direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×D11, C176⋊3C2, D22.2C8, C8.19D22, C88.19C22, Dic11.2C8, C11⋊C16⋊6C2, C11⋊1(C2×C16), C11⋊C8.3C4, C22.1(C2×C8), C2.1(C8×D11), C44.21(C2×C4), (C4×D11).4C4, (C8×D11).3C2, C4.16(C4×D11), SmallGroup(352,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C16×D11 |
Generators and relations for C16×D11
G = < a,b,c | a16=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C16×D11
►On 176 points
Generators in S176
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 91 31 56 34 169 146 97 120 75 131)(2 92 32 57 35 170 147 98 121 76 132)(3 93 17 58 36 171 148 99 122 77 133)(4 94 18 59 37 172 149 100 123 78 134)(5 95 19 60 38 173 150 101 124 79 135)(6 96 20 61 39 174 151 102 125 80 136)(7 81 21 62 40 175 152 103 126 65 137)(8 82 22 63 41 176 153 104 127 66 138)(9 83 23 64 42 161 154 105 128 67 139)(10 84 24 49 43 162 155 106 113 68 140)(11 85 25 50 44 163 156 107 114 69 141)(12 86 26 51 45 164 157 108 115 70 142)(13 87 27 52 46 165 158 109 116 71 143)(14 88 28 53 47 166 159 110 117 72 144)(15 89 29 54 48 167 160 111 118 73 129)(16 90 30 55 33 168 145 112 119 74 130)
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 113)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 97)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(161 169)(162 170)(163 171)(164 172)(165 173)(166 174)(167 175)(168 176)
G:=sub<Sym(176)| (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,91,31,56,34,169,146,97,120,75,131)(2,92,32,57,35,170,147,98,121,76,132)(3,93,17,58,36,171,148,99,122,77,133)(4,94,18,59,37,172,149,100,123,78,134)(5,95,19,60,38,173,150,101,124,79,135)(6,96,20,61,39,174,151,102,125,80,136)(7,81,21,62,40,175,152,103,126,65,137)(8,82,22,63,41,176,153,104,127,66,138)(9,83,23,64,42,161,154,105,128,67,139)(10,84,24,49,43,162,155,106,113,68,140)(11,85,25,50,44,163,156,107,114,69,141)(12,86,26,51,45,164,157,108,115,70,142)(13,87,27,52,46,165,158,109,116,71,143)(14,88,28,53,47,166,159,110,117,72,144)(15,89,29,54,48,167,160,111,118,73,129)(16,90,30,55,33,168,145,112,119,74,130), (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,113)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,97)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)>;
G:=Group( (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,91,31,56,34,169,146,97,120,75,131)(2,92,32,57,35,170,147,98,121,76,132)(3,93,17,58,36,171,148,99,122,77,133)(4,94,18,59,37,172,149,100,123,78,134)(5,95,19,60,38,173,150,101,124,79,135)(6,96,20,61,39,174,151,102,125,80,136)(7,81,21,62,40,175,152,103,126,65,137)(8,82,22,63,41,176,153,104,127,66,138)(9,83,23,64,42,161,154,105,128,67,139)(10,84,24,49,43,162,155,106,113,68,140)(11,85,25,50,44,163,156,107,114,69,141)(12,86,26,51,45,164,157,108,115,70,142)(13,87,27,52,46,165,158,109,116,71,143)(14,88,28,53,47,166,159,110,117,72,144)(15,89,29,54,48,167,160,111,118,73,129)(16,90,30,55,33,168,145,112,119,74,130), (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,113)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,97)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176) );
G=PermutationGroup([[(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,91,31,56,34,169,146,97,120,75,131),(2,92,32,57,35,170,147,98,121,76,132),(3,93,17,58,36,171,148,99,122,77,133),(4,94,18,59,37,172,149,100,123,78,134),(5,95,19,60,38,173,150,101,124,79,135),(6,96,20,61,39,174,151,102,125,80,136),(7,81,21,62,40,175,152,103,126,65,137),(8,82,22,63,41,176,153,104,127,66,138),(9,83,23,64,42,161,154,105,128,67,139),(10,84,24,49,43,162,155,106,113,68,140),(11,85,25,50,44,163,156,107,114,69,141),(12,86,26,51,45,164,157,108,115,70,142),(13,87,27,52,46,165,158,109,116,71,143),(14,88,28,53,47,166,159,110,117,72,144),(15,89,29,54,48,167,160,111,118,73,129),(16,90,30,55,33,168,145,112,119,74,130)], [(1,139),(2,140),(3,141),(4,142),(5,143),(6,144),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,121),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,113),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,97),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(161,169),(162,170),(163,171),(164,172),(165,173),(166,174),(167,175),(168,176)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 16A | ··· | 16H | 16I | ··· | 16P | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 16 | ··· | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D11 | D22 | C4×D11 | C8×D11 | C16×D11 |
| kernel | C16×D11 | C11⋊C16 | C176 | C8×D11 | C11⋊C8 | C4×D11 | Dic11 | D22 | D11 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 5 | 5 | 10 | 20 | 40 |
Matrix representation of C16×D11 ►in GL2(𝔽353) generated by
| 100 | 0 |
| 0 | 100 |
| 95 | 1 |
| 352 | 0 |
| 0 | 352 |
| 352 | 0 |
G:=sub<GL(2,GF(353))| [100,0,0,100],[95,352,1,0],[0,352,352,0] >;
C16×D11 in GAP, Magma, Sage, TeX
C_{16}\times D_{11} % in TeX
G:=Group("C16xD11"); // GroupNames label
G:=SmallGroup(352,3);
// by ID
G=gap.SmallGroup(352,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,31,50,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^16=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export