direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C42⋊C2, (C2×C42)⋊3C10, (C22×C20)⋊32C4, (C4×C20)⋊48C22, (C22×C4)⋊10C20, C42⋊13(C2×C10), (C23×C4).9C10, C2.3(C23×C20), C24.29(C2×C10), C10.76(C23×C4), C23.34(C2×C20), C4.30(C22×C20), (C23×C20).24C2, C20.247(C22×C4), (C2×C20).706C23, (C2×C10).334C24, C22.7(C23×C10), C22.25(C22×C20), (C23×C10).89C22, C23.67(C22×C10), (C22×C10).467C23, (C22×C20).609C22, (C2×C4×C20)⋊5C2, (C2×C4⋊C4)⋊24C10, (C10×C4⋊C4)⋊51C2, C4⋊C4⋊18(C2×C10), (C2×C4)⋊11(C2×C20), (C2×C20)⋊53(C2×C4), C2.1(C10×C4○D4), (C5×C4⋊C4)⋊75C22, C10.220(C2×C4○D4), C22.26(C5×C4○D4), (C10×C22⋊C4).35C2, C22⋊C4.27(C2×C10), (C2×C22⋊C4).15C10, (C22×C4).97(C2×C10), (C2×C10).226(C4○D4), (C2×C4).133(C22×C10), (C2×C10).265(C22×C4), (C22×C10).188(C2×C4), (C5×C22⋊C4).158C22, SmallGroup(320,1516)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C42⋊C2
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 402 in 330 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C42⋊C2, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C23×C10, C2×C4×C20, C10×C22⋊C4, C10×C4⋊C4, C5×C42⋊C2, C23×C20, C10×C42⋊C2
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C4○D4, C24, C20, C2×C10, C42⋊C2, C23×C4, C2×C4○D4, C2×C20, C22×C10, C2×C42⋊C2, C22×C20, C5×C4○D4, C23×C10, C5×C42⋊C2, C23×C20, C10×C4○D4, C10×C42⋊C2
►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)
(1 142 48 136)(2 143 49 137)(3 144 50 138)(4 145 41 139)(5 146 42 140)(6 147 43 131)(7 148 44 132)(8 149 45 133)(9 150 46 134)(10 141 47 135)(11 74 21 88)(12 75 22 89)(13 76 23 90)(14 77 24 81)(15 78 25 82)(16 79 26 83)(17 80 27 84)(18 71 28 85)(19 72 29 86)(20 73 30 87)(31 92 160 108)(32 93 151 109)(33 94 152 110)(34 95 153 101)(35 96 154 102)(36 97 155 103)(37 98 156 104)(38 99 157 105)(39 100 158 106)(40 91 159 107)(51 119 67 123)(52 120 68 124)(53 111 69 125)(54 112 70 126)(55 113 61 127)(56 114 62 128)(57 115 63 129)(58 116 64 130)(59 117 65 121)(60 118 66 122)
(1 79 67 96)(2 80 68 97)(3 71 69 98)(4 72 70 99)(5 73 61 100)(6 74 62 91)(7 75 63 92)(8 76 64 93)(9 77 65 94)(10 78 66 95)(11 114 40 131)(12 115 31 132)(13 116 32 133)(14 117 33 134)(15 118 34 135)(16 119 35 136)(17 120 36 137)(18 111 37 138)(19 112 38 139)(20 113 39 140)(21 128 159 147)(22 129 160 148)(23 130 151 149)(24 121 152 150)(25 122 153 141)(26 123 154 142)(27 124 155 143)(28 125 156 144)(29 126 157 145)(30 127 158 146)(41 86 54 105)(42 87 55 106)(43 88 56 107)(44 89 57 108)(45 90 58 109)(46 81 59 110)(47 82 60 101)(48 83 51 102)(49 84 52 103)(50 85 53 104)
(1 48)(2 49)(3 50)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 159)(12 160)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 144)(112 145)(113 146)(114 147)(115 148)(116 149)(117 150)(118 141)(119 142)(120 143)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 131)(129 132)(130 133)
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), (1,142,48,136)(2,143,49,137)(3,144,50,138)(4,145,41,139)(5,146,42,140)(6,147,43,131)(7,148,44,132)(8,149,45,133)(9,150,46,134)(10,141,47,135)(11,74,21,88)(12,75,22,89)(13,76,23,90)(14,77,24,81)(15,78,25,82)(16,79,26,83)(17,80,27,84)(18,71,28,85)(19,72,29,86)(20,73,30,87)(31,92,160,108)(32,93,151,109)(33,94,152,110)(34,95,153,101)(35,96,154,102)(36,97,155,103)(37,98,156,104)(38,99,157,105)(39,100,158,106)(40,91,159,107)(51,119,67,123)(52,120,68,124)(53,111,69,125)(54,112,70,126)(55,113,61,127)(56,114,62,128)(57,115,63,129)(58,116,64,130)(59,117,65,121)(60,118,66,122), (1,79,67,96)(2,80,68,97)(3,71,69,98)(4,72,70,99)(5,73,61,100)(6,74,62,91)(7,75,63,92)(8,76,64,93)(9,77,65,94)(10,78,66,95)(11,114,40,131)(12,115,31,132)(13,116,32,133)(14,117,33,134)(15,118,34,135)(16,119,35,136)(17,120,36,137)(18,111,37,138)(19,112,38,139)(20,113,39,140)(21,128,159,147)(22,129,160,148)(23,130,151,149)(24,121,152,150)(25,122,153,141)(26,123,154,142)(27,124,155,143)(28,125,156,144)(29,126,157,145)(30,127,158,146)(41,86,54,105)(42,87,55,106)(43,88,56,107)(44,89,57,108)(45,90,58,109)(46,81,59,110)(47,82,60,101)(48,83,51,102)(49,84,52,103)(50,85,53,104), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,159)(12,160)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,141)(119,142)(120,143)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,131)(129,132)(130,133)>;
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), (1,142,48,136)(2,143,49,137)(3,144,50,138)(4,145,41,139)(5,146,42,140)(6,147,43,131)(7,148,44,132)(8,149,45,133)(9,150,46,134)(10,141,47,135)(11,74,21,88)(12,75,22,89)(13,76,23,90)(14,77,24,81)(15,78,25,82)(16,79,26,83)(17,80,27,84)(18,71,28,85)(19,72,29,86)(20,73,30,87)(31,92,160,108)(32,93,151,109)(33,94,152,110)(34,95,153,101)(35,96,154,102)(36,97,155,103)(37,98,156,104)(38,99,157,105)(39,100,158,106)(40,91,159,107)(51,119,67,123)(52,120,68,124)(53,111,69,125)(54,112,70,126)(55,113,61,127)(56,114,62,128)(57,115,63,129)(58,116,64,130)(59,117,65,121)(60,118,66,122), (1,79,67,96)(2,80,68,97)(3,71,69,98)(4,72,70,99)(5,73,61,100)(6,74,62,91)(7,75,63,92)(8,76,64,93)(9,77,65,94)(10,78,66,95)(11,114,40,131)(12,115,31,132)(13,116,32,133)(14,117,33,134)(15,118,34,135)(16,119,35,136)(17,120,36,137)(18,111,37,138)(19,112,38,139)(20,113,39,140)(21,128,159,147)(22,129,160,148)(23,130,151,149)(24,121,152,150)(25,122,153,141)(26,123,154,142)(27,124,155,143)(28,125,156,144)(29,126,157,145)(30,127,158,146)(41,86,54,105)(42,87,55,106)(43,88,56,107)(44,89,57,108)(45,90,58,109)(46,81,59,110)(47,82,60,101)(48,83,51,102)(49,84,52,103)(50,85,53,104), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,159)(12,160)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,141)(119,142)(120,143)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,131)(129,132)(130,133) );
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)], [(1,142,48,136),(2,143,49,137),(3,144,50,138),(4,145,41,139),(5,146,42,140),(6,147,43,131),(7,148,44,132),(8,149,45,133),(9,150,46,134),(10,141,47,135),(11,74,21,88),(12,75,22,89),(13,76,23,90),(14,77,24,81),(15,78,25,82),(16,79,26,83),(17,80,27,84),(18,71,28,85),(19,72,29,86),(20,73,30,87),(31,92,160,108),(32,93,151,109),(33,94,152,110),(34,95,153,101),(35,96,154,102),(36,97,155,103),(37,98,156,104),(38,99,157,105),(39,100,158,106),(40,91,159,107),(51,119,67,123),(52,120,68,124),(53,111,69,125),(54,112,70,126),(55,113,61,127),(56,114,62,128),(57,115,63,129),(58,116,64,130),(59,117,65,121),(60,118,66,122)], [(1,79,67,96),(2,80,68,97),(3,71,69,98),(4,72,70,99),(5,73,61,100),(6,74,62,91),(7,75,63,92),(8,76,64,93),(9,77,65,94),(10,78,66,95),(11,114,40,131),(12,115,31,132),(13,116,32,133),(14,117,33,134),(15,118,34,135),(16,119,35,136),(17,120,36,137),(18,111,37,138),(19,112,38,139),(20,113,39,140),(21,128,159,147),(22,129,160,148),(23,130,151,149),(24,121,152,150),(25,122,153,141),(26,123,154,142),(27,124,155,143),(28,125,156,144),(29,126,157,145),(30,127,158,146),(41,86,54,105),(42,87,55,106),(43,88,56,107),(44,89,57,108),(45,90,58,109),(46,81,59,110),(47,82,60,101),(48,83,51,102),(49,84,52,103),(50,85,53,104)], [(1,48),(2,49),(3,50),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,159),(12,160),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,144),(112,145),(113,146),(114,147),(115,148),(116,149),(117,150),(118,141),(119,142),(120,143),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,131),(129,132),(130,133)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20DH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C10×C42⋊C2 | C2×C4×C20 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C42⋊C2 | C23×C20 | C22×C20 | C2×C42⋊C2 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C23×C4 | C22×C4 | C2×C10 | C22 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 16 | 4 | 8 | 8 | 8 | 32 | 4 | 64 | 8 | 32 |
Matrix representation of C10×C42⋊C2 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 32 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,16,0,0,0,0,16],[32,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C10×C42⋊C2 in GAP, Magma, Sage, TeX
C_{10}\times C_4^2\rtimes C_2 % in TeX
G:=Group("C10xC4^2:C2"); // GroupNames label
G:=SmallGroup(320,1516);
// by ID
G=gap.SmallGroup(320,1516);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,436]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations