direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D17, C17⋊C22, C187⋊2C2, SmallGroup(374,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C11×D17 |
Generators and relations for C11×D17
G = < a,b,c | a11=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D17
►On 187 points
Generators in S187
(1 185 158 145 120 106 87 69 65 46 20)(2 186 159 146 121 107 88 70 66 47 21)(3 187 160 147 122 108 89 71 67 48 22)(4 171 161 148 123 109 90 72 68 49 23)(5 172 162 149 124 110 91 73 52 50 24)(6 173 163 150 125 111 92 74 53 51 25)(7 174 164 151 126 112 93 75 54 35 26)(8 175 165 152 127 113 94 76 55 36 27)(9 176 166 153 128 114 95 77 56 37 28)(10 177 167 137 129 115 96 78 57 38 29)(11 178 168 138 130 116 97 79 58 39 30)(12 179 169 139 131 117 98 80 59 40 31)(13 180 170 140 132 118 99 81 60 41 32)(14 181 154 141 133 119 100 82 61 42 33)(15 182 155 142 134 103 101 83 62 43 34)(16 183 156 143 135 104 102 84 63 44 18)(17 184 157 144 136 105 86 85 64 45 19)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 39)(36 38)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(52 60)(53 59)(54 58)(55 57)(61 68)(62 67)(63 66)(64 65)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(86 87)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(103 108)(104 107)(105 106)(109 119)(110 118)(111 117)(112 116)(113 115)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(154 161)(155 160)(156 159)(157 158)(162 170)(163 169)(164 168)(165 167)(171 181)(172 180)(173 179)(174 178)(175 177)(182 187)(183 186)(184 185)
G:=sub<Sym(187)| (1,185,158,145,120,106,87,69,65,46,20)(2,186,159,146,121,107,88,70,66,47,21)(3,187,160,147,122,108,89,71,67,48,22)(4,171,161,148,123,109,90,72,68,49,23)(5,172,162,149,124,110,91,73,52,50,24)(6,173,163,150,125,111,92,74,53,51,25)(7,174,164,151,126,112,93,75,54,35,26)(8,175,165,152,127,113,94,76,55,36,27)(9,176,166,153,128,114,95,77,56,37,28)(10,177,167,137,129,115,96,78,57,38,29)(11,178,168,138,130,116,97,79,58,39,30)(12,179,169,139,131,117,98,80,59,40,31)(13,180,170,140,132,118,99,81,60,41,32)(14,181,154,141,133,119,100,82,61,42,33)(15,182,155,142,134,103,101,83,62,43,34)(16,183,156,143,135,104,102,84,63,44,18)(17,184,157,144,136,105,86,85,64,45,19), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,39)(36,38)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(52,60)(53,59)(54,58)(55,57)(61,68)(62,67)(63,66)(64,65)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,108)(104,107)(105,106)(109,119)(110,118)(111,117)(112,116)(113,115)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(154,161)(155,160)(156,159)(157,158)(162,170)(163,169)(164,168)(165,167)(171,181)(172,180)(173,179)(174,178)(175,177)(182,187)(183,186)(184,185)>;
G:=Group( (1,185,158,145,120,106,87,69,65,46,20)(2,186,159,146,121,107,88,70,66,47,21)(3,187,160,147,122,108,89,71,67,48,22)(4,171,161,148,123,109,90,72,68,49,23)(5,172,162,149,124,110,91,73,52,50,24)(6,173,163,150,125,111,92,74,53,51,25)(7,174,164,151,126,112,93,75,54,35,26)(8,175,165,152,127,113,94,76,55,36,27)(9,176,166,153,128,114,95,77,56,37,28)(10,177,167,137,129,115,96,78,57,38,29)(11,178,168,138,130,116,97,79,58,39,30)(12,179,169,139,131,117,98,80,59,40,31)(13,180,170,140,132,118,99,81,60,41,32)(14,181,154,141,133,119,100,82,61,42,33)(15,182,155,142,134,103,101,83,62,43,34)(16,183,156,143,135,104,102,84,63,44,18)(17,184,157,144,136,105,86,85,64,45,19), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,39)(36,38)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(52,60)(53,59)(54,58)(55,57)(61,68)(62,67)(63,66)(64,65)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,108)(104,107)(105,106)(109,119)(110,118)(111,117)(112,116)(113,115)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(154,161)(155,160)(156,159)(157,158)(162,170)(163,169)(164,168)(165,167)(171,181)(172,180)(173,179)(174,178)(175,177)(182,187)(183,186)(184,185) );
G=PermutationGroup([[(1,185,158,145,120,106,87,69,65,46,20),(2,186,159,146,121,107,88,70,66,47,21),(3,187,160,147,122,108,89,71,67,48,22),(4,171,161,148,123,109,90,72,68,49,23),(5,172,162,149,124,110,91,73,52,50,24),(6,173,163,150,125,111,92,74,53,51,25),(7,174,164,151,126,112,93,75,54,35,26),(8,175,165,152,127,113,94,76,55,36,27),(9,176,166,153,128,114,95,77,56,37,28),(10,177,167,137,129,115,96,78,57,38,29),(11,178,168,138,130,116,97,79,58,39,30),(12,179,169,139,131,117,98,80,59,40,31),(13,180,170,140,132,118,99,81,60,41,32),(14,181,154,141,133,119,100,82,61,42,33),(15,182,155,142,134,103,101,83,62,43,34),(16,183,156,143,135,104,102,84,63,44,18),(17,184,157,144,136,105,86,85,64,45,19)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,21),(19,20),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(35,39),(36,38),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(52,60),(53,59),(54,58),(55,57),(61,68),(62,67),(63,66),(64,65),(69,85),(70,84),(71,83),(72,82),(73,81),(74,80),(75,79),(76,78),(86,87),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(103,108),(104,107),(105,106),(109,119),(110,118),(111,117),(112,116),(113,115),(120,136),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(137,152),(138,151),(139,150),(140,149),(141,148),(142,147),(143,146),(144,145),(154,161),(155,160),(156,159),(157,158),(162,170),(163,169),(164,168),(165,167),(171,181),(172,180),(173,179),(174,178),(175,177),(182,187),(183,186),(184,185)]])
110 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11J | 17A | ··· | 17H | 22A | ··· | 22J | 187A | ··· | 187CB |
| order | 1 | 2 | 11 | ··· | 11 | 17 | ··· | 17 | 22 | ··· | 22 | 187 | ··· | 187 |
| size | 1 | 17 | 1 | ··· | 1 | 2 | ··· | 2 | 17 | ··· | 17 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | D17 | C11×D17 |
| kernel | C11×D17 | C187 | D17 | C17 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 8 | 80 |
Matrix representation of C11×D17 ►in GL2(𝔽1123) generated by
| 155 | 0 |
| 0 | 155 |
| 281 | 1 |
| 85 | 344 |
| 985 | 464 |
| 155 | 138 |
G:=sub<GL(2,GF(1123))| [155,0,0,155],[281,85,1,344],[985,155,464,138] >;
C11×D17 in GAP, Magma, Sage, TeX
C_{11}\times D_{17} % in TeX
G:=Group("C11xD17"); // GroupNames label
G:=SmallGroup(374,2);
// by ID
G=gap.SmallGroup(374,2);
# by ID
G:=PCGroup([3,-2,-11,-17,3170]);
// Polycyclic
G:=Group<a,b,c|a^11=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export