metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2C16, C10.2M5(2), C20.13M4(2), (C2×C8).7F5, (C2×C40).4C4, C5⋊1(C22⋊C16), C10.4(C2×C16), C5⋊2C8.53D4, C2.4(D5⋊C16), (C22×D5).6C8, C4.11(C4.F5), C10.3(C22⋊C8), C22.9(D5⋊C8), (C2×Dic5).10C8, C4.35(C22⋊F5), C2.2(C8.F5), C2.1(D10⋊C8), C20.33(C22⋊C4), (C2×C5⋊C16)⋊5C2, (C2×C4×D5).41C4, (D5×C2×C8).10C2, (C2×C10).5(C2×C8), (C2×C4).155(C2×F5), (C2×C20).161(C2×C4), (C2×C5⋊2C8).345C22, SmallGroup(320,225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊C16
G = < a,b,c | a10=b2=c16=1, bab=a-1, cac-1=a3, cbc-1=a7b >
Subgroups: 226 in 66 conjugacy classes, 30 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22⋊C16, C5⋊C16, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C5⋊C16, D5×C2×C8, D10⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C16, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×C16, M5(2), C2×F5, C22⋊C16, D5⋊C8, C4.F5, C22⋊F5, D5⋊C16, C8.F5, D10⋊C8, D10⋊C16
►On 160 points
Generators in S160
(1 79 95 103 141 54 150 120 18 39)(2 104 151 40 96 55 19 80 142 121)(3 41 20 122 152 56 143 105 81 65)(4 123 144 66 21 57 82 42 153 106)(5 67 83 107 129 58 154 124 22 43)(6 108 155 44 84 59 23 68 130 125)(7 45 24 126 156 60 131 109 85 69)(8 127 132 70 25 61 86 46 157 110)(9 71 87 111 133 62 158 128 26 47)(10 112 159 48 88 63 27 72 134 113)(11 33 28 114 160 64 135 97 89 73)(12 115 136 74 29 49 90 34 145 98)(13 75 91 99 137 50 146 116 30 35)(14 100 147 36 92 51 31 76 138 117)(15 37 32 118 148 52 139 101 93 77)(16 119 140 78 17 53 94 38 149 102)
(1 47)(2 88)(3 73)(4 29)(5 35)(6 92)(7 77)(8 17)(9 39)(10 96)(11 65)(12 21)(13 43)(14 84)(15 69)(16 25)(18 71)(19 134)(20 97)(22 75)(23 138)(24 101)(26 79)(27 142)(28 105)(30 67)(31 130)(32 109)(33 81)(34 42)(36 108)(37 85)(38 46)(40 112)(41 89)(44 100)(45 93)(48 104)(49 106)(50 129)(51 125)(52 156)(53 110)(54 133)(55 113)(56 160)(57 98)(58 137)(59 117)(60 148)(61 102)(62 141)(63 121)(64 152)(66 115)(68 76)(70 119)(72 80)(74 123)(78 127)(82 145)(83 116)(86 149)(87 120)(90 153)(91 124)(94 157)(95 128)(99 154)(103 158)(107 146)(111 150)(114 143)(118 131)(122 135)(126 139)(132 140)(136 144)(147 155)(151 159)
(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)| (1,79,95,103,141,54,150,120,18,39)(2,104,151,40,96,55,19,80,142,121)(3,41,20,122,152,56,143,105,81,65)(4,123,144,66,21,57,82,42,153,106)(5,67,83,107,129,58,154,124,22,43)(6,108,155,44,84,59,23,68,130,125)(7,45,24,126,156,60,131,109,85,69)(8,127,132,70,25,61,86,46,157,110)(9,71,87,111,133,62,158,128,26,47)(10,112,159,48,88,63,27,72,134,113)(11,33,28,114,160,64,135,97,89,73)(12,115,136,74,29,49,90,34,145,98)(13,75,91,99,137,50,146,116,30,35)(14,100,147,36,92,51,31,76,138,117)(15,37,32,118,148,52,139,101,93,77)(16,119,140,78,17,53,94,38,149,102), (1,47)(2,88)(3,73)(4,29)(5,35)(6,92)(7,77)(8,17)(9,39)(10,96)(11,65)(12,21)(13,43)(14,84)(15,69)(16,25)(18,71)(19,134)(20,97)(22,75)(23,138)(24,101)(26,79)(27,142)(28,105)(30,67)(31,130)(32,109)(33,81)(34,42)(36,108)(37,85)(38,46)(40,112)(41,89)(44,100)(45,93)(48,104)(49,106)(50,129)(51,125)(52,156)(53,110)(54,133)(55,113)(56,160)(57,98)(58,137)(59,117)(60,148)(61,102)(62,141)(63,121)(64,152)(66,115)(68,76)(70,119)(72,80)(74,123)(78,127)(82,145)(83,116)(86,149)(87,120)(90,153)(91,124)(94,157)(95,128)(99,154)(103,158)(107,146)(111,150)(114,143)(118,131)(122,135)(126,139)(132,140)(136,144)(147,155)(151,159), (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( (1,79,95,103,141,54,150,120,18,39)(2,104,151,40,96,55,19,80,142,121)(3,41,20,122,152,56,143,105,81,65)(4,123,144,66,21,57,82,42,153,106)(5,67,83,107,129,58,154,124,22,43)(6,108,155,44,84,59,23,68,130,125)(7,45,24,126,156,60,131,109,85,69)(8,127,132,70,25,61,86,46,157,110)(9,71,87,111,133,62,158,128,26,47)(10,112,159,48,88,63,27,72,134,113)(11,33,28,114,160,64,135,97,89,73)(12,115,136,74,29,49,90,34,145,98)(13,75,91,99,137,50,146,116,30,35)(14,100,147,36,92,51,31,76,138,117)(15,37,32,118,148,52,139,101,93,77)(16,119,140,78,17,53,94,38,149,102), (1,47)(2,88)(3,73)(4,29)(5,35)(6,92)(7,77)(8,17)(9,39)(10,96)(11,65)(12,21)(13,43)(14,84)(15,69)(16,25)(18,71)(19,134)(20,97)(22,75)(23,138)(24,101)(26,79)(27,142)(28,105)(30,67)(31,130)(32,109)(33,81)(34,42)(36,108)(37,85)(38,46)(40,112)(41,89)(44,100)(45,93)(48,104)(49,106)(50,129)(51,125)(52,156)(53,110)(54,133)(55,113)(56,160)(57,98)(58,137)(59,117)(60,148)(61,102)(62,141)(63,121)(64,152)(66,115)(68,76)(70,119)(72,80)(74,123)(78,127)(82,145)(83,116)(86,149)(87,120)(90,153)(91,124)(94,157)(95,128)(99,154)(103,158)(107,146)(111,150)(114,143)(118,131)(122,135)(126,139)(132,140)(136,144)(147,155)(151,159), (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([[(1,79,95,103,141,54,150,120,18,39),(2,104,151,40,96,55,19,80,142,121),(3,41,20,122,152,56,143,105,81,65),(4,123,144,66,21,57,82,42,153,106),(5,67,83,107,129,58,154,124,22,43),(6,108,155,44,84,59,23,68,130,125),(7,45,24,126,156,60,131,109,85,69),(8,127,132,70,25,61,86,46,157,110),(9,71,87,111,133,62,158,128,26,47),(10,112,159,48,88,63,27,72,134,113),(11,33,28,114,160,64,135,97,89,73),(12,115,136,74,29,49,90,34,145,98),(13,75,91,99,137,50,146,116,30,35),(14,100,147,36,92,51,31,76,138,117),(15,37,32,118,148,52,139,101,93,77),(16,119,140,78,17,53,94,38,149,102)], [(1,47),(2,88),(3,73),(4,29),(5,35),(6,92),(7,77),(8,17),(9,39),(10,96),(11,65),(12,21),(13,43),(14,84),(15,69),(16,25),(18,71),(19,134),(20,97),(22,75),(23,138),(24,101),(26,79),(27,142),(28,105),(30,67),(31,130),(32,109),(33,81),(34,42),(36,108),(37,85),(38,46),(40,112),(41,89),(44,100),(45,93),(48,104),(49,106),(50,129),(51,125),(52,156),(53,110),(54,133),(55,113),(56,160),(57,98),(58,137),(59,117),(60,148),(61,102),(62,141),(63,121),(64,152),(66,115),(68,76),(70,119),(72,80),(74,123),(78,127),(82,145),(83,116),(86,149),(87,120),(90,153),(91,124),(94,157),(95,128),(99,154),(103,158),(107,146),(111,150),(114,143),(118,131),(122,135),(126,139),(132,140),(136,144),(147,155),(151,159)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | M4(2) | M5(2) | F5 | C2×F5 | C4.F5 | C22⋊F5 | D5⋊C8 | D5⋊C16 | C8.F5 |
| kernel | D10⋊C16 | C2×C5⋊C16 | D5×C2×C8 | C2×C40 | C2×C4×D5 | C2×Dic5 | C22×D5 | D10 | C5⋊2C8 | C20 | C10 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊C16 ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 240 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 165 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 1 |
| 165 | 239 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 76 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 114 | 127 | 141 | 0 |
| 0 | 0 | 0 | 0 | 14 | 127 | 0 | 114 |
| 0 | 0 | 0 | 0 | 114 | 0 | 127 | 14 |
| 0 | 0 | 0 | 0 | 0 | 141 | 127 | 114 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0],[1,165,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,239,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1],[165,0,0,0,0,0,0,0,239,76,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,114,14,114,0,0,0,0,0,127,127,0,141,0,0,0,0,141,0,127,127,0,0,0,0,0,114,14,114] >;
D10⋊C16 in GAP, Magma, Sage, TeX
D_{10}\rtimes C_{16} % in TeX
G:=Group("D10:C16"); // GroupNames label
G:=SmallGroup(320,225);
// by ID
G=gap.SmallGroup(320,225);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^10=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^7*b>;
// generators/relations