direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×D20, C42⋊38D10, C10⋊2(C4×D4), (C2×C20)⋊32D4, C20⋊11(C2×D4), (C2×C42)⋊7D5, C20⋊8(C22×C4), (C4×C20)⋊53C22, D10⋊2(C22×C4), C2.1(C22×D20), C10.2(C22×D4), (C2×C10).17C24, C10.25(C23×C4), C4⋊Dic5⋊81C22, C22.63(C2×D20), (C2×C20).875C23, (C22×D20).21C2, (C22×C4).468D10, D10⋊C4⋊75C22, C22.14(C23×D5), (C2×D20).292C22, C22.68(C4○D20), C23.314(C22×D5), (C22×C20).503C22, (C22×C10).379C23, (C2×Dic5).183C23, (C23×D5).104C22, (C22×D5).156C23, (C22×Dic5).224C22, C5⋊2(C2×C4×D4), C4⋊3(C2×C4×D5), (C2×C4×C20)⋊11C2, (C2×C4)⋊12(C4×D5), (C2×C20)⋊42(C2×C4), C2.6(D5×C22×C4), C10.5(C2×C4○D4), C2.3(C2×C4○D20), (C2×C4×D5)⋊62C22, (D5×C22×C4)⋊14C2, C22.69(C2×C4×D5), (C2×C4⋊Dic5)⋊49C2, (C2×C10).169(C2×D4), (C22×D5)⋊14(C2×C4), (C2×D10⋊C4)⋊45C2, (C2×C10).96(C4○D4), (C2×C4).817(C22×D5), (C2×C10).248(C22×C4), SmallGroup(320,1145)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1518 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×6], C22 [×32], C5, C2×C4 [×14], C2×C4 [×26], D4 [×16], C23, C23 [×20], D5 [×8], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C24 [×2], Dic5 [×4], C20 [×8], C20 [×2], D10 [×8], D10 [×24], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×16], D20 [×16], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×14], C2×C20 [×2], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×C4×D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×4], C2×C4×D5 [×8], C2×C4×D5 [×8], C2×D20 [×12], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C4×D20 [×8], C2×C4⋊Dic5, C2×D10⋊C4 [×2], C2×C4×C20, D5×C22×C4 [×2], C22×D20, C2×C4×D20
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], D20 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], C2×D20 [×6], C4○D20 [×2], C23×D5, C4×D20 [×4], D5×C22×C4, C22×D20, C2×C4○D20, C2×C4×D20
Generators and relations
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 128)(42 129)(43 130)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 137)(51 138)(52 139)(53 140)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(1 85 41 29)(2 86 42 30)(3 87 43 31)(4 88 44 32)(5 89 45 33)(6 90 46 34)(7 91 47 35)(8 92 48 36)(9 93 49 37)(10 94 50 38)(11 95 51 39)(12 96 52 40)(13 97 53 21)(14 98 54 22)(15 99 55 23)(16 100 56 24)(17 81 57 25)(18 82 58 26)(19 83 59 27)(20 84 60 28)(61 131 160 110)(62 132 141 111)(63 133 142 112)(64 134 143 113)(65 135 144 114)(66 136 145 115)(67 137 146 116)(68 138 147 117)(69 139 148 118)(70 140 149 119)(71 121 150 120)(72 122 151 101)(73 123 152 102)(74 124 153 103)(75 125 154 104)(76 126 155 105)(77 127 156 106)(78 128 157 107)(79 129 158 108)(80 130 159 109)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 120)(18 119)(19 118)(20 117)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 129)(50 128)(51 127)(52 126)(53 125)(54 124)(55 123)(56 122)(57 121)(58 140)(59 139)(60 138)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)
G:=sub<Sym(160)| (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(41,128)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (1,85,41,29)(2,86,42,30)(3,87,43,31)(4,88,44,32)(5,89,45,33)(6,90,46,34)(7,91,47,35)(8,92,48,36)(9,93,49,37)(10,94,50,38)(11,95,51,39)(12,96,52,40)(13,97,53,21)(14,98,54,22)(15,99,55,23)(16,100,56,24)(17,81,57,25)(18,82,58,26)(19,83,59,27)(20,84,60,28)(61,131,160,110)(62,132,141,111)(63,133,142,112)(64,134,143,113)(65,135,144,114)(66,136,145,115)(67,137,146,116)(68,138,147,117)(69,139,148,118)(70,140,149,119)(71,121,150,120)(72,122,151,101)(73,123,152,102)(74,124,153,103)(75,125,154,104)(76,126,155,105)(77,127,156,106)(78,128,157,107)(79,129,158,108)(80,130,159,109), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,120)(18,119)(19,118)(20,117)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,129)(50,128)(51,127)(52,126)(53,125)(54,124)(55,123)(56,122)(57,121)(58,140)(59,139)(60,138)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)>;
G:=Group( (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(41,128)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(49,136)(50,137)(51,138)(52,139)(53,140)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (1,85,41,29)(2,86,42,30)(3,87,43,31)(4,88,44,32)(5,89,45,33)(6,90,46,34)(7,91,47,35)(8,92,48,36)(9,93,49,37)(10,94,50,38)(11,95,51,39)(12,96,52,40)(13,97,53,21)(14,98,54,22)(15,99,55,23)(16,100,56,24)(17,81,57,25)(18,82,58,26)(19,83,59,27)(20,84,60,28)(61,131,160,110)(62,132,141,111)(63,133,142,112)(64,134,143,113)(65,135,144,114)(66,136,145,115)(67,137,146,116)(68,138,147,117)(69,139,148,118)(70,140,149,119)(71,121,150,120)(72,122,151,101)(73,123,152,102)(74,124,153,103)(75,125,154,104)(76,126,155,105)(77,127,156,106)(78,128,157,107)(79,129,158,108)(80,130,159,109), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,120)(18,119)(19,118)(20,117)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,129)(50,128)(51,127)(52,126)(53,125)(54,124)(55,123)(56,122)(57,121)(58,140)(59,139)(60,138)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92) );
G=PermutationGroup([(1,107),(2,108),(3,109),(4,110),(5,111),(6,112),(7,113),(8,114),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,149),(22,150),(23,151),(24,152),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,141),(34,142),(35,143),(36,144),(37,145),(38,146),(39,147),(40,148),(41,128),(42,129),(43,130),(44,131),(45,132),(46,133),(47,134),(48,135),(49,136),(50,137),(51,138),(52,139),(53,140),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)], [(1,85,41,29),(2,86,42,30),(3,87,43,31),(4,88,44,32),(5,89,45,33),(6,90,46,34),(7,91,47,35),(8,92,48,36),(9,93,49,37),(10,94,50,38),(11,95,51,39),(12,96,52,40),(13,97,53,21),(14,98,54,22),(15,99,55,23),(16,100,56,24),(17,81,57,25),(18,82,58,26),(19,83,59,27),(20,84,60,28),(61,131,160,110),(62,132,141,111),(63,133,142,112),(64,134,143,113),(65,135,144,114),(66,136,145,115),(67,137,146,116),(68,138,147,117),(69,139,148,118),(70,140,149,119),(71,121,150,120),(72,122,151,101),(73,123,152,102),(74,124,153,103),(75,125,154,104),(76,126,155,105),(77,127,156,106),(78,128,157,107),(79,129,158,108),(80,130,159,109)], [(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)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,120),(18,119),(19,118),(20,117),(21,154),(22,153),(23,152),(24,151),(25,150),(26,149),(27,148),(28,147),(29,146),(30,145),(31,144),(32,143),(33,142),(34,141),(35,160),(36,159),(37,158),(38,157),(39,156),(40,155),(41,137),(42,136),(43,135),(44,134),(45,133),(46,132),(47,131),(48,130),(49,129),(50,128),(51,127),(52,126),(53,125),(54,124),(55,123),(56,122),(57,121),(58,140),(59,139),(60,138),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,100),(73,99),(74,98),(75,97),(76,96),(77,95),(78,94),(79,93),(80,92)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 36 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | ··· | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C4○D20 |
| kernel | C2×C4×D20 | C4×D20 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C4×C20 | D5×C22×C4 | C22×D20 | C2×D20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 4 | 2 | 4 | 8 | 6 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times D_{20} % in TeX
G:=Group("C2xC4xD20"); // GroupNames label
G:=SmallGroup(320,1145);
// by ID
G=gap.SmallGroup(320,1145);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,80,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀