metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C11⋊2D16, D8⋊1D11, D88⋊3C2, C22.8D8, C44.3D4, C8.4D22, C88.2C22, C11⋊C16⋊1C2, (C11×D8)⋊1C2, C2.4(D4⋊D11), C4.1(C11⋊D4), SmallGroup(352,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11⋊D16
G = < a,b,c | a11=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C11⋊D16
►On 176 points
Generators in S176
(1 153 73 123 17 39 102 86 60 166 135)(2 136 167 61 87 103 40 18 124 74 154)(3 155 75 125 19 41 104 88 62 168 137)(4 138 169 63 89 105 42 20 126 76 156)(5 157 77 127 21 43 106 90 64 170 139)(6 140 171 49 91 107 44 22 128 78 158)(7 159 79 113 23 45 108 92 50 172 141)(8 142 173 51 93 109 46 24 114 80 160)(9 145 65 115 25 47 110 94 52 174 143)(10 144 175 53 95 111 48 26 116 66 146)(11 147 67 117 27 33 112 96 54 176 129)(12 130 161 55 81 97 34 28 118 68 148)(13 149 69 119 29 35 98 82 56 162 131)(14 132 163 57 83 99 36 30 120 70 150)(15 151 71 121 31 37 100 84 58 164 133)(16 134 165 59 85 101 38 32 122 72 152)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(43 98)(44 97)(45 112)(46 111)(47 110)(48 109)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 128)(56 127)(57 126)(58 125)(59 124)(60 123)(61 122)(62 121)(63 120)(64 119)(65 174)(66 173)(67 172)(68 171)(69 170)(70 169)(71 168)(72 167)(73 166)(74 165)(75 164)(76 163)(77 162)(78 161)(79 176)(80 175)(129 159)(130 158)(131 157)(132 156)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(144 160)
G:=sub<Sym(176)| (1,153,73,123,17,39,102,86,60,166,135)(2,136,167,61,87,103,40,18,124,74,154)(3,155,75,125,19,41,104,88,62,168,137)(4,138,169,63,89,105,42,20,126,76,156)(5,157,77,127,21,43,106,90,64,170,139)(6,140,171,49,91,107,44,22,128,78,158)(7,159,79,113,23,45,108,92,50,172,141)(8,142,173,51,93,109,46,24,114,80,160)(9,145,65,115,25,47,110,94,52,174,143)(10,144,175,53,95,111,48,26,116,66,146)(11,147,67,117,27,33,112,96,54,176,129)(12,130,161,55,81,97,34,28,118,68,148)(13,149,69,119,29,35,98,82,56,162,131)(14,132,163,57,83,99,36,30,120,70,150)(15,151,71,121,31,37,100,84,58,164,133)(16,134,165,59,85,101,38,32,122,72,152), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,112)(46,111)(47,110)(48,109)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,176)(80,175)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(144,160)>;
G:=Group( (1,153,73,123,17,39,102,86,60,166,135)(2,136,167,61,87,103,40,18,124,74,154)(3,155,75,125,19,41,104,88,62,168,137)(4,138,169,63,89,105,42,20,126,76,156)(5,157,77,127,21,43,106,90,64,170,139)(6,140,171,49,91,107,44,22,128,78,158)(7,159,79,113,23,45,108,92,50,172,141)(8,142,173,51,93,109,46,24,114,80,160)(9,145,65,115,25,47,110,94,52,174,143)(10,144,175,53,95,111,48,26,116,66,146)(11,147,67,117,27,33,112,96,54,176,129)(12,130,161,55,81,97,34,28,118,68,148)(13,149,69,119,29,35,98,82,56,162,131)(14,132,163,57,83,99,36,30,120,70,150)(15,151,71,121,31,37,100,84,58,164,133)(16,134,165,59,85,101,38,32,122,72,152), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,112)(46,111)(47,110)(48,109)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,176)(80,175)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(144,160) );
G=PermutationGroup([[(1,153,73,123,17,39,102,86,60,166,135),(2,136,167,61,87,103,40,18,124,74,154),(3,155,75,125,19,41,104,88,62,168,137),(4,138,169,63,89,105,42,20,126,76,156),(5,157,77,127,21,43,106,90,64,170,139),(6,140,171,49,91,107,44,22,128,78,158),(7,159,79,113,23,45,108,92,50,172,141),(8,142,173,51,93,109,46,24,114,80,160),(9,145,65,115,25,47,110,94,52,174,143),(10,144,175,53,95,111,48,26,116,66,146),(11,147,67,117,27,33,112,96,54,176,129),(12,130,161,55,81,97,34,28,118,68,148),(13,149,69,119,29,35,98,82,56,162,131),(14,132,163,57,83,99,36,30,120,70,150),(15,151,71,121,31,37,100,84,58,164,133),(16,134,165,59,85,101,38,32,122,72,152)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,86),(18,85),(19,84),(20,83),(21,82),(22,81),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,90),(30,89),(31,88),(32,87),(33,108),(34,107),(35,106),(36,105),(37,104),(38,103),(39,102),(40,101),(41,100),(42,99),(43,98),(44,97),(45,112),(46,111),(47,110),(48,109),(49,118),(50,117),(51,116),(52,115),(53,114),(54,113),(55,128),(56,127),(57,126),(58,125),(59,124),(60,123),(61,122),(62,121),(63,120),(64,119),(65,174),(66,173),(67,172),(68,171),(69,170),(70,169),(71,168),(72,167),(73,166),(74,165),(75,164),(76,163),(77,162),(78,161),(79,176),(80,175),(129,159),(130,158),(131,157),(132,156),(133,155),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148),(141,147),(142,146),(143,145),(144,160)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 22A | ··· | 22E | 22F | ··· | 22O | 44A | ··· | 44E | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 8 | 88 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D11 | D16 | D22 | C11⋊D4 | D4⋊D11 | C11⋊D16 |
| kernel | C11⋊D16 | C11⋊C16 | D88 | C11×D8 | C44 | C22 | D8 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 4 | 5 | 10 | 5 | 10 |
Matrix representation of C11⋊D16 ►in GL4(𝔽353) generated by
| 204 | 1 | 0 | 0 |
| 202 | 155 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 280 | 204 | 0 | 0 |
| 140 | 73 | 0 | 0 |
| 0 | 0 | 333 | 183 |
| 0 | 0 | 170 | 333 |
| 73 | 149 | 0 | 0 |
| 213 | 280 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(353))| [204,202,0,0,1,155,0,0,0,0,1,0,0,0,0,1],[280,140,0,0,204,73,0,0,0,0,333,170,0,0,183,333],[73,213,0,0,149,280,0,0,0,0,0,1,0,0,1,0] >;
C11⋊D16 in GAP, Magma, Sage, TeX
C_{11}\rtimes D_{16} % in TeX
G:=Group("C11:D16"); // GroupNames label
G:=SmallGroup(352,32);
// by ID
G=gap.SmallGroup(352,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,218,116,122,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^11=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export