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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, 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, C2×C5⋊2C16, C2×C80, D5×C2×C8, D10⋊1C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C16, 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 40 133 109 51 146 86 124 23 66)(2 41 134 110 52 147 87 125 24 67)(3 42 135 111 53 148 88 126 25 68)(4 43 136 112 54 149 89 127 26 69)(5 44 137 97 55 150 90 128 27 70)(6 45 138 98 56 151 91 113 28 71)(7 46 139 99 57 152 92 114 29 72)(8 47 140 100 58 153 93 115 30 73)(9 48 141 101 59 154 94 116 31 74)(10 33 142 102 60 155 95 117 32 75)(11 34 143 103 61 156 96 118 17 76)(12 35 144 104 62 157 81 119 18 77)(13 36 129 105 63 158 82 120 19 78)(14 37 130 106 64 159 83 121 20 79)(15 38 131 107 49 160 84 122 21 80)(16 39 132 108 50 145 85 123 22 65)
(1 66)(2 52)(3 68)(4 54)(5 70)(6 56)(7 72)(8 58)(9 74)(10 60)(11 76)(12 62)(13 78)(14 64)(15 80)(16 50)(17 34)(18 81)(19 36)(20 83)(21 38)(22 85)(23 40)(24 87)(25 42)(26 89)(27 44)(28 91)(29 46)(30 93)(31 48)(32 95)(33 102)(35 104)(37 106)(39 108)(41 110)(43 112)(45 98)(47 100)(49 160)(51 146)(53 148)(55 150)(57 152)(59 154)(61 156)(63 158)(65 145)(67 147)(69 149)(71 151)(73 153)(75 155)(77 157)(79 159)(82 105)(84 107)(86 109)(88 111)(90 97)(92 99)(94 101)(96 103)(114 139)(116 141)(118 143)(120 129)(122 131)(124 133)(126 135)(128 137)
(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)| (1,40,133,109,51,146,86,124,23,66)(2,41,134,110,52,147,87,125,24,67)(3,42,135,111,53,148,88,126,25,68)(4,43,136,112,54,149,89,127,26,69)(5,44,137,97,55,150,90,128,27,70)(6,45,138,98,56,151,91,113,28,71)(7,46,139,99,57,152,92,114,29,72)(8,47,140,100,58,153,93,115,30,73)(9,48,141,101,59,154,94,116,31,74)(10,33,142,102,60,155,95,117,32,75)(11,34,143,103,61,156,96,118,17,76)(12,35,144,104,62,157,81,119,18,77)(13,36,129,105,63,158,82,120,19,78)(14,37,130,106,64,159,83,121,20,79)(15,38,131,107,49,160,84,122,21,80)(16,39,132,108,50,145,85,123,22,65), (1,66)(2,52)(3,68)(4,54)(5,70)(6,56)(7,72)(8,58)(9,74)(10,60)(11,76)(12,62)(13,78)(14,64)(15,80)(16,50)(17,34)(18,81)(19,36)(20,83)(21,38)(22,85)(23,40)(24,87)(25,42)(26,89)(27,44)(28,91)(29,46)(30,93)(31,48)(32,95)(33,102)(35,104)(37,106)(39,108)(41,110)(43,112)(45,98)(47,100)(49,160)(51,146)(53,148)(55,150)(57,152)(59,154)(61,156)(63,158)(65,145)(67,147)(69,149)(71,151)(73,153)(75,155)(77,157)(79,159)(82,105)(84,107)(86,109)(88,111)(90,97)(92,99)(94,101)(96,103)(114,139)(116,141)(118,143)(120,129)(122,131)(124,133)(126,135)(128,137), (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( (1,40,133,109,51,146,86,124,23,66)(2,41,134,110,52,147,87,125,24,67)(3,42,135,111,53,148,88,126,25,68)(4,43,136,112,54,149,89,127,26,69)(5,44,137,97,55,150,90,128,27,70)(6,45,138,98,56,151,91,113,28,71)(7,46,139,99,57,152,92,114,29,72)(8,47,140,100,58,153,93,115,30,73)(9,48,141,101,59,154,94,116,31,74)(10,33,142,102,60,155,95,117,32,75)(11,34,143,103,61,156,96,118,17,76)(12,35,144,104,62,157,81,119,18,77)(13,36,129,105,63,158,82,120,19,78)(14,37,130,106,64,159,83,121,20,79)(15,38,131,107,49,160,84,122,21,80)(16,39,132,108,50,145,85,123,22,65), (1,66)(2,52)(3,68)(4,54)(5,70)(6,56)(7,72)(8,58)(9,74)(10,60)(11,76)(12,62)(13,78)(14,64)(15,80)(16,50)(17,34)(18,81)(19,36)(20,83)(21,38)(22,85)(23,40)(24,87)(25,42)(26,89)(27,44)(28,91)(29,46)(30,93)(31,48)(32,95)(33,102)(35,104)(37,106)(39,108)(41,110)(43,112)(45,98)(47,100)(49,160)(51,146)(53,148)(55,150)(57,152)(59,154)(61,156)(63,158)(65,145)(67,147)(69,149)(71,151)(73,153)(75,155)(77,157)(79,159)(82,105)(84,107)(86,109)(88,111)(90,97)(92,99)(94,101)(96,103)(114,139)(116,141)(118,143)(120,129)(122,131)(124,133)(126,135)(128,137), (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([[(1,40,133,109,51,146,86,124,23,66),(2,41,134,110,52,147,87,125,24,67),(3,42,135,111,53,148,88,126,25,68),(4,43,136,112,54,149,89,127,26,69),(5,44,137,97,55,150,90,128,27,70),(6,45,138,98,56,151,91,113,28,71),(7,46,139,99,57,152,92,114,29,72),(8,47,140,100,58,153,93,115,30,73),(9,48,141,101,59,154,94,116,31,74),(10,33,142,102,60,155,95,117,32,75),(11,34,143,103,61,156,96,118,17,76),(12,35,144,104,62,157,81,119,18,77),(13,36,129,105,63,158,82,120,19,78),(14,37,130,106,64,159,83,121,20,79),(15,38,131,107,49,160,84,122,21,80),(16,39,132,108,50,145,85,123,22,65)], [(1,66),(2,52),(3,68),(4,54),(5,70),(6,56),(7,72),(8,58),(9,74),(10,60),(11,76),(12,62),(13,78),(14,64),(15,80),(16,50),(17,34),(18,81),(19,36),(20,83),(21,38),(22,85),(23,40),(24,87),(25,42),(26,89),(27,44),(28,91),(29,46),(30,93),(31,48),(32,95),(33,102),(35,104),(37,106),(39,108),(41,110),(43,112),(45,98),(47,100),(49,160),(51,146),(53,148),(55,150),(57,152),(59,154),(61,156),(63,158),(65,145),(67,147),(69,149),(71,151),(73,153),(75,155),(77,157),(79,159),(82,105),(84,107),(86,109),(88,111),(90,97),(92,99),(94,101),(96,103),(114,139),(116,141),(118,143),(120,129),(122,131),(124,133),(126,135),(128,137)], [(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