metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1C16, C40.89D4, C8.30D20, C10.9M5(2), C20.40M4(2), (C2×C80)⋊1C2, (C2×C16)⋊1D5, C2.5(D5×C16), C5⋊3(C22⋊C16), C10.15(C2×C16), (C2×C8).335D10, C8.49(C5⋊D4), (C2×Dic5).5C8, C2.3(C80⋊C2), (C22×D5).3C8, C22.11(C8×D5), C4.15(C8⋊D5), C2.1(D10⋊1C8), C10.22(C22⋊C8), (C2×C40).401C22, C4.40(D10⋊C4), C20.102(C22⋊C4), (D5×C2×C8).9C2, (C2×C4×D5).15C4, (C2×C5⋊2C16)⋊9C2, (C2×C5⋊2C8).17C4, (C2×C10).40(C2×C8), (C2×C4).169(C4×D5), (C2×C20).416(C2×C4), SmallGroup(320,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊1C16
G = < a,b,c | a10=b2=c16=1, bab=a-1, ac=ca, cbc-1=a5b >
Subgroups: 214 in 66 conjugacy classes, 33 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4, C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×3], C23, D5 [×2], C10 [×3], C16 [×2], C2×C8, C2×C8 [×3], C22×C4, Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16, C2×C16, C22×C8, C5⋊2C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C22⋊C16, C5⋊2C16, C80, C8×D5 [×2], C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C5⋊2C16, C2×C80, D5×C2×C8, D10⋊1C16
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C16 [×2], C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C2×C16, M5(2), C4×D5, D20, C5⋊D4, C22⋊C16, C8×D5, C8⋊D5, D10⋊C4, D5×C16, C80⋊C2, D10⋊1C8, D10⋊1C16
►On 160 points
Generators in S160
(1 133 146 62 105 26 127 33 87 66)(2 134 147 63 106 27 128 34 88 67)(3 135 148 64 107 28 113 35 89 68)(4 136 149 49 108 29 114 36 90 69)(5 137 150 50 109 30 115 37 91 70)(6 138 151 51 110 31 116 38 92 71)(7 139 152 52 111 32 117 39 93 72)(8 140 153 53 112 17 118 40 94 73)(9 141 154 54 97 18 119 41 95 74)(10 142 155 55 98 19 120 42 96 75)(11 143 156 56 99 20 121 43 81 76)(12 144 157 57 100 21 122 44 82 77)(13 129 158 58 101 22 123 45 83 78)(14 130 159 59 102 23 124 46 84 79)(15 131 160 60 103 24 125 47 85 80)(16 132 145 61 104 25 126 48 86 65)
(1 66)(2 106)(3 68)(4 108)(5 70)(6 110)(7 72)(8 112)(9 74)(10 98)(11 76)(12 100)(13 78)(14 102)(15 80)(16 104)(17 73)(18 97)(19 75)(20 99)(21 77)(22 101)(23 79)(24 103)(25 65)(26 105)(27 67)(28 107)(29 69)(30 109)(31 71)(32 111)(33 146)(35 148)(37 150)(39 152)(41 154)(43 156)(45 158)(47 160)(49 136)(50 115)(51 138)(52 117)(53 140)(54 119)(55 142)(56 121)(57 144)(58 123)(59 130)(60 125)(61 132)(62 127)(63 134)(64 113)(81 143)(82 122)(83 129)(84 124)(85 131)(86 126)(87 133)(88 128)(89 135)(90 114)(91 137)(92 116)(93 139)(94 118)(95 141)(96 120)
(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)| (1,133,146,62,105,26,127,33,87,66)(2,134,147,63,106,27,128,34,88,67)(3,135,148,64,107,28,113,35,89,68)(4,136,149,49,108,29,114,36,90,69)(5,137,150,50,109,30,115,37,91,70)(6,138,151,51,110,31,116,38,92,71)(7,139,152,52,111,32,117,39,93,72)(8,140,153,53,112,17,118,40,94,73)(9,141,154,54,97,18,119,41,95,74)(10,142,155,55,98,19,120,42,96,75)(11,143,156,56,99,20,121,43,81,76)(12,144,157,57,100,21,122,44,82,77)(13,129,158,58,101,22,123,45,83,78)(14,130,159,59,102,23,124,46,84,79)(15,131,160,60,103,24,125,47,85,80)(16,132,145,61,104,25,126,48,86,65), (1,66)(2,106)(3,68)(4,108)(5,70)(6,110)(7,72)(8,112)(9,74)(10,98)(11,76)(12,100)(13,78)(14,102)(15,80)(16,104)(17,73)(18,97)(19,75)(20,99)(21,77)(22,101)(23,79)(24,103)(25,65)(26,105)(27,67)(28,107)(29,69)(30,109)(31,71)(32,111)(33,146)(35,148)(37,150)(39,152)(41,154)(43,156)(45,158)(47,160)(49,136)(50,115)(51,138)(52,117)(53,140)(54,119)(55,142)(56,121)(57,144)(58,123)(59,130)(60,125)(61,132)(62,127)(63,134)(64,113)(81,143)(82,122)(83,129)(84,124)(85,131)(86,126)(87,133)(88,128)(89,135)(90,114)(91,137)(92,116)(93,139)(94,118)(95,141)(96,120), (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( (1,133,146,62,105,26,127,33,87,66)(2,134,147,63,106,27,128,34,88,67)(3,135,148,64,107,28,113,35,89,68)(4,136,149,49,108,29,114,36,90,69)(5,137,150,50,109,30,115,37,91,70)(6,138,151,51,110,31,116,38,92,71)(7,139,152,52,111,32,117,39,93,72)(8,140,153,53,112,17,118,40,94,73)(9,141,154,54,97,18,119,41,95,74)(10,142,155,55,98,19,120,42,96,75)(11,143,156,56,99,20,121,43,81,76)(12,144,157,57,100,21,122,44,82,77)(13,129,158,58,101,22,123,45,83,78)(14,130,159,59,102,23,124,46,84,79)(15,131,160,60,103,24,125,47,85,80)(16,132,145,61,104,25,126,48,86,65), (1,66)(2,106)(3,68)(4,108)(5,70)(6,110)(7,72)(8,112)(9,74)(10,98)(11,76)(12,100)(13,78)(14,102)(15,80)(16,104)(17,73)(18,97)(19,75)(20,99)(21,77)(22,101)(23,79)(24,103)(25,65)(26,105)(27,67)(28,107)(29,69)(30,109)(31,71)(32,111)(33,146)(35,148)(37,150)(39,152)(41,154)(43,156)(45,158)(47,160)(49,136)(50,115)(51,138)(52,117)(53,140)(54,119)(55,142)(56,121)(57,144)(58,123)(59,130)(60,125)(61,132)(62,127)(63,134)(64,113)(81,143)(82,122)(83,129)(84,124)(85,131)(86,126)(87,133)(88,128)(89,135)(90,114)(91,137)(92,116)(93,139)(94,118)(95,141)(96,120), (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([(1,133,146,62,105,26,127,33,87,66),(2,134,147,63,106,27,128,34,88,67),(3,135,148,64,107,28,113,35,89,68),(4,136,149,49,108,29,114,36,90,69),(5,137,150,50,109,30,115,37,91,70),(6,138,151,51,110,31,116,38,92,71),(7,139,152,52,111,32,117,39,93,72),(8,140,153,53,112,17,118,40,94,73),(9,141,154,54,97,18,119,41,95,74),(10,142,155,55,98,19,120,42,96,75),(11,143,156,56,99,20,121,43,81,76),(12,144,157,57,100,21,122,44,82,77),(13,129,158,58,101,22,123,45,83,78),(14,130,159,59,102,23,124,46,84,79),(15,131,160,60,103,24,125,47,85,80),(16,132,145,61,104,25,126,48,86,65)], [(1,66),(2,106),(3,68),(4,108),(5,70),(6,110),(7,72),(8,112),(9,74),(10,98),(11,76),(12,100),(13,78),(14,102),(15,80),(16,104),(17,73),(18,97),(19,75),(20,99),(21,77),(22,101),(23,79),(24,103),(25,65),(26,105),(27,67),(28,107),(29,69),(30,109),(31,71),(32,111),(33,146),(35,148),(37,150),(39,152),(41,154),(43,156),(45,158),(47,160),(49,136),(50,115),(51,138),(52,117),(53,140),(54,119),(55,142),(56,121),(57,144),(58,123),(59,130),(60,125),(61,132),(62,127),(63,134),(64,113),(81,143),(82,122),(83,129),(84,124),(85,131),(86,126),(87,133),(88,128),(89,135),(90,114),(91,137),(92,116),(93,139),(94,118),(95,141),(96,120)], [(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)])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 16A | ··· | 16H | 16I | ··· | 16P | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 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 | C4 | C8 | C8 | C16 | D4 | D5 | M4(2) | D10 | M5(2) | D20 | C5⋊D4 | C4×D5 | C8⋊D5 | C8×D5 | D5×C16 | C80⋊C2 |
| kernel | D10⋊1C16 | C2×C5⋊2C16 | C2×C80 | D5×C2×C8 | C2×C5⋊2C8 | C2×C4×D5 | C2×Dic5 | C22×D5 | D10 | C40 | C2×C16 | C20 | C2×C8 | C10 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of D10⋊1C16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 51 | 51 |
| 0 | 190 | 1 |
| 240 | 0 | 0 |
| 0 | 51 | 51 |
| 0 | 1 | 190 |
| 126 | 0 | 0 |
| 0 | 165 | 192 |
| 0 | 49 | 76 |
G:=sub<GL(3,GF(241))| [1,0,0,0,51,190,0,51,1],[240,0,0,0,51,1,0,51,190],[126,0,0,0,165,49,0,192,76] >;
D10⋊1C16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_1C_{16} % in TeX
G:=Group("D10:1C16"); // GroupNames label
G:=SmallGroup(320,65);
// by ID
G=gap.SmallGroup(320,65);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,100,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^10=b^2=c^16=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations