direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×D5⋊C16, D10⋊3C16, D5⋊(C2×C16), C10⋊1(C2×C16), C5⋊C16⋊4C22, C5⋊1(C22×C16), (C4×D5).7C8, C8.41(C2×F5), (C2×C8).24F5, C20.23(C2×C8), C40.38(C2×C4), (C2×C40).22C4, (C8×D5).10C4, C4.18(D5⋊C8), D10.16(C2×C8), C10.8(C22×C8), (C22×D5).7C8, C4.45(C22×F5), C20.85(C22×C4), C5⋊2C8.35C23, (C2×Dic5).12C8, Dic5.16(C2×C8), (C8×D5).61C22, C22.12(D5⋊C8), (C2×C5⋊C16)⋊9C2, C2.2(C2×D5⋊C8), (D5×C2×C8).29C2, (C2×C4×D5).45C4, (C2×C10).10(C2×C8), C5⋊2C8.51(C2×C4), (C4×D5).91(C2×C4), (C2×C4).163(C2×F5), (C2×C20).172(C2×C4), (C2×C5⋊2C8).348C22, SmallGroup(320,1051)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×D5⋊C16 |
Generators and relations for C2×D5⋊C16
G = < a,b,c,d | a2=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >
Subgroups: 250 in 98 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C2×C16, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C16, C5⋊C16, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, D5⋊C16, C2×C5⋊C16, D5×C2×C8, C2×D5⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C2×C8, C22×C4, F5, C2×C16, C22×C8, C2×F5, C22×C16, D5⋊C8, C22×F5, D5⋊C16, C2×D5⋊C8, C2×D5⋊C16
►On 160 points
Generators in S160
(1 112)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 80)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 129)(46 130)(47 131)(48 132)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 113)(64 114)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)
(1 84 131 115 65)(2 116 85 66 132)(3 67 117 133 86)(4 134 68 87 118)(5 88 135 119 69)(6 120 89 70 136)(7 71 121 137 90)(8 138 72 91 122)(9 92 139 123 73)(10 124 93 74 140)(11 75 125 141 94)(12 142 76 95 126)(13 96 143 127 77)(14 128 81 78 144)(15 79 113 129 82)(16 130 80 83 114)(17 147 64 111 46)(18 112 148 47 49)(19 48 97 50 149)(20 51 33 150 98)(21 151 52 99 34)(22 100 152 35 53)(23 36 101 54 153)(24 55 37 154 102)(25 155 56 103 38)(26 104 156 39 57)(27 40 105 58 157)(28 59 41 158 106)(29 159 60 107 42)(30 108 160 43 61)(31 44 109 62 145)(32 63 45 146 110)
(1 65)(2 132)(3 86)(4 118)(5 69)(6 136)(7 90)(8 122)(9 73)(10 140)(11 94)(12 126)(13 77)(14 144)(15 82)(16 114)(18 112)(19 50)(20 33)(22 100)(23 54)(24 37)(26 104)(27 58)(28 41)(30 108)(31 62)(32 45)(34 151)(36 101)(38 155)(40 105)(42 159)(44 109)(46 147)(48 97)(49 148)(52 99)(53 152)(56 103)(57 156)(60 107)(61 160)(64 111)(66 116)(67 133)(70 120)(71 137)(74 124)(75 141)(78 128)(79 129)(83 130)(84 115)(87 134)(88 119)(91 138)(92 123)(95 142)(96 127)(98 150)(102 154)(106 158)(110 146)
(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)
G:=sub<Sym(160)| (1,112)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,80)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,129)(46,130)(47,131)(48,132)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,113)(64,114)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160), (1,84,131,115,65)(2,116,85,66,132)(3,67,117,133,86)(4,134,68,87,118)(5,88,135,119,69)(6,120,89,70,136)(7,71,121,137,90)(8,138,72,91,122)(9,92,139,123,73)(10,124,93,74,140)(11,75,125,141,94)(12,142,76,95,126)(13,96,143,127,77)(14,128,81,78,144)(15,79,113,129,82)(16,130,80,83,114)(17,147,64,111,46)(18,112,148,47,49)(19,48,97,50,149)(20,51,33,150,98)(21,151,52,99,34)(22,100,152,35,53)(23,36,101,54,153)(24,55,37,154,102)(25,155,56,103,38)(26,104,156,39,57)(27,40,105,58,157)(28,59,41,158,106)(29,159,60,107,42)(30,108,160,43,61)(31,44,109,62,145)(32,63,45,146,110), (1,65)(2,132)(3,86)(4,118)(5,69)(6,136)(7,90)(8,122)(9,73)(10,140)(11,94)(12,126)(13,77)(14,144)(15,82)(16,114)(18,112)(19,50)(20,33)(22,100)(23,54)(24,37)(26,104)(27,58)(28,41)(30,108)(31,62)(32,45)(34,151)(36,101)(38,155)(40,105)(42,159)(44,109)(46,147)(48,97)(49,148)(52,99)(53,152)(56,103)(57,156)(60,107)(61,160)(64,111)(66,116)(67,133)(70,120)(71,137)(74,124)(75,141)(78,128)(79,129)(83,130)(84,115)(87,134)(88,119)(91,138)(92,123)(95,142)(96,127)(98,150)(102,154)(106,158)(110,146), (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)>;
G:=Group( (1,112)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,80)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,129)(46,130)(47,131)(48,132)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,113)(64,114)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160), (1,84,131,115,65)(2,116,85,66,132)(3,67,117,133,86)(4,134,68,87,118)(5,88,135,119,69)(6,120,89,70,136)(7,71,121,137,90)(8,138,72,91,122)(9,92,139,123,73)(10,124,93,74,140)(11,75,125,141,94)(12,142,76,95,126)(13,96,143,127,77)(14,128,81,78,144)(15,79,113,129,82)(16,130,80,83,114)(17,147,64,111,46)(18,112,148,47,49)(19,48,97,50,149)(20,51,33,150,98)(21,151,52,99,34)(22,100,152,35,53)(23,36,101,54,153)(24,55,37,154,102)(25,155,56,103,38)(26,104,156,39,57)(27,40,105,58,157)(28,59,41,158,106)(29,159,60,107,42)(30,108,160,43,61)(31,44,109,62,145)(32,63,45,146,110), (1,65)(2,132)(3,86)(4,118)(5,69)(6,136)(7,90)(8,122)(9,73)(10,140)(11,94)(12,126)(13,77)(14,144)(15,82)(16,114)(18,112)(19,50)(20,33)(22,100)(23,54)(24,37)(26,104)(27,58)(28,41)(30,108)(31,62)(32,45)(34,151)(36,101)(38,155)(40,105)(42,159)(44,109)(46,147)(48,97)(49,148)(52,99)(53,152)(56,103)(57,156)(60,107)(61,160)(64,111)(66,116)(67,133)(70,120)(71,137)(74,124)(75,141)(78,128)(79,129)(83,130)(84,115)(87,134)(88,119)(91,138)(92,123)(95,142)(96,127)(98,150)(102,154)(106,158)(110,146), (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) );
G=PermutationGroup([[(1,112),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,80),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,129),(46,130),(47,131),(48,132),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,113),(64,114),(81,145),(82,146),(83,147),(84,148),(85,149),(86,150),(87,151),(88,152),(89,153),(90,154),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160)], [(1,84,131,115,65),(2,116,85,66,132),(3,67,117,133,86),(4,134,68,87,118),(5,88,135,119,69),(6,120,89,70,136),(7,71,121,137,90),(8,138,72,91,122),(9,92,139,123,73),(10,124,93,74,140),(11,75,125,141,94),(12,142,76,95,126),(13,96,143,127,77),(14,128,81,78,144),(15,79,113,129,82),(16,130,80,83,114),(17,147,64,111,46),(18,112,148,47,49),(19,48,97,50,149),(20,51,33,150,98),(21,151,52,99,34),(22,100,152,35,53),(23,36,101,54,153),(24,55,37,154,102),(25,155,56,103,38),(26,104,156,39,57),(27,40,105,58,157),(28,59,41,158,106),(29,159,60,107,42),(30,108,160,43,61),(31,44,109,62,145),(32,63,45,146,110)], [(1,65),(2,132),(3,86),(4,118),(5,69),(6,136),(7,90),(8,122),(9,73),(10,140),(11,94),(12,126),(13,77),(14,144),(15,82),(16,114),(18,112),(19,50),(20,33),(22,100),(23,54),(24,37),(26,104),(27,58),(28,41),(30,108),(31,62),(32,45),(34,151),(36,101),(38,155),(40,105),(42,159),(44,109),(46,147),(48,97),(49,148),(52,99),(53,152),(56,103),(57,156),(60,107),(61,160),(64,111),(66,116),(67,133),(70,120),(71,137),(74,124),(75,141),(78,128),(79,129),(83,130),(84,115),(87,134),(88,119),(91,138),(92,123),(95,142),(96,127),(98,150),(102,154),(106,158),(110,146)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 16A | ··· | 16AF | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | D5⋊C16 |
| kernel | C2×D5⋊C16 | D5⋊C16 | C2×C5⋊C16 | D5×C2×C8 | C8×D5 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 32 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×D5⋊C16 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 240 | 240 | 240 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 154 | 0 | 76 | 76 |
| 0 | 76 | 76 | 0 | 154 |
| 0 | 165 | 78 | 165 | 0 |
| 0 | 87 | 163 | 163 | 87 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,240,0,0,0],[240,0,0,0,0,0,240,0,0,0,0,240,0,0,1,0,240,0,1,0,0,240,1,0,0],[240,0,0,0,0,0,154,76,165,87,0,0,76,78,163,0,76,0,165,163,0,76,154,0,87] >;
C2×D5⋊C16 in GAP, Magma, Sage, TeX
C_2\times D_5\rtimes C_{16} % in TeX
G:=Group("C2xD5:C16"); // GroupNames label
G:=SmallGroup(320,1051);
// by ID
G=gap.SmallGroup(320,1051);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations