direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊2D20, C4⋊3(C2×D20), (C2×C20)⋊8D4, C20⋊4(C2×D4), C4⋊C4⋊38D10, D10⋊2(C2×D4), (C2×C4)⋊10D20, C10⋊2(C4⋊D4), (C22×D20)⋊7C2, (C22×D5)⋊10D4, C10.9(C22×D4), (C2×D20)⋊45C22, (C2×C10).50C24, C22.67(C2×D20), C22.133(D4×D5), C2.11(C22×D20), (C2×C20).487C23, (C22×C4).360D10, D10⋊C4⋊50C22, C22.84(C23×D5), C23.328(C22×D5), (C22×C10).399C23, (C22×C20).217C22, C22.36(Q8⋊2D5), (C2×Dic5).198C23, (C22×D5).166C23, (C23×D5).112C22, (C22×Dic5).236C22, C5⋊2(C2×C4⋊D4), C2.15(C2×D4×D5), (C2×C4⋊C4)⋊15D5, (C10×C4⋊C4)⋊12C2, (D5×C22×C4)⋊1C2, (C2×C4×D5)⋊55C22, (C5×C4⋊C4)⋊46C22, C2.7(C2×Q8⋊2D5), C10.109(C2×C4○D4), (C2×C10).174(C2×D4), (C2×D10⋊C4)⋊20C2, (C2×C4).141(C22×D5), (C2×C10).197(C4○D4), SmallGroup(320,1178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊2D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1902 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×36], C5, C2×C4 [×10], C2×C4 [×16], D4 [×24], C23, C23 [×26], D5 [×8], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4 [×24], C24 [×3], Dic5 [×2], C20 [×4], C20 [×4], D10 [×4], D10 [×32], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], C4×D5 [×8], D20 [×24], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×10], C2×C20 [×4], C22×D5 [×10], C22×D5 [×16], C22×C10, C2×C4⋊D4, D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C2×D20 [×12], C2×D20 [×12], C22×Dic5, C22×C20, C22×C20 [×2], C23×D5, C23×D5 [×2], C4⋊2D20 [×8], C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, C22×D20, C22×D20 [×2], C2×C4⋊2D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D20 [×4], C22×D5 [×7], C2×C4⋊D4, C2×D20 [×6], D4×D5 [×2], Q8⋊2D5 [×2], C23×D5, C4⋊2D20 [×4], C22×D20, C2×D4×D5, C2×Q8⋊2D5, C2×C4⋊2D20
►On 160 points
Generators in S160
(1 138)(2 139)(3 140)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 119)(22 120)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)
(1 160 76 109)(2 110 77 141)(3 142 78 111)(4 112 79 143)(5 144 80 113)(6 114 61 145)(7 146 62 115)(8 116 63 147)(9 148 64 117)(10 118 65 149)(11 150 66 119)(12 120 67 151)(13 152 68 101)(14 102 69 153)(15 154 70 103)(16 104 71 155)(17 156 72 105)(18 106 73 157)(19 158 74 107)(20 108 75 159)(21 128 82 52)(22 53 83 129)(23 130 84 54)(24 55 85 131)(25 132 86 56)(26 57 87 133)(27 134 88 58)(28 59 89 135)(29 136 90 60)(30 41 91 137)(31 138 92 42)(32 43 93 139)(33 140 94 44)(34 45 95 121)(35 122 96 46)(36 47 97 123)(37 124 98 48)(38 49 99 125)(39 126 100 50)(40 51 81 127)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 140)(69 139)(70 138)(71 137)(72 136)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(81 145)(82 144)(83 143)(84 142)(85 141)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 147)(100 146)
G:=sub<Sym(160)| (1,138)(2,139)(3,140)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148), (1,160,76,109)(2,110,77,141)(3,142,78,111)(4,112,79,143)(5,144,80,113)(6,114,61,145)(7,146,62,115)(8,116,63,147)(9,148,64,117)(10,118,65,149)(11,150,66,119)(12,120,67,151)(13,152,68,101)(14,102,69,153)(15,154,70,103)(16,104,71,155)(17,156,72,105)(18,106,73,157)(19,158,74,107)(20,108,75,159)(21,128,82,52)(22,53,83,129)(23,130,84,54)(24,55,85,131)(25,132,86,56)(26,57,87,133)(27,134,88,58)(28,59,89,135)(29,136,90,60)(30,41,91,137)(31,138,92,42)(32,43,93,139)(33,140,94,44)(34,45,95,121)(35,122,96,46)(36,47,97,123)(37,124,98,48)(38,49,99,125)(39,126,100,50)(40,51,81,127), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146)>;
G:=Group( (1,138)(2,139)(3,140)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148), (1,160,76,109)(2,110,77,141)(3,142,78,111)(4,112,79,143)(5,144,80,113)(6,114,61,145)(7,146,62,115)(8,116,63,147)(9,148,64,117)(10,118,65,149)(11,150,66,119)(12,120,67,151)(13,152,68,101)(14,102,69,153)(15,154,70,103)(16,104,71,155)(17,156,72,105)(18,106,73,157)(19,158,74,107)(20,108,75,159)(21,128,82,52)(22,53,83,129)(23,130,84,54)(24,55,85,131)(25,132,86,56)(26,57,87,133)(27,134,88,58)(28,59,89,135)(29,136,90,60)(30,41,91,137)(31,138,92,42)(32,43,93,139)(33,140,94,44)(34,45,95,121)(35,122,96,46)(36,47,97,123)(37,124,98,48)(38,49,99,125)(39,126,100,50)(40,51,81,127), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,140)(69,139)(70,138)(71,137)(72,136)(73,135)(74,134)(75,133)(76,132)(77,131)(78,130)(79,129)(80,128)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146) );
G=PermutationGroup([(1,138),(2,139),(3,140),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,119),(22,120),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,73),(60,74),(81,149),(82,150),(83,151),(84,152),(85,153),(86,154),(87,155),(88,156),(89,157),(90,158),(91,159),(92,160),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148)], [(1,160,76,109),(2,110,77,141),(3,142,78,111),(4,112,79,143),(5,144,80,113),(6,114,61,145),(7,146,62,115),(8,116,63,147),(9,148,64,117),(10,118,65,149),(11,150,66,119),(12,120,67,151),(13,152,68,101),(14,102,69,153),(15,154,70,103),(16,104,71,155),(17,156,72,105),(18,106,73,157),(19,158,74,107),(20,108,75,159),(21,128,82,52),(22,53,83,129),(23,130,84,54),(24,55,85,131),(25,132,86,56),(26,57,87,133),(27,134,88,58),(28,59,89,135),(29,136,90,60),(30,41,91,137),(31,138,92,42),(32,43,93,139),(33,140,94,44),(34,45,95,121),(35,122,96,46),(36,47,97,123),(37,124,98,48),(38,49,99,125),(39,126,100,50),(40,51,81,127)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,60),(18,59),(19,58),(20,57),(21,113),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,120),(35,119),(36,118),(37,117),(38,116),(39,115),(40,114),(61,127),(62,126),(63,125),(64,124),(65,123),(66,122),(67,121),(68,140),(69,139),(70,138),(71,137),(72,136),(73,135),(74,134),(75,133),(76,132),(77,131),(78,130),(79,129),(80,128),(81,145),(82,144),(83,143),(84,142),(85,141),(86,160),(87,159),(88,158),(89,157),(90,156),(91,155),(92,154),(93,153),(94,152),(95,151),(96,150),(97,149),(98,148),(99,147),(100,146)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | D4×D5 | Q8⋊2D5 |
| kernel | C2×C4⋊2D20 | C4⋊2D20 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | C22×D20 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 4 | 8 | 6 | 16 | 4 | 4 |
Matrix representation of C2×C4⋊2D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 36 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 37 |
| 0 | 0 | 0 | 0 | 36 | 26 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,29,0,0,0,0,36,12],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,15,36,0,0,0,0,37,26],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40] >;
C2×C4⋊2D20 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes_2D_{20} % in TeX
G:=Group("C2xC4:2D20"); // GroupNames label
G:=SmallGroup(320,1178);
// by ID
G=gap.SmallGroup(320,1178);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations