,82,62), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,137)(82,138)(83,139)(84,140)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)>;
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), (1,135,149,23)(2,124,150,32)(3,133,151,21)(4,122,152,30)(5,131,153,39)(6,140,154,28)(7,129,155,37)(8,138,156,26)(9,127,157,35)(10,136,158,24)(11,125,159,33)(12,134,160,22)(13,123,141,31)(14,132,142,40)(15,121,143,29)(16,130,144,38)(17,139,145,27)(18,128,146,36)(19,137,147,25)(20,126,148,34)(41,109,83,65)(42,118,84,74)(43,107,85,63)(44,116,86,72)(45,105,87,61)(46,114,88,70)(47,103,89,79)(48,112,90,68)(49,101,91,77)(50,110,92,66)(51,119,93,75)(52,108,94,64)(53,117,95,73)(54,106,96,62)(55,115,97,71)(56,104,98,80)(57,113,99,69)(58,102,100,78)(59,111,81,67)(60,120,82,76), (1,23,149,135)(2,34,150,126)(3,25,151,137)(4,36,152,128)(5,27,153,139)(6,38,154,130)(7,29,155,121)(8,40,156,132)(9,31,157,123)(10,22,158,134)(11,33,159,125)(12,24,160,136)(13,35,141,127)(14,26,142,138)(15,37,143,129)(16,28,144,140)(17,39,145,131)(18,30,146,122)(19,21,147,133)(20,32,148,124)(41,117,83,73)(42,108,84,64)(43,119,85,75)(44,110,86,66)(45,101,87,77)(46,112,88,68)(47,103,89,79)(48,114,90,70)(49,105,91,61)(50,116,92,72)(51,107,93,63)(52,118,94,74)(53,109,95,65)(54,120,96,76)(55,111,97,67)(56,102,98,78)(57,113,99,69)(58,104,100,80)(59,115,81,71)(60,106,82,62), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,137)(82,138)(83,139)(84,140)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136) );
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)], [(1,135,149,23),(2,124,150,32),(3,133,151,21),(4,122,152,30),(5,131,153,39),(6,140,154,28),(7,129,155,37),(8,138,156,26),(9,127,157,35),(10,136,158,24),(11,125,159,33),(12,134,160,22),(13,123,141,31),(14,132,142,40),(15,121,143,29),(16,130,144,38),(17,139,145,27),(18,128,146,36),(19,137,147,25),(20,126,148,34),(41,109,83,65),(42,118,84,74),(43,107,85,63),(44,116,86,72),(45,105,87,61),(46,114,88,70),(47,103,89,79),(48,112,90,68),(49,101,91,77),(50,110,92,66),(51,119,93,75),(52,108,94,64),(53,117,95,73),(54,106,96,62),(55,115,97,71),(56,104,98,80),(57,113,99,69),(58,102,100,78),(59,111,81,67),(60,120,82,76)], [(1,23,149,135),(2,34,150,126),(3,25,151,137),(4,36,152,128),(5,27,153,139),(6,38,154,130),(7,29,155,121),(8,40,156,132),(9,31,157,123),(10,22,158,134),(11,33,159,125),(12,24,160,136),(13,35,141,127),(14,26,142,138),(15,37,143,129),(16,28,144,140),(17,39,145,131),(18,30,146,122),(19,21,147,133),(20,32,148,124),(41,117,83,73),(42,108,84,64),(43,119,85,75),(44,110,86,66),(45,101,87,77),(46,112,88,68),(47,103,89,79),(48,114,90,70),(49,105,91,61),(50,116,92,72),(51,107,93,63),(52,118,94,74),(53,109,95,65),(54,120,96,76),(55,111,97,67),(56,102,98,78),(57,113,99,69),(58,104,100,80),(59,115,81,71),(60,106,82,62)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160),(81,137),(82,138),(83,139),(84,140),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 | 0 | 0 |
| 0 | 0 | 32 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,35,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 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 | 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 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4⋊2D5 | D4×D5 | D5×C4○D4 |
| kernel | C20⋊(C4○D4) | Dic5⋊4D4 | Dic5.5D4 | C20⋊Q8 | D20⋊8C4 | C2×C4×Dic5 | C20⋊7D4 | D4×Dic5 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊D4 | C2×D4⋊2D5 | C2×Dic5 | C4⋊D4 | Dic5 | C20 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes (C_4\circ D_4) % in TeX
G:=Group("C20:(C4oD4)"); // GroupNames label
G:=SmallGroup(320,1268);
// by ID
G=gap.SmallGroup(320,1268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,b*a*b^-1=a^9,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀