direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C2.D16, D8⋊1C20, C40.94D4, C10.13D16, C10.9SD32, C20.30SD16, (C2×C80)⋊7C2, (C2×C16)⋊3C10, (C5×D8)⋊13C4, C8.7(C2×C20), C2.D8⋊1C10, C2.1(C5×D16), C8.14(C5×D4), (C10×D8).7C2, (C2×D8).1C10, (C2×C10).49D8, C4.1(C5×SD16), C2.1(C5×SD32), C22.8(C5×D8), C40.104(C2×C4), (C2×C20).405D4, (C2×C40).416C22, C10.52(D4⋊C4), C20.116(C22⋊C4), (C5×C2.D8)⋊10C2, (C2×C4).59(C5×D4), C4.1(C5×C22⋊C4), (C2×C8).71(C2×C10), C2.6(C5×D4⋊C4), SmallGroup(320,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C2.D16
G = < a,b,c,d | a5=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 178 in 66 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, C20, C20, C2×C10, C2×C10, C2.D8, C2×C16, C2×D8, C40, C2×C20, C2×C20, C5×D4, C22×C10, C2.D16, C80, C5×C4⋊C4, C2×C40, C5×D8, C5×D8, D4×C10, C5×C2.D8, C2×C80, C10×D8, C5×C2.D16
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, D16, SD32, C2×C20, C5×D4, C2.D16, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×D16, C5×SD32, C5×C2.D16
►On 160 points
Generators in S160
(1 110 96 127 41)(2 111 81 128 42)(3 112 82 113 43)(4 97 83 114 44)(5 98 84 115 45)(6 99 85 116 46)(7 100 86 117 47)(8 101 87 118 48)(9 102 88 119 33)(10 103 89 120 34)(11 104 90 121 35)(12 105 91 122 36)(13 106 92 123 37)(14 107 93 124 38)(15 108 94 125 39)(16 109 95 126 40)(17 74 61 151 141)(18 75 62 152 142)(19 76 63 153 143)(20 77 64 154 144)(21 78 49 155 129)(22 79 50 156 130)(23 80 51 157 131)(24 65 52 158 132)(25 66 53 159 133)(26 67 54 160 134)(27 68 55 145 135)(28 69 56 146 136)(29 70 57 147 137)(30 71 58 148 138)(31 72 59 149 139)(32 73 60 150 140)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 97)(64 98)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(113 142)(114 143)(115 144)(116 129)(117 130)(118 131)(119 132)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 141)
(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 16 73 72)(2 71 74 15)(3 14 75 70)(4 69 76 13)(5 12 77 68)(6 67 78 11)(7 10 79 66)(8 65 80 9)(17 39 42 30)(18 29 43 38)(19 37 44 28)(20 27 45 36)(21 35 46 26)(22 25 47 34)(23 33 48 24)(31 41 40 32)(49 104 99 54)(50 53 100 103)(51 102 101 52)(55 98 105 64)(56 63 106 97)(57 112 107 62)(58 61 108 111)(59 110 109 60)(81 148 151 94)(82 93 152 147)(83 146 153 92)(84 91 154 145)(85 160 155 90)(86 89 156 159)(87 158 157 88)(95 150 149 96)(113 124 142 137)(114 136 143 123)(115 122 144 135)(116 134 129 121)(117 120 130 133)(118 132 131 119)(125 128 138 141)(126 140 139 127)
G:=sub<Sym(160)| (1,110,96,127,41)(2,111,81,128,42)(3,112,82,113,43)(4,97,83,114,44)(5,98,84,115,45)(6,99,85,116,46)(7,100,86,117,47)(8,101,87,118,48)(9,102,88,119,33)(10,103,89,120,34)(11,104,90,121,35)(12,105,91,122,36)(13,106,92,123,37)(14,107,93,124,38)(15,108,94,125,39)(16,109,95,126,40)(17,74,61,151,141)(18,75,62,152,142)(19,76,63,153,143)(20,77,64,154,144)(21,78,49,155,129)(22,79,50,156,130)(23,80,51,157,131)(24,65,52,158,132)(25,66,53,159,133)(26,67,54,160,134)(27,68,55,145,135)(28,69,56,146,136)(29,70,57,147,137)(30,71,58,148,138)(31,72,59,149,139)(32,73,60,150,140), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,97)(64,98)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(113,142)(114,143)(115,144)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141), (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), (1,16,73,72)(2,71,74,15)(3,14,75,70)(4,69,76,13)(5,12,77,68)(6,67,78,11)(7,10,79,66)(8,65,80,9)(17,39,42,30)(18,29,43,38)(19,37,44,28)(20,27,45,36)(21,35,46,26)(22,25,47,34)(23,33,48,24)(31,41,40,32)(49,104,99,54)(50,53,100,103)(51,102,101,52)(55,98,105,64)(56,63,106,97)(57,112,107,62)(58,61,108,111)(59,110,109,60)(81,148,151,94)(82,93,152,147)(83,146,153,92)(84,91,154,145)(85,160,155,90)(86,89,156,159)(87,158,157,88)(95,150,149,96)(113,124,142,137)(114,136,143,123)(115,122,144,135)(116,134,129,121)(117,120,130,133)(118,132,131,119)(125,128,138,141)(126,140,139,127)>;
G:=Group( (1,110,96,127,41)(2,111,81,128,42)(3,112,82,113,43)(4,97,83,114,44)(5,98,84,115,45)(6,99,85,116,46)(7,100,86,117,47)(8,101,87,118,48)(9,102,88,119,33)(10,103,89,120,34)(11,104,90,121,35)(12,105,91,122,36)(13,106,92,123,37)(14,107,93,124,38)(15,108,94,125,39)(16,109,95,126,40)(17,74,61,151,141)(18,75,62,152,142)(19,76,63,153,143)(20,77,64,154,144)(21,78,49,155,129)(22,79,50,156,130)(23,80,51,157,131)(24,65,52,158,132)(25,66,53,159,133)(26,67,54,160,134)(27,68,55,145,135)(28,69,56,146,136)(29,70,57,147,137)(30,71,58,148,138)(31,72,59,149,139)(32,73,60,150,140), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,97)(64,98)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(113,142)(114,143)(115,144)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141), (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), (1,16,73,72)(2,71,74,15)(3,14,75,70)(4,69,76,13)(5,12,77,68)(6,67,78,11)(7,10,79,66)(8,65,80,9)(17,39,42,30)(18,29,43,38)(19,37,44,28)(20,27,45,36)(21,35,46,26)(22,25,47,34)(23,33,48,24)(31,41,40,32)(49,104,99,54)(50,53,100,103)(51,102,101,52)(55,98,105,64)(56,63,106,97)(57,112,107,62)(58,61,108,111)(59,110,109,60)(81,148,151,94)(82,93,152,147)(83,146,153,92)(84,91,154,145)(85,160,155,90)(86,89,156,159)(87,158,157,88)(95,150,149,96)(113,124,142,137)(114,136,143,123)(115,122,144,135)(116,134,129,121)(117,120,130,133)(118,132,131,119)(125,128,138,141)(126,140,139,127) );
G=PermutationGroup([[(1,110,96,127,41),(2,111,81,128,42),(3,112,82,113,43),(4,97,83,114,44),(5,98,84,115,45),(6,99,85,116,46),(7,100,86,117,47),(8,101,87,118,48),(9,102,88,119,33),(10,103,89,120,34),(11,104,90,121,35),(12,105,91,122,36),(13,106,92,123,37),(14,107,93,124,38),(15,108,94,125,39),(16,109,95,126,40),(17,74,61,151,141),(18,75,62,152,142),(19,76,63,153,143),(20,77,64,154,144),(21,78,49,155,129),(22,79,50,156,130),(23,80,51,157,131),(24,65,52,158,132),(25,66,53,159,133),(26,67,54,160,134),(27,68,55,145,135),(28,69,56,146,136),(29,70,57,147,137),(30,71,58,148,138),(31,72,59,149,139),(32,73,60,150,140)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,41),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,97),(64,98),(81,151),(82,152),(83,153),(84,154),(85,155),(86,156),(87,157),(88,158),(89,159),(90,160),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(113,142),(114,143),(115,144),(116,129),(117,130),(118,131),(119,132),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,141)], [(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)], [(1,16,73,72),(2,71,74,15),(3,14,75,70),(4,69,76,13),(5,12,77,68),(6,67,78,11),(7,10,79,66),(8,65,80,9),(17,39,42,30),(18,29,43,38),(19,37,44,28),(20,27,45,36),(21,35,46,26),(22,25,47,34),(23,33,48,24),(31,41,40,32),(49,104,99,54),(50,53,100,103),(51,102,101,52),(55,98,105,64),(56,63,106,97),(57,112,107,62),(58,61,108,111),(59,110,109,60),(81,148,151,94),(82,93,152,147),(83,146,153,92),(84,91,154,145),(85,160,155,90),(86,89,156,159),(87,158,157,88),(95,150,149,96),(113,124,142,137),(114,136,143,123),(115,122,144,135),(116,134,129,121),(117,120,130,133),(118,132,131,119),(125,128,138,141),(126,140,139,127)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | D4 | SD16 | D8 | D16 | SD32 | C5×D4 | C5×D4 | C5×SD16 | C5×D8 | C5×D16 | C5×SD32 |
| kernel | C5×C2.D16 | C5×C2.D8 | C2×C80 | C10×D8 | C5×D8 | C2.D16 | C2.D8 | C2×C16 | C2×D8 | D8 | C40 | C2×C20 | C20 | C2×C10 | C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C5×C2.D16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 205 | 0 |
| 0 | 0 | 205 |
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 64 | 0 | 0 |
| 0 | 62 | 35 |
| 0 | 103 | 97 |
| 64 | 0 | 0 |
| 0 | 179 | 206 |
| 0 | 41 | 62 |
G:=sub<GL(3,GF(241))| [1,0,0,0,205,0,0,0,205],[240,0,0,0,240,0,0,0,240],[64,0,0,0,62,103,0,35,97],[64,0,0,0,179,41,0,206,62] >;
C5×C2.D16 in GAP, Magma, Sage, TeX
C_5\times C_2.D_{16} % in TeX
G:=Group("C5xC2.D16"); // GroupNames label
G:=SmallGroup(320,162);
// by ID
G=gap.SmallGroup(320,162);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,1410,360,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations