direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C22, C14⋊C22, C154⋊3C2, C77⋊4C22, C7⋊(C2×C22), SmallGroup(308,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C22 |
Generators and relations for D7×C22
G = < a,b,c | a22=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C22
►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 32 78 117 108 135 45)(2 33 79 118 109 136 46)(3 34 80 119 110 137 47)(4 35 81 120 89 138 48)(5 36 82 121 90 139 49)(6 37 83 122 91 140 50)(7 38 84 123 92 141 51)(8 39 85 124 93 142 52)(9 40 86 125 94 143 53)(10 41 87 126 95 144 54)(11 42 88 127 96 145 55)(12 43 67 128 97 146 56)(13 44 68 129 98 147 57)(14 23 69 130 99 148 58)(15 24 70 131 100 149 59)(16 25 71 132 101 150 60)(17 26 72 111 102 151 61)(18 27 73 112 103 152 62)(19 28 74 113 104 153 63)(20 29 75 114 105 154 64)(21 30 76 115 106 133 65)(22 31 77 116 107 134 66)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 133)(42 134)(43 135)(44 136)(67 108)(68 109)(69 110)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)(121 132)
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), (1,32,78,117,108,135,45)(2,33,79,118,109,136,46)(3,34,80,119,110,137,47)(4,35,81,120,89,138,48)(5,36,82,121,90,139,49)(6,37,83,122,91,140,50)(7,38,84,123,92,141,51)(8,39,85,124,93,142,52)(9,40,86,125,94,143,53)(10,41,87,126,95,144,54)(11,42,88,127,96,145,55)(12,43,67,128,97,146,56)(13,44,68,129,98,147,57)(14,23,69,130,99,148,58)(15,24,70,131,100,149,59)(16,25,71,132,101,150,60)(17,26,72,111,102,151,61)(18,27,73,112,103,152,62)(19,28,74,113,104,153,63)(20,29,75,114,105,154,64)(21,30,76,115,106,133,65)(22,31,77,116,107,134,66), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,133)(42,134)(43,135)(44,136)(67,108)(68,109)(69,110)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)>;
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), (1,32,78,117,108,135,45)(2,33,79,118,109,136,46)(3,34,80,119,110,137,47)(4,35,81,120,89,138,48)(5,36,82,121,90,139,49)(6,37,83,122,91,140,50)(7,38,84,123,92,141,51)(8,39,85,124,93,142,52)(9,40,86,125,94,143,53)(10,41,87,126,95,144,54)(11,42,88,127,96,145,55)(12,43,67,128,97,146,56)(13,44,68,129,98,147,57)(14,23,69,130,99,148,58)(15,24,70,131,100,149,59)(16,25,71,132,101,150,60)(17,26,72,111,102,151,61)(18,27,73,112,103,152,62)(19,28,74,113,104,153,63)(20,29,75,114,105,154,64)(21,30,76,115,106,133,65)(22,31,77,116,107,134,66), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,133)(42,134)(43,135)(44,136)(67,108)(68,109)(69,110)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132) );
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)], [(1,32,78,117,108,135,45),(2,33,79,118,109,136,46),(3,34,80,119,110,137,47),(4,35,81,120,89,138,48),(5,36,82,121,90,139,49),(6,37,83,122,91,140,50),(7,38,84,123,92,141,51),(8,39,85,124,93,142,52),(9,40,86,125,94,143,53),(10,41,87,126,95,144,54),(11,42,88,127,96,145,55),(12,43,67,128,97,146,56),(13,44,68,129,98,147,57),(14,23,69,130,99,148,58),(15,24,70,131,100,149,59),(16,25,71,132,101,150,60),(17,26,72,111,102,151,61),(18,27,73,112,103,152,62),(19,28,74,113,104,153,63),(20,29,75,114,105,154,64),(21,30,76,115,106,133,65),(22,31,77,116,107,134,66)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,133),(42,134),(43,135),(44,136),(67,108),(68,109),(69,110),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,131),(121,132)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 11A | ··· | 11J | 14A | 14B | 14C | 22A | ··· | 22J | 22K | ··· | 22AD | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 14 | 22 | ··· | 22 | 22 | ··· | 22 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D7 | D14 | C11×D7 | D7×C22 |
| kernel | D7×C22 | C11×D7 | C154 | D14 | D7 | C14 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 3 | 3 | 30 | 30 |
Matrix representation of D7×C22 ►in GL3(𝔽463) generated by
| 462 | 0 | 0 |
| 0 | 247 | 0 |
| 0 | 0 | 247 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 462 | 75 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(463))| [462,0,0,0,247,0,0,0,247],[1,0,0,0,0,462,0,1,75],[1,0,0,0,0,1,0,1,0] >;
D7×C22 in GAP, Magma, Sage, TeX
D_7\times C_{22} % in TeX
G:=Group("D7xC22"); // GroupNames label
G:=SmallGroup(308,7);
// by ID
G=gap.SmallGroup(308,7);
# by ID
G:=PCGroup([4,-2,-2,-11,-7,4227]);
// Polycyclic
G:=Group<a,b,c|a^22=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export