direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C16⋊D5, C16⋊8D10, C4.7D40, C80⋊9C22, C10⋊1SD32, C20.32D8, C40.61D4, C8.11D20, C40.56C23, D40.6C22, C22.13D40, Dic20⋊6C22, (C2×C16)⋊7D5, C5⋊1(C2×SD32), (C2×C80)⋊11C2, (C2×D40).5C2, C2.12(C2×D40), C4.37(C2×D20), (C2×C4).84D20, C10.10(C2×D8), (C2×C10).19D8, (C2×Dic20)⋊8C2, (C2×C8).304D10, C20.280(C2×D4), (C2×C20).381D4, C8.46(C22×D5), (C2×C40).377C22, SmallGroup(320,530)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C16⋊D5
G = < a,b,c,d | a2=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b7, dcd=c-1 >
Subgroups: 574 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, Q16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C2×SD32, C80, D40, D40, Dic20, Dic20, C2×C40, C2×Dic10, C2×D20, C16⋊D5, C2×C80, C2×D40, C2×Dic20, C2×C16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, D20, C22×D5, C2×SD32, D40, C2×D20, C16⋊D5, C2×D40, C2×C16⋊D5
►On 160 points
Generators in S160
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 145)(13 146)(14 147)(15 148)(16 149)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 65)(31 66)(32 67)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(97 140)(98 141)(99 142)(100 143)(101 144)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)
(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 33 62 103 20)(2 34 63 104 21)(3 35 64 105 22)(4 36 49 106 23)(5 37 50 107 24)(6 38 51 108 25)(7 39 52 109 26)(8 40 53 110 27)(9 41 54 111 28)(10 42 55 112 29)(11 43 56 97 30)(12 44 57 98 31)(13 45 58 99 32)(14 46 59 100 17)(15 47 60 101 18)(16 48 61 102 19)(65 160 82 114 140)(66 145 83 115 141)(67 146 84 116 142)(68 147 85 117 143)(69 148 86 118 144)(70 149 87 119 129)(71 150 88 120 130)(72 151 89 121 131)(73 152 90 122 132)(74 153 91 123 133)(75 154 92 124 134)(76 155 93 125 135)(77 156 94 126 136)(78 157 95 127 137)(79 158 96 128 138)(80 159 81 113 139)
(1 71)(2 78)(3 69)(4 76)(5 67)(6 74)(7 65)(8 72)(9 79)(10 70)(11 77)(12 68)(13 75)(14 66)(15 73)(16 80)(17 145)(18 152)(19 159)(20 150)(21 157)(22 148)(23 155)(24 146)(25 153)(26 160)(27 151)(28 158)(29 149)(30 156)(31 147)(32 154)(33 130)(34 137)(35 144)(36 135)(37 142)(38 133)(39 140)(40 131)(41 138)(42 129)(43 136)(44 143)(45 134)(46 141)(47 132)(48 139)(49 125)(50 116)(51 123)(52 114)(53 121)(54 128)(55 119)(56 126)(57 117)(58 124)(59 115)(60 122)(61 113)(62 120)(63 127)(64 118)(81 102)(82 109)(83 100)(84 107)(85 98)(86 105)(87 112)(88 103)(89 110)(90 101)(91 108)(92 99)(93 106)(94 97)(95 104)(96 111)
G:=sub<Sym(160)| (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,145)(13,146)(14,147)(15,148)(16,149)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,65)(31,66)(32,67)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(97,140)(98,141)(99,142)(100,143)(101,144)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139), (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), (1,33,62,103,20)(2,34,63,104,21)(3,35,64,105,22)(4,36,49,106,23)(5,37,50,107,24)(6,38,51,108,25)(7,39,52,109,26)(8,40,53,110,27)(9,41,54,111,28)(10,42,55,112,29)(11,43,56,97,30)(12,44,57,98,31)(13,45,58,99,32)(14,46,59,100,17)(15,47,60,101,18)(16,48,61,102,19)(65,160,82,114,140)(66,145,83,115,141)(67,146,84,116,142)(68,147,85,117,143)(69,148,86,118,144)(70,149,87,119,129)(71,150,88,120,130)(72,151,89,121,131)(73,152,90,122,132)(74,153,91,123,133)(75,154,92,124,134)(76,155,93,125,135)(77,156,94,126,136)(78,157,95,127,137)(79,158,96,128,138)(80,159,81,113,139), (1,71)(2,78)(3,69)(4,76)(5,67)(6,74)(7,65)(8,72)(9,79)(10,70)(11,77)(12,68)(13,75)(14,66)(15,73)(16,80)(17,145)(18,152)(19,159)(20,150)(21,157)(22,148)(23,155)(24,146)(25,153)(26,160)(27,151)(28,158)(29,149)(30,156)(31,147)(32,154)(33,130)(34,137)(35,144)(36,135)(37,142)(38,133)(39,140)(40,131)(41,138)(42,129)(43,136)(44,143)(45,134)(46,141)(47,132)(48,139)(49,125)(50,116)(51,123)(52,114)(53,121)(54,128)(55,119)(56,126)(57,117)(58,124)(59,115)(60,122)(61,113)(62,120)(63,127)(64,118)(81,102)(82,109)(83,100)(84,107)(85,98)(86,105)(87,112)(88,103)(89,110)(90,101)(91,108)(92,99)(93,106)(94,97)(95,104)(96,111)>;
G:=Group( (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,145)(13,146)(14,147)(15,148)(16,149)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,65)(31,66)(32,67)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(97,140)(98,141)(99,142)(100,143)(101,144)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139), (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), (1,33,62,103,20)(2,34,63,104,21)(3,35,64,105,22)(4,36,49,106,23)(5,37,50,107,24)(6,38,51,108,25)(7,39,52,109,26)(8,40,53,110,27)(9,41,54,111,28)(10,42,55,112,29)(11,43,56,97,30)(12,44,57,98,31)(13,45,58,99,32)(14,46,59,100,17)(15,47,60,101,18)(16,48,61,102,19)(65,160,82,114,140)(66,145,83,115,141)(67,146,84,116,142)(68,147,85,117,143)(69,148,86,118,144)(70,149,87,119,129)(71,150,88,120,130)(72,151,89,121,131)(73,152,90,122,132)(74,153,91,123,133)(75,154,92,124,134)(76,155,93,125,135)(77,156,94,126,136)(78,157,95,127,137)(79,158,96,128,138)(80,159,81,113,139), (1,71)(2,78)(3,69)(4,76)(5,67)(6,74)(7,65)(8,72)(9,79)(10,70)(11,77)(12,68)(13,75)(14,66)(15,73)(16,80)(17,145)(18,152)(19,159)(20,150)(21,157)(22,148)(23,155)(24,146)(25,153)(26,160)(27,151)(28,158)(29,149)(30,156)(31,147)(32,154)(33,130)(34,137)(35,144)(36,135)(37,142)(38,133)(39,140)(40,131)(41,138)(42,129)(43,136)(44,143)(45,134)(46,141)(47,132)(48,139)(49,125)(50,116)(51,123)(52,114)(53,121)(54,128)(55,119)(56,126)(57,117)(58,124)(59,115)(60,122)(61,113)(62,120)(63,127)(64,118)(81,102)(82,109)(83,100)(84,107)(85,98)(86,105)(87,112)(88,103)(89,110)(90,101)(91,108)(92,99)(93,106)(94,97)(95,104)(96,111) );
G=PermutationGroup([[(1,150),(2,151),(3,152),(4,153),(5,154),(6,155),(7,156),(8,157),(9,158),(10,159),(11,160),(12,145),(13,146),(14,147),(15,148),(16,149),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,65),(31,66),(32,67),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,121),(64,122),(97,140),(98,141),(99,142),(100,143),(101,144),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139)], [(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)], [(1,33,62,103,20),(2,34,63,104,21),(3,35,64,105,22),(4,36,49,106,23),(5,37,50,107,24),(6,38,51,108,25),(7,39,52,109,26),(8,40,53,110,27),(9,41,54,111,28),(10,42,55,112,29),(11,43,56,97,30),(12,44,57,98,31),(13,45,58,99,32),(14,46,59,100,17),(15,47,60,101,18),(16,48,61,102,19),(65,160,82,114,140),(66,145,83,115,141),(67,146,84,116,142),(68,147,85,117,143),(69,148,86,118,144),(70,149,87,119,129),(71,150,88,120,130),(72,151,89,121,131),(73,152,90,122,132),(74,153,91,123,133),(75,154,92,124,134),(76,155,93,125,135),(77,156,94,126,136),(78,157,95,127,137),(79,158,96,128,138),(80,159,81,113,139)], [(1,71),(2,78),(3,69),(4,76),(5,67),(6,74),(7,65),(8,72),(9,79),(10,70),(11,77),(12,68),(13,75),(14,66),(15,73),(16,80),(17,145),(18,152),(19,159),(20,150),(21,157),(22,148),(23,155),(24,146),(25,153),(26,160),(27,151),(28,158),(29,149),(30,156),(31,147),(32,154),(33,130),(34,137),(35,144),(36,135),(37,142),(38,133),(39,140),(40,131),(41,138),(42,129),(43,136),(44,143),(45,134),(46,141),(47,132),(48,139),(49,125),(50,116),(51,123),(52,114),(53,121),(54,128),(55,119),(56,126),(57,117),(58,124),(59,115),(60,122),(61,113),(62,120),(63,127),(64,118),(81,102),(82,109),(83,100),(84,107),(85,98),(86,105),(87,112),(88,103),(89,110),(90,101),(91,108),(92,99),(93,106),(94,97),(95,104),(96,111)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | SD32 | D20 | D20 | D40 | D40 | C16⋊D5 |
| kernel | C2×C16⋊D5 | C16⋊D5 | C2×C80 | C2×D40 | C2×Dic20 | C40 | C2×C20 | C2×C16 | C20 | C2×C10 | C16 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C2×C16⋊D5 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 240 | 0 | 0 |
| 0 | 220 | 130 |
| 0 | 234 | 227 |
| 1 | 0 | 0 |
| 0 | 190 | 240 |
| 0 | 191 | 240 |
| 1 | 0 | 0 |
| 0 | 0 | 189 |
| 0 | 190 | 0 |
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[240,0,0,0,220,234,0,130,227],[1,0,0,0,190,191,0,240,240],[1,0,0,0,0,190,0,189,0] >;
C2×C16⋊D5 in GAP, Magma, Sage, TeX
C_2\times C_{16}\rtimes D_5 % in TeX
G:=Group("C2xC16:D5"); // GroupNames label
G:=SmallGroup(320,530);
// by ID
G=gap.SmallGroup(320,530);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,142,1571,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^16=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^7,d*c*d=c^-1>;
// generators/relations