direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C16, C80⋊11C22, C40.63C23, (C2×C80)⋊14C2, C10⋊3(C2×C16), C5⋊3(C22×C16), C8.43(C4×D5), C4.23(C8×D5), C20.62(C2×C8), (C4×D5).10C8, (C8×D5).14C4, C40.101(C2×C4), D10.21(C2×C8), (C2×C8).341D10, C22.13(C8×D5), C8.57(C22×D5), C5⋊2C16⋊13C22, C10.36(C22×C8), (C2×Dic5).16C8, Dic5.22(C2×C8), (C22×D5).10C8, (C8×D5).67C22, (C2×C40).408C22, C20.187(C22×C4), C2.2(D5×C2×C8), (D5×C2×C8).36C2, (C2×C4×D5).49C4, C4.102(C2×C4×D5), (C2×C5⋊2C16)⋊17C2, (C2×C5⋊2C8).39C4, (C2×C10).42(C2×C8), C5⋊2C8.57(C2×C4), (C2×C4).175(C4×D5), (C2×C20).421(C2×C4), (C4×D5).100(C2×C4), SmallGroup(320,526)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C16 |
Generators and relations for D5×C2×C16
G = < a,b,c,d | a2=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 238 in 98 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C2×C16, C2×C16, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C16, C5⋊2C16, C80, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, D5×C16, C2×C5⋊2C16, C2×C80, D5×C2×C8, D5×C2×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C16, C2×C8, C22×C4, D10, C2×C16, C22×C8, C4×D5, C22×D5, C22×C16, C8×D5, C2×C4×D5, D5×C16, D5×C2×C8, D5×C2×C16
►On 160 points
Generators in S160
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 81)(14 82)(15 83)(16 84)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 145)(47 146)(48 147)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 97)
(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 54 26 152 75)(2 55 27 153 76)(3 56 28 154 77)(4 57 29 155 78)(5 58 30 156 79)(6 59 31 157 80)(7 60 32 158 65)(8 61 17 159 66)(9 62 18 160 67)(10 63 19 145 68)(11 64 20 146 69)(12 49 21 147 70)(13 50 22 148 71)(14 51 23 149 72)(15 52 24 150 73)(16 53 25 151 74)(33 104 81 140 118)(34 105 82 141 119)(35 106 83 142 120)(36 107 84 143 121)(37 108 85 144 122)(38 109 86 129 123)(39 110 87 130 124)(40 111 88 131 125)(41 112 89 132 126)(42 97 90 133 127)(43 98 91 134 128)(44 99 92 135 113)(45 100 93 136 114)(46 101 94 137 115)(47 102 95 138 116)(48 103 96 139 117)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 65)(16 66)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 129)(47 130)(48 131)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 154)(81 112)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
G:=sub<Sym(160)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,145)(47,146)(48,147)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,97), (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), (1,54,26,152,75)(2,55,27,153,76)(3,56,28,154,77)(4,57,29,155,78)(5,58,30,156,79)(6,59,31,157,80)(7,60,32,158,65)(8,61,17,159,66)(9,62,18,160,67)(10,63,19,145,68)(11,64,20,146,69)(12,49,21,147,70)(13,50,22,148,71)(14,51,23,149,72)(15,52,24,150,73)(16,53,25,151,74)(33,104,81,140,118)(34,105,82,141,119)(35,106,83,142,120)(36,107,84,143,121)(37,108,85,144,122)(38,109,86,129,123)(39,110,87,130,124)(40,111,88,131,125)(41,112,89,132,126)(42,97,90,133,127)(43,98,91,134,128)(44,99,92,135,113)(45,100,93,136,114)(46,101,94,137,115)(47,102,95,138,116)(48,103,96,139,117), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,129)(47,130)(48,131)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(81,112)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,145)(47,146)(48,147)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,97), (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), (1,54,26,152,75)(2,55,27,153,76)(3,56,28,154,77)(4,57,29,155,78)(5,58,30,156,79)(6,59,31,157,80)(7,60,32,158,65)(8,61,17,159,66)(9,62,18,160,67)(10,63,19,145,68)(11,64,20,146,69)(12,49,21,147,70)(13,50,22,148,71)(14,51,23,149,72)(15,52,24,150,73)(16,53,25,151,74)(33,104,81,140,118)(34,105,82,141,119)(35,106,83,142,120)(36,107,84,143,121)(37,108,85,144,122)(38,109,86,129,123)(39,110,87,130,124)(40,111,88,131,125)(41,112,89,132,126)(42,97,90,133,127)(43,98,91,134,128)(44,99,92,135,113)(45,100,93,136,114)(46,101,94,137,115)(47,102,95,138,116)(48,103,96,139,117), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,129)(47,130)(48,131)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(81,112)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128) );
G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,81),(14,82),(15,83),(16,84),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,156),(42,157),(43,158),(44,159),(45,160),(46,145),(47,146),(48,147),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,97)], [(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)], [(1,54,26,152,75),(2,55,27,153,76),(3,56,28,154,77),(4,57,29,155,78),(5,58,30,156,79),(6,59,31,157,80),(7,60,32,158,65),(8,61,17,159,66),(9,62,18,160,67),(10,63,19,145,68),(11,64,20,146,69),(12,49,21,147,70),(13,50,22,148,71),(14,51,23,149,72),(15,52,24,150,73),(16,53,25,151,74),(33,104,81,140,118),(34,105,82,141,119),(35,106,83,142,120),(36,107,84,143,121),(37,108,85,144,122),(38,109,86,129,123),(39,110,87,130,124),(40,111,88,131,125),(41,112,89,132,126),(42,97,90,133,127),(43,98,91,134,128),(44,99,92,135,113),(45,100,93,136,114),(46,101,94,137,115),(47,102,95,138,116),(48,103,96,139,117)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,65),(16,66),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,132),(34,133),(35,134),(36,135),(37,136),(38,137),(39,138),(40,139),(41,140),(42,141),(43,142),(44,143),(45,144),(46,129),(47,130),(48,131),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,151),(62,152),(63,153),(64,154),(81,112),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 16A | ··· | 16P | 16Q | ··· | 16AF | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | D5 | D10 | D10 | C4×D5 | C4×D5 | C8×D5 | C8×D5 | D5×C16 |
| kernel | D5×C2×C16 | D5×C16 | C2×C5⋊2C16 | C2×C80 | D5×C2×C8 | C8×D5 | C2×C5⋊2C8 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 32 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of D5×C2×C16 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 130 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 189 | 1 |
| 0 | 240 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 189 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[130,0,0,0,8,0,0,0,8],[1,0,0,0,189,240,0,1,0],[1,0,0,0,1,0,0,189,240] >;
D5×C2×C16 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_{16} % in TeX
G:=Group("D5xC2xC16"); // GroupNames label
G:=SmallGroup(320,526);
// by ID
G=gap.SmallGroup(320,526);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,58,80,102,12550]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations