direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C10×D16, C40.72D4, C20.44D8, C80⋊10C22, C40.72C23, C4.6(C5×D8), C8.9(C5×D4), C16⋊2(C2×C10), (C2×C80)⋊12C2, (C2×C16)⋊5C10, (C2×D8)⋊6C10, D8⋊1(C2×C10), C4.7(D4×C10), (C10×D8)⋊20C2, (C2×C10).55D8, C10.84(C2×D8), C2.12(C10×D8), C20.314(C2×D4), (C2×C20).426D4, (C5×D8)⋊17C22, C8.3(C22×C10), C22.14(C5×D8), (C2×C40).426C22, (C2×C4).82(C5×D4), (C2×C8).84(C2×C10), SmallGroup(320,1006)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C8 [×2], C2×C4, D4 [×6], C23 [×2], C10, C10 [×2], C10 [×4], C16 [×2], C2×C8, D8 [×4], D8 [×2], C2×D4 [×2], C20 [×2], C2×C10, C2×C10 [×8], C2×C16, D16 [×4], C2×D8 [×2], C40 [×2], C2×C20, C5×D4 [×6], C22×C10 [×2], C2×D16, C80 [×2], C2×C40, C5×D8 [×4], C5×D8 [×2], D4×C10 [×2], C2×C80, C5×D16 [×4], C10×D8 [×2], C10×D16
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], D16 [×2], C2×D8, C5×D4 [×2], C22×C10, C2×D16, C5×D8 [×2], D4×C10, C5×D16 [×2], C10×D8, C10×D16
Generators and relations
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
►On 160 points
Generators in S160
(1 101 74 114 33 54 19 131 91 147)(2 102 75 115 34 55 20 132 92 148)(3 103 76 116 35 56 21 133 93 149)(4 104 77 117 36 57 22 134 94 150)(5 105 78 118 37 58 23 135 95 151)(6 106 79 119 38 59 24 136 96 152)(7 107 80 120 39 60 25 137 81 153)(8 108 65 121 40 61 26 138 82 154)(9 109 66 122 41 62 27 139 83 155)(10 110 67 123 42 63 28 140 84 156)(11 111 68 124 43 64 29 141 85 157)(12 112 69 125 44 49 30 142 86 158)(13 97 70 126 45 50 31 143 87 159)(14 98 71 127 46 51 32 144 88 160)(15 99 72 128 47 52 17 129 89 145)(16 100 73 113 48 53 18 130 90 146)
(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)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 64)(15 63)(16 62)(17 110)(18 109)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 112)(32 111)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 146)(42 145)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(65 131)(66 130)(67 129)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 132)(81 115)(82 114)(83 113)(84 128)(85 127)(86 126)(87 125)(88 124)(89 123)(90 122)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)
G:=sub<Sym(160)| (1,101,74,114,33,54,19,131,91,147)(2,102,75,115,34,55,20,132,92,148)(3,103,76,116,35,56,21,133,93,149)(4,104,77,117,36,57,22,134,94,150)(5,105,78,118,37,58,23,135,95,151)(6,106,79,119,38,59,24,136,96,152)(7,107,80,120,39,60,25,137,81,153)(8,108,65,121,40,61,26,138,82,154)(9,109,66,122,41,62,27,139,83,155)(10,110,67,123,42,63,28,140,84,156)(11,111,68,124,43,64,29,141,85,157)(12,112,69,125,44,49,30,142,86,158)(13,97,70,126,45,50,31,143,87,159)(14,98,71,127,46,51,32,144,88,160)(15,99,72,128,47,52,17,129,89,145)(16,100,73,113,48,53,18,130,90,146), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,64)(15,63)(16,62)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,112)(32,111)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(65,131)(66,130)(67,129)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,132)(81,115)(82,114)(83,113)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)>;
G:=Group( (1,101,74,114,33,54,19,131,91,147)(2,102,75,115,34,55,20,132,92,148)(3,103,76,116,35,56,21,133,93,149)(4,104,77,117,36,57,22,134,94,150)(5,105,78,118,37,58,23,135,95,151)(6,106,79,119,38,59,24,136,96,152)(7,107,80,120,39,60,25,137,81,153)(8,108,65,121,40,61,26,138,82,154)(9,109,66,122,41,62,27,139,83,155)(10,110,67,123,42,63,28,140,84,156)(11,111,68,124,43,64,29,141,85,157)(12,112,69,125,44,49,30,142,86,158)(13,97,70,126,45,50,31,143,87,159)(14,98,71,127,46,51,32,144,88,160)(15,99,72,128,47,52,17,129,89,145)(16,100,73,113,48,53,18,130,90,146), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,64)(15,63)(16,62)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,112)(32,111)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(65,131)(66,130)(67,129)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,132)(81,115)(82,114)(83,113)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116) );
G=PermutationGroup([(1,101,74,114,33,54,19,131,91,147),(2,102,75,115,34,55,20,132,92,148),(3,103,76,116,35,56,21,133,93,149),(4,104,77,117,36,57,22,134,94,150),(5,105,78,118,37,58,23,135,95,151),(6,106,79,119,38,59,24,136,96,152),(7,107,80,120,39,60,25,137,81,153),(8,108,65,121,40,61,26,138,82,154),(9,109,66,122,41,62,27,139,83,155),(10,110,67,123,42,63,28,140,84,156),(11,111,68,124,43,64,29,141,85,157),(12,112,69,125,44,49,30,142,86,158),(13,97,70,126,45,50,31,143,87,159),(14,98,71,127,46,51,32,144,88,160),(15,99,72,128,47,52,17,129,89,145),(16,100,73,113,48,53,18,130,90,146)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,64),(15,63),(16,62),(17,110),(18,109),(19,108),(20,107),(21,106),(22,105),(23,104),(24,103),(25,102),(26,101),(27,100),(28,99),(29,98),(30,97),(31,112),(32,111),(33,154),(34,153),(35,152),(36,151),(37,150),(38,149),(39,148),(40,147),(41,146),(42,145),(43,160),(44,159),(45,158),(46,157),(47,156),(48,155),(65,131),(66,130),(67,129),(68,144),(69,143),(70,142),(71,141),(72,140),(73,139),(74,138),(75,137),(76,136),(77,135),(78,134),(79,133),(80,132),(81,115),(82,114),(83,113),(84,128),(85,127),(86,126),(87,125),(88,124),(89,123),(90,122),(91,121),(92,120),(93,119),(94,118),(95,117),(96,116)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 1 | 0 | 0 |
| 0 | 156 | 27 |
| 0 | 214 | 156 |
| 1 | 0 | 0 |
| 0 | 85 | 214 |
| 0 | 214 | 156 |
G:=sub<GL(3,GF(241))| [240,0,0,0,91,0,0,0,91],[1,0,0,0,156,214,0,27,156],[1,0,0,0,85,214,0,214,156] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10AB | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | D16 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | C5×D16 |
| kernel | C10×D16 | C2×C80 | C5×D16 | C10×D8 | C2×D16 | C2×C16 | D16 | C2×D8 | C40 | C2×C20 | C20 | C2×C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times D_{16} % in TeX
G:=Group("C10xD16"); // GroupNames label
G:=SmallGroup(320,1006);
// by ID
G=gap.SmallGroup(320,1006);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀