direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×D11, C22⋊C14, C154⋊2C2, C77⋊3C22, C11⋊(C2×C14), SmallGroup(308,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C14×D11 |
Generators and relations for C14×D11
G = < a,b,c | a14=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C14×D11
►On 154 points
Generators in S154
(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)
(1 41 149 21 140 89 49 123 79 102 66)(2 42 150 22 127 90 50 124 80 103 67)(3 29 151 23 128 91 51 125 81 104 68)(4 30 152 24 129 92 52 126 82 105 69)(5 31 153 25 130 93 53 113 83 106 70)(6 32 154 26 131 94 54 114 84 107 57)(7 33 141 27 132 95 55 115 71 108 58)(8 34 142 28 133 96 56 116 72 109 59)(9 35 143 15 134 97 43 117 73 110 60)(10 36 144 16 135 98 44 118 74 111 61)(11 37 145 17 136 85 45 119 75 112 62)(12 38 146 18 137 86 46 120 76 99 63)(13 39 147 19 138 87 47 121 77 100 64)(14 40 148 20 139 88 48 122 78 101 65)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 57)(14 58)(15 124)(16 125)(17 126)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 111)(30 112)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
G:=sub<Sym(154)| (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), (1,41,149,21,140,89,49,123,79,102,66)(2,42,150,22,127,90,50,124,80,103,67)(3,29,151,23,128,91,51,125,81,104,68)(4,30,152,24,129,92,52,126,82,105,69)(5,31,153,25,130,93,53,113,83,106,70)(6,32,154,26,131,94,54,114,84,107,57)(7,33,141,27,132,95,55,115,71,108,58)(8,34,142,28,133,96,56,116,72,109,59)(9,35,143,15,134,97,43,117,73,110,60)(10,36,144,16,135,98,44,118,74,111,61)(11,37,145,17,136,85,45,119,75,112,62)(12,38,146,18,137,86,46,120,76,99,63)(13,39,147,19,138,87,47,121,77,100,64)(14,40,148,20,139,88,48,122,78,101,65), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,57)(14,58)(15,124)(16,125)(17,126)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)>;
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), (1,41,149,21,140,89,49,123,79,102,66)(2,42,150,22,127,90,50,124,80,103,67)(3,29,151,23,128,91,51,125,81,104,68)(4,30,152,24,129,92,52,126,82,105,69)(5,31,153,25,130,93,53,113,83,106,70)(6,32,154,26,131,94,54,114,84,107,57)(7,33,141,27,132,95,55,115,71,108,58)(8,34,142,28,133,96,56,116,72,109,59)(9,35,143,15,134,97,43,117,73,110,60)(10,36,144,16,135,98,44,118,74,111,61)(11,37,145,17,136,85,45,119,75,112,62)(12,38,146,18,137,86,46,120,76,99,63)(13,39,147,19,138,87,47,121,77,100,64)(14,40,148,20,139,88,48,122,78,101,65), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,57)(14,58)(15,124)(16,125)(17,126)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98) );
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)], [(1,41,149,21,140,89,49,123,79,102,66),(2,42,150,22,127,90,50,124,80,103,67),(3,29,151,23,128,91,51,125,81,104,68),(4,30,152,24,129,92,52,126,82,105,69),(5,31,153,25,130,93,53,113,83,106,70),(6,32,154,26,131,94,54,114,84,107,57),(7,33,141,27,132,95,55,115,71,108,58),(8,34,142,28,133,96,56,116,72,109,59),(9,35,143,15,134,97,43,117,73,110,60),(10,36,144,16,135,98,44,118,74,111,61),(11,37,145,17,136,85,45,119,75,112,62),(12,38,146,18,137,86,46,120,76,99,63),(13,39,147,19,138,87,47,121,77,100,64),(14,40,148,20,139,88,48,122,78,101,65)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,57),(14,58),(15,124),(16,125),(17,126),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,111),(30,112),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | ··· | 7F | 11A | ··· | 11E | 14A | ··· | 14F | 14G | ··· | 14R | 22A | ··· | 22E | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 14 | ··· | 14 | 22 | ··· | 22 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D11 | D22 | C7×D11 | C14×D11 |
| kernel | C14×D11 | C7×D11 | C154 | D22 | D11 | C22 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 5 | 5 | 30 | 30 |
Matrix representation of C14×D11 ►in GL2(𝔽43) generated by
| 39 | 0 |
| 0 | 39 |
| 42 | 9 |
| 28 | 5 |
| 38 | 36 |
| 28 | 5 |
G:=sub<GL(2,GF(43))| [39,0,0,39],[42,28,9,5],[38,28,36,5] >;
C14×D11 in GAP, Magma, Sage, TeX
C_{14}\times D_{11} % in TeX
G:=Group("C14xD11"); // GroupNames label
G:=SmallGroup(308,6);
// by ID
G=gap.SmallGroup(308,6);
# by ID
G:=PCGroup([4,-2,-2,-7,-11,4483]);
// Polycyclic
G:=Group<a,b,c|a^14=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export