direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17×D11, C11⋊C34, C187⋊3C2, SmallGroup(374,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C17×D11 |
Generators and relations for C17×D11
G = < a,b,c | a17=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C17×D11
►On 187 points
Generators in S187
(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)(171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187)
(1 75 55 145 90 30 173 160 134 35 115)(2 76 56 146 91 31 174 161 135 36 116)(3 77 57 147 92 32 175 162 136 37 117)(4 78 58 148 93 33 176 163 120 38 118)(5 79 59 149 94 34 177 164 121 39 119)(6 80 60 150 95 18 178 165 122 40 103)(7 81 61 151 96 19 179 166 123 41 104)(8 82 62 152 97 20 180 167 124 42 105)(9 83 63 153 98 21 181 168 125 43 106)(10 84 64 137 99 22 182 169 126 44 107)(11 85 65 138 100 23 183 170 127 45 108)(12 69 66 139 101 24 184 154 128 46 109)(13 70 67 140 102 25 185 155 129 47 110)(14 71 68 141 86 26 186 156 130 48 111)(15 72 52 142 87 27 187 157 131 49 112)(16 73 53 143 88 28 171 158 132 50 113)(17 74 54 144 89 29 172 159 133 51 114)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 120)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(86 186)(87 187)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(97 180)(98 181)(99 182)(100 183)(101 184)(102 185)(137 169)(138 170)(139 154)(140 155)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)
G:=sub<Sym(187)| (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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187), (1,75,55,145,90,30,173,160,134,35,115)(2,76,56,146,91,31,174,161,135,36,116)(3,77,57,147,92,32,175,162,136,37,117)(4,78,58,148,93,33,176,163,120,38,118)(5,79,59,149,94,34,177,164,121,39,119)(6,80,60,150,95,18,178,165,122,40,103)(7,81,61,151,96,19,179,166,123,41,104)(8,82,62,152,97,20,180,167,124,42,105)(9,83,63,153,98,21,181,168,125,43,106)(10,84,64,137,99,22,182,169,126,44,107)(11,85,65,138,100,23,183,170,127,45,108)(12,69,66,139,101,24,184,154,128,46,109)(13,70,67,140,102,25,185,155,129,47,110)(14,71,68,141,86,26,186,156,130,48,111)(15,72,52,142,87,27,187,157,131,49,112)(16,73,53,143,88,28,171,158,132,50,113)(17,74,54,144,89,29,172,159,133,51,114), (1,115)(2,116)(3,117)(4,118)(5,119)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(86,186)(87,187)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(137,169)(138,170)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168)>;
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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187), (1,75,55,145,90,30,173,160,134,35,115)(2,76,56,146,91,31,174,161,135,36,116)(3,77,57,147,92,32,175,162,136,37,117)(4,78,58,148,93,33,176,163,120,38,118)(5,79,59,149,94,34,177,164,121,39,119)(6,80,60,150,95,18,178,165,122,40,103)(7,81,61,151,96,19,179,166,123,41,104)(8,82,62,152,97,20,180,167,124,42,105)(9,83,63,153,98,21,181,168,125,43,106)(10,84,64,137,99,22,182,169,126,44,107)(11,85,65,138,100,23,183,170,127,45,108)(12,69,66,139,101,24,184,154,128,46,109)(13,70,67,140,102,25,185,155,129,47,110)(14,71,68,141,86,26,186,156,130,48,111)(15,72,52,142,87,27,187,157,131,49,112)(16,73,53,143,88,28,171,158,132,50,113)(17,74,54,144,89,29,172,159,133,51,114), (1,115)(2,116)(3,117)(4,118)(5,119)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(86,186)(87,187)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(137,169)(138,170)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168) );
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),(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187)], [(1,75,55,145,90,30,173,160,134,35,115),(2,76,56,146,91,31,174,161,135,36,116),(3,77,57,147,92,32,175,162,136,37,117),(4,78,58,148,93,33,176,163,120,38,118),(5,79,59,149,94,34,177,164,121,39,119),(6,80,60,150,95,18,178,165,122,40,103),(7,81,61,151,96,19,179,166,123,41,104),(8,82,62,152,97,20,180,167,124,42,105),(9,83,63,153,98,21,181,168,125,43,106),(10,84,64,137,99,22,182,169,126,44,107),(11,85,65,138,100,23,183,170,127,45,108),(12,69,66,139,101,24,184,154,128,46,109),(13,70,67,140,102,25,185,155,129,47,110),(14,71,68,141,86,26,186,156,130,48,111),(15,72,52,142,87,27,187,157,131,49,112),(16,73,53,143,88,28,171,158,132,50,113),(17,74,54,144,89,29,172,159,133,51,114)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,120),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(86,186),(87,187),(88,171),(89,172),(90,173),(91,174),(92,175),(93,176),(94,177),(95,178),(96,179),(97,180),(98,181),(99,182),(100,183),(101,184),(102,185),(137,169),(138,170),(139,154),(140,155),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,162),(148,163),(149,164),(150,165),(151,166),(152,167),(153,168)]])
119 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11E | 17A | ··· | 17P | 34A | ··· | 34P | 187A | ··· | 187CB |
| order | 1 | 2 | 11 | ··· | 11 | 17 | ··· | 17 | 34 | ··· | 34 | 187 | ··· | 187 |
| size | 1 | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | D11 | C17×D11 |
| kernel | C17×D11 | C187 | D11 | C11 | C17 | C1 |
| # reps | 1 | 1 | 16 | 16 | 5 | 80 |
Matrix representation of C17×D11 ►in GL2(𝔽1123) generated by
| 309 | 0 |
| 0 | 309 |
| 0 | 1 |
| 1122 | 819 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1123))| [309,0,0,309],[0,1122,1,819],[0,1,1,0] >;
C17×D11 in GAP, Magma, Sage, TeX
C_{17}\times D_{11} % in TeX
G:=Group("C17xD11"); // GroupNames label
G:=SmallGroup(374,1);
// by ID
G=gap.SmallGroup(374,1);
# by ID
G:=PCGroup([3,-2,-17,-11,3062]);
// Polycyclic
G:=Group<a,b,c|a^17=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export