metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4⋊2Dic5, (C2×C20).501D4, C20.213(C2×D4), D4.6(C2×Dic5), Q8.6(C2×Dic5), (C2×D4).202D10, Q8⋊Dic5⋊44C2, D4⋊Dic5⋊44C2, (C2×Q8).171D10, C10.111(C4○D8), (C2×C20).480C23, C20.143(C22×C4), (C22×C10).112D4, (C22×C4).355D10, C23.44(C5⋊D4), C5⋊6(C23.24D4), C4.32(C23.D5), C4.14(C22×Dic5), C20.145(C22⋊C4), C2.7(D4.8D10), (D4×C10).243C22, C4⋊Dic5.355C22, (Q8×C10).206C22, C22.3(C23.D5), C23.21D10⋊19C2, (C22×C20).206C22, (C5×C4○D4)⋊8C4, (C2×C4○D4).2D5, C4.95(C2×C5⋊D4), (C22×C5⋊2C8)⋊8C2, (C10×C4○D4).2C2, (C5×D4).37(C2×C4), (C5×Q8).39(C2×C4), (C2×C20).296(C2×C4), (C2×C10).566(C2×D4), (C2×C4).52(C2×Dic5), C22.97(C2×C5⋊D4), C2.18(C2×C23.D5), (C2×C4).281(C5⋊D4), C10.123(C2×C22⋊C4), (C2×C4).565(C22×D5), (C2×C10).89(C22⋊C4), (C2×C5⋊2C8).290C22, SmallGroup(320,860)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.(C2×D4)
G = < a,b,c,d | a20=c4=d2=1, b2=a10, ab=ba, cac-1=a-1, dad=a11, bc=cb, bd=db, dcd=a5c-1 >
Subgroups: 398 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C5⋊2C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C23.24D4, C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], D4⋊Dic5 [×2], Q8⋊Dic5 [×2], C22×C5⋊2C8, C23.21D10, C10×C4○D4, C20.(C2×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C4○D8 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.24D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4.8D10 [×2], C2×C23.D5, C20.(C2×D4)
►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)
(1 112 11 102)(2 113 12 103)(3 114 13 104)(4 115 14 105)(5 116 15 106)(6 117 16 107)(7 118 17 108)(8 119 18 109)(9 120 19 110)(10 101 20 111)(21 132 31 122)(22 133 32 123)(23 134 33 124)(24 135 34 125)(25 136 35 126)(26 137 36 127)(27 138 37 128)(28 139 38 129)(29 140 39 130)(30 121 40 131)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 61 56 71)(47 62 57 72)(48 63 58 73)(49 64 59 74)(50 65 60 75)(81 151 91 141)(82 152 92 142)(83 153 93 143)(84 154 94 144)(85 155 95 145)(86 156 96 146)(87 157 97 147)(88 158 98 148)(89 159 99 149)(90 160 100 150)
(1 70 100 123)(2 69 81 122)(3 68 82 121)(4 67 83 140)(5 66 84 139)(6 65 85 138)(7 64 86 137)(8 63 87 136)(9 62 88 135)(10 61 89 134)(11 80 90 133)(12 79 91 132)(13 78 92 131)(14 77 93 130)(15 76 94 129)(16 75 95 128)(17 74 96 127)(18 73 97 126)(19 72 98 125)(20 71 99 124)(21 113 44 151)(22 112 45 150)(23 111 46 149)(24 110 47 148)(25 109 48 147)(26 108 49 146)(27 107 50 145)(28 106 51 144)(29 105 52 143)(30 104 53 142)(31 103 54 141)(32 102 55 160)(33 101 56 159)(34 120 57 158)(35 119 58 157)(36 118 59 156)(37 117 60 155)(38 116 41 154)(39 115 42 153)(40 114 43 152)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 44)(22 55)(23 46)(24 57)(25 48)(26 59)(27 50)(28 41)(29 52)(30 43)(31 54)(32 45)(33 56)(34 47)(35 58)(36 49)(37 60)(38 51)(39 42)(40 53)(61 134)(62 125)(63 136)(64 127)(65 138)(66 129)(67 140)(68 131)(69 122)(70 133)(71 124)(72 135)(73 126)(74 137)(75 128)(76 139)(77 130)(78 121)(79 132)(80 123)(81 96)(82 87)(83 98)(84 89)(85 100)(86 91)(88 93)(90 95)(92 97)(94 99)(101 116)(102 107)(103 118)(104 109)(105 120)(106 111)(108 113)(110 115)(112 117)(114 119)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)
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), (1,112,11,102)(2,113,12,103)(3,114,13,104)(4,115,14,105)(5,116,15,106)(6,117,16,107)(7,118,17,108)(8,119,18,109)(9,120,19,110)(10,101,20,111)(21,132,31,122)(22,133,32,123)(23,134,33,124)(24,135,34,125)(25,136,35,126)(26,137,36,127)(27,138,37,128)(28,139,38,129)(29,140,39,130)(30,121,40,131)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75)(81,151,91,141)(82,152,92,142)(83,153,93,143)(84,154,94,144)(85,155,95,145)(86,156,96,146)(87,157,97,147)(88,158,98,148)(89,159,99,149)(90,160,100,150), (1,70,100,123)(2,69,81,122)(3,68,82,121)(4,67,83,140)(5,66,84,139)(6,65,85,138)(7,64,86,137)(8,63,87,136)(9,62,88,135)(10,61,89,134)(11,80,90,133)(12,79,91,132)(13,78,92,131)(14,77,93,130)(15,76,94,129)(16,75,95,128)(17,74,96,127)(18,73,97,126)(19,72,98,125)(20,71,99,124)(21,113,44,151)(22,112,45,150)(23,111,46,149)(24,110,47,148)(25,109,48,147)(26,108,49,146)(27,107,50,145)(28,106,51,144)(29,105,52,143)(30,104,53,142)(31,103,54,141)(32,102,55,160)(33,101,56,159)(34,120,57,158)(35,119,58,157)(36,118,59,156)(37,117,60,155)(38,116,41,154)(39,115,42,153)(40,114,43,152), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,44)(22,55)(23,46)(24,57)(25,48)(26,59)(27,50)(28,41)(29,52)(30,43)(31,54)(32,45)(33,56)(34,47)(35,58)(36,49)(37,60)(38,51)(39,42)(40,53)(61,134)(62,125)(63,136)(64,127)(65,138)(66,129)(67,140)(68,131)(69,122)(70,133)(71,124)(72,135)(73,126)(74,137)(75,128)(76,139)(77,130)(78,121)(79,132)(80,123)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,116)(102,107)(103,118)(104,109)(105,120)(106,111)(108,113)(110,115)(112,117)(114,119)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159)>;
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), (1,112,11,102)(2,113,12,103)(3,114,13,104)(4,115,14,105)(5,116,15,106)(6,117,16,107)(7,118,17,108)(8,119,18,109)(9,120,19,110)(10,101,20,111)(21,132,31,122)(22,133,32,123)(23,134,33,124)(24,135,34,125)(25,136,35,126)(26,137,36,127)(27,138,37,128)(28,139,38,129)(29,140,39,130)(30,121,40,131)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75)(81,151,91,141)(82,152,92,142)(83,153,93,143)(84,154,94,144)(85,155,95,145)(86,156,96,146)(87,157,97,147)(88,158,98,148)(89,159,99,149)(90,160,100,150), (1,70,100,123)(2,69,81,122)(3,68,82,121)(4,67,83,140)(5,66,84,139)(6,65,85,138)(7,64,86,137)(8,63,87,136)(9,62,88,135)(10,61,89,134)(11,80,90,133)(12,79,91,132)(13,78,92,131)(14,77,93,130)(15,76,94,129)(16,75,95,128)(17,74,96,127)(18,73,97,126)(19,72,98,125)(20,71,99,124)(21,113,44,151)(22,112,45,150)(23,111,46,149)(24,110,47,148)(25,109,48,147)(26,108,49,146)(27,107,50,145)(28,106,51,144)(29,105,52,143)(30,104,53,142)(31,103,54,141)(32,102,55,160)(33,101,56,159)(34,120,57,158)(35,119,58,157)(36,118,59,156)(37,117,60,155)(38,116,41,154)(39,115,42,153)(40,114,43,152), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,44)(22,55)(23,46)(24,57)(25,48)(26,59)(27,50)(28,41)(29,52)(30,43)(31,54)(32,45)(33,56)(34,47)(35,58)(36,49)(37,60)(38,51)(39,42)(40,53)(61,134)(62,125)(63,136)(64,127)(65,138)(66,129)(67,140)(68,131)(69,122)(70,133)(71,124)(72,135)(73,126)(74,137)(75,128)(76,139)(77,130)(78,121)(79,132)(80,123)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,116)(102,107)(103,118)(104,109)(105,120)(106,111)(108,113)(110,115)(112,117)(114,119)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159) );
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)], [(1,112,11,102),(2,113,12,103),(3,114,13,104),(4,115,14,105),(5,116,15,106),(6,117,16,107),(7,118,17,108),(8,119,18,109),(9,120,19,110),(10,101,20,111),(21,132,31,122),(22,133,32,123),(23,134,33,124),(24,135,34,125),(25,136,35,126),(26,137,36,127),(27,138,37,128),(28,139,38,129),(29,140,39,130),(30,121,40,131),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,61,56,71),(47,62,57,72),(48,63,58,73),(49,64,59,74),(50,65,60,75),(81,151,91,141),(82,152,92,142),(83,153,93,143),(84,154,94,144),(85,155,95,145),(86,156,96,146),(87,157,97,147),(88,158,98,148),(89,159,99,149),(90,160,100,150)], [(1,70,100,123),(2,69,81,122),(3,68,82,121),(4,67,83,140),(5,66,84,139),(6,65,85,138),(7,64,86,137),(8,63,87,136),(9,62,88,135),(10,61,89,134),(11,80,90,133),(12,79,91,132),(13,78,92,131),(14,77,93,130),(15,76,94,129),(16,75,95,128),(17,74,96,127),(18,73,97,126),(19,72,98,125),(20,71,99,124),(21,113,44,151),(22,112,45,150),(23,111,46,149),(24,110,47,148),(25,109,48,147),(26,108,49,146),(27,107,50,145),(28,106,51,144),(29,105,52,143),(30,104,53,142),(31,103,54,141),(32,102,55,160),(33,101,56,159),(34,120,57,158),(35,119,58,157),(36,118,59,156),(37,117,60,155),(38,116,41,154),(39,115,42,153),(40,114,43,152)], [(1,6),(2,17),(3,8),(4,19),(5,10),(7,12),(9,14),(11,16),(13,18),(15,20),(21,44),(22,55),(23,46),(24,57),(25,48),(26,59),(27,50),(28,41),(29,52),(30,43),(31,54),(32,45),(33,56),(34,47),(35,58),(36,49),(37,60),(38,51),(39,42),(40,53),(61,134),(62,125),(63,136),(64,127),(65,138),(66,129),(67,140),(68,131),(69,122),(70,133),(71,124),(72,135),(73,126),(74,137),(75,128),(76,139),(77,130),(78,121),(79,132),(80,123),(81,96),(82,87),(83,98),(84,89),(85,100),(86,91),(88,93),(90,95),(92,97),(94,99),(101,116),(102,107),(103,118),(104,109),(105,120),(106,111),(108,113),(110,115),(112,117),(114,119),(141,156),(142,147),(143,158),(144,149),(145,160),(146,151),(148,153),(150,155),(152,157),(154,159)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | D10 | Dic5 | C4○D8 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | C20.(C2×D4) | D4⋊Dic5 | Q8⋊Dic5 | C22×C5⋊2C8 | C23.21D10 | C10×C4○D4 | C5×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 12 | 4 | 8 |
Matrix representation of C20.(C2×D4) ►in GL4(𝔽41) generated by
| 23 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 29 | 29 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [23,0,0,0,0,25,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[0,40,0,0,1,0,0,0,0,0,12,29,0,0,29,29],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
C20.(C2×D4) in GAP, Magma, Sage, TeX
C_{20}.(C_2\times D_4) % in TeX
G:=Group("C20.(C2xD4)"); // GroupNames label
G:=SmallGroup(320,860);
// by ID
G=gap.SmallGroup(320,860);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=a^5*c^-1>;
// generators/relations