direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D49, C49⋊C23, C98⋊C22, C14.11D14, (C2×C98)⋊3C2, C7.(C22×D7), (C2×C14).3D7, SmallGroup(392,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C22×D49 |
Generators and relations for C22×D49
G = < a,b,c,d | a2=b2=c49=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
49C2
49C2
49C2
49C2
49C22
49C22
49C22
49C22
49C22
49C22
7D7
7D7
7D7
7D7
49C23
7D14
7D14
7D14
7D14
7D14
7D14
Smallest permutation representation of C22×D49
►On 196 points
Generators in S196
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 120)(16 119)(17 118)(18 117)(19 116)(20 115)(21 114)(22 113)(23 112)(24 111)(25 110)(26 109)(27 108)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 99)(37 147)(38 146)(39 145)(40 144)(41 143)(42 142)(43 141)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 151)(51 150)(52 149)(53 148)(54 196)(55 195)(56 194)(57 193)(58 192)(59 191)(60 190)(61 189)(62 188)(63 187)(64 186)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(79 171)(80 170)(81 169)(82 168)(83 167)(84 166)(85 165)(86 164)(87 163)(88 162)(89 161)(90 160)(91 159)(92 158)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)
G:=sub<Sym(196)| (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,141)(8,142)(9,143)(10,144)(11,145)(12,146)(13,147)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173), (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,147)(38,146)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,151)(51,150)(52,149)(53,148)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)>;
G:=Group( (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,141)(8,142)(9,143)(10,144)(11,145)(12,146)(13,147)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173), (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,147)(38,146)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,151)(51,150)(52,149)(53,148)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152) );
G=PermutationGroup([[(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,141),(8,142),(9,143),(10,144),(11,145),(12,146),(13,147),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,161),(91,162),(92,163),(93,164),(94,165),(95,166),(96,167),(97,168),(98,169)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,183),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(121,196),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(131,157),(132,158),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166),(141,167),(142,168),(143,169),(144,170),(145,171),(146,172),(147,173)], [(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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)], [(1,134),(2,133),(3,132),(4,131),(5,130),(6,129),(7,128),(8,127),(9,126),(10,125),(11,124),(12,123),(13,122),(14,121),(15,120),(16,119),(17,118),(18,117),(19,116),(20,115),(21,114),(22,113),(23,112),(24,111),(25,110),(26,109),(27,108),(28,107),(29,106),(30,105),(31,104),(32,103),(33,102),(34,101),(35,100),(36,99),(37,147),(38,146),(39,145),(40,144),(41,143),(42,142),(43,141),(44,140),(45,139),(46,138),(47,137),(48,136),(49,135),(50,151),(51,150),(52,149),(53,148),(54,196),(55,195),(56,194),(57,193),(58,192),(59,191),(60,190),(61,189),(62,188),(63,187),(64,186),(65,185),(66,184),(67,183),(68,182),(69,181),(70,180),(71,179),(72,178),(73,177),(74,176),(75,175),(76,174),(77,173),(78,172),(79,171),(80,170),(81,169),(82,168),(83,167),(84,166),(85,165),(86,164),(87,163),(88,162),(89,161),(90,160),(91,159),(92,158),(93,157),(94,156),(95,155),(96,154),(97,153),(98,152)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 7A | 7B | 7C | 14A | ··· | 14I | 49A | ··· | 49U | 98A | ··· | 98BK |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 14 | ··· | 14 | 49 | ··· | 49 | 98 | ··· | 98 |
| size | 1 | 1 | 1 | 1 | 49 | 49 | 49 | 49 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D7 | D14 | D49 | D98 |
| kernel | C22×D49 | D98 | C2×C98 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 6 | 1 | 3 | 9 | 21 | 63 |
Matrix representation of C22×D49 ►in GL4(𝔽197) generated by
| 1 | 0 | 0 | 0 |
| 0 | 196 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 196 | 0 | 0 | 0 |
| 0 | 196 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 195 | 21 |
| 0 | 0 | 176 | 23 |
| 196 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 195 | 21 |
| 0 | 0 | 28 | 2 |
G:=sub<GL(4,GF(197))| [1,0,0,0,0,196,0,0,0,0,1,0,0,0,0,1],[196,0,0,0,0,196,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,195,176,0,0,21,23],[196,0,0,0,0,1,0,0,0,0,195,28,0,0,21,2] >;
C22×D49 in GAP, Magma, Sage, TeX
C_2^2\times D_{49} % in TeX
G:=Group("C2^2xD49"); // GroupNames label
G:=SmallGroup(392,12);
// by ID
G=gap.SmallGroup(392,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,2083,858,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^49=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export