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
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
Generators and relations
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
►On 160 points
Generators in S160
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(73 159)(74 160)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)
(1 21 76 46)(2 47 77 22)(3 23 78 48)(4 49 79 24)(5 25 80 50)(6 51 61 26)(7 27 62 52)(8 53 63 28)(9 29 64 54)(10 55 65 30)(11 31 66 56)(12 57 67 32)(13 33 68 58)(14 59 69 34)(15 35 70 60)(16 41 71 36)(17 37 72 42)(18 43 73 38)(19 39 74 44)(20 45 75 40)(81 133 142 115)(82 116 143 134)(83 135 144 117)(84 118 145 136)(85 137 146 119)(86 120 147 138)(87 139 148 101)(88 102 149 140)(89 121 150 103)(90 104 151 122)(91 123 152 105)(92 106 153 124)(93 125 154 107)(94 108 155 126)(95 127 156 109)(96 110 157 128)(97 129 158 111)(98 112 159 130)(99 131 160 113)(100 114 141 132)
(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)
(1 156)(2 155)(3 154)(4 153)(5 152)(6 151)(7 150)(8 149)(9 148)(10 147)(11 146)(12 145)(13 144)(14 143)(15 142)(16 141)(17 160)(18 159)(19 158)(20 157)(21 127)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 133)(36 132)(37 131)(38 130)(39 129)(40 128)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)
G:=sub<Sym(160)| (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146), (1,21,76,46)(2,47,77,22)(3,23,78,48)(4,49,79,24)(5,25,80,50)(6,51,61,26)(7,27,62,52)(8,53,63,28)(9,29,64,54)(10,55,65,30)(11,31,66,56)(12,57,67,32)(13,33,68,58)(14,59,69,34)(15,35,70,60)(16,41,71,36)(17,37,72,42)(18,43,73,38)(19,39,74,44)(20,45,75,40)(81,133,142,115)(82,116,143,134)(83,135,144,117)(84,118,145,136)(85,137,146,119)(86,120,147,138)(87,139,148,101)(88,102,149,140)(89,121,150,103)(90,104,151,122)(91,123,152,105)(92,106,153,124)(93,125,154,107)(94,108,155,126)(95,127,156,109)(96,110,157,128)(97,129,158,111)(98,112,159,130)(99,131,160,113)(100,114,141,132), (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), (1,156)(2,155)(3,154)(4,153)(5,152)(6,151)(7,150)(8,149)(9,148)(10,147)(11,146)(12,145)(13,144)(14,143)(15,142)(16,141)(17,160)(18,159)(19,158)(20,157)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)>;
G:=Group( (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146), (1,21,76,46)(2,47,77,22)(3,23,78,48)(4,49,79,24)(5,25,80,50)(6,51,61,26)(7,27,62,52)(8,53,63,28)(9,29,64,54)(10,55,65,30)(11,31,66,56)(12,57,67,32)(13,33,68,58)(14,59,69,34)(15,35,70,60)(16,41,71,36)(17,37,72,42)(18,43,73,38)(19,39,74,44)(20,45,75,40)(81,133,142,115)(82,116,143,134)(83,135,144,117)(84,118,145,136)(85,137,146,119)(86,120,147,138)(87,139,148,101)(88,102,149,140)(89,121,150,103)(90,104,151,122)(91,123,152,105)(92,106,153,124)(93,125,154,107)(94,108,155,126)(95,127,156,109)(96,110,157,128)(97,129,158,111)(98,112,159,130)(99,131,160,113)(100,114,141,132), (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), (1,156)(2,155)(3,154)(4,153)(5,152)(6,151)(7,150)(8,149)(9,148)(10,147)(11,146)(12,145)(13,144)(14,143)(15,142)(16,141)(17,160)(18,159)(19,158)(20,157)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91) );
G=PermutationGroup([(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(73,159),(74,160),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146)], [(1,21,76,46),(2,47,77,22),(3,23,78,48),(4,49,79,24),(5,25,80,50),(6,51,61,26),(7,27,62,52),(8,53,63,28),(9,29,64,54),(10,55,65,30),(11,31,66,56),(12,57,67,32),(13,33,68,58),(14,59,69,34),(15,35,70,60),(16,41,71,36),(17,37,72,42),(18,43,73,38),(19,39,74,44),(20,45,75,40),(81,133,142,115),(82,116,143,134),(83,135,144,117),(84,118,145,136),(85,137,146,119),(86,120,147,138),(87,139,148,101),(88,102,149,140),(89,121,150,103),(90,104,151,122),(91,123,152,105),(92,106,153,124),(93,125,154,107),(94,108,155,126),(95,127,156,109),(96,110,157,128),(97,129,158,111),(98,112,159,130),(99,131,160,113),(100,114,141,132)], [(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)], [(1,156),(2,155),(3,154),(4,153),(5,152),(6,151),(7,150),(8,149),(9,148),(10,147),(11,146),(12,145),(13,144),(14,143),(15,142),(16,141),(17,160),(18,159),(19,158),(20,157),(21,127),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,140),(29,139),(30,138),(31,137),(32,136),(33,135),(34,134),(35,133),(36,132),(37,131),(38,130),(39,129),(40,128),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,120),(56,119),(57,118),(58,117),(59,116),(60,115),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀