direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D49, D98.C2, C196⋊2C2, C28.5D7, C2.1D98, C14.7D14, Dic49⋊2C2, C98.2C22, C7.(C4×D7), C49⋊1(C2×C4), SmallGroup(392,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C4×D49 |
Generators and relations for C4×D49
G = < a,b,c | a4=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D49
►On 196 points
Generators in S196
(1 177 73 147)(2 178 74 99)(3 179 75 100)(4 180 76 101)(5 181 77 102)(6 182 78 103)(7 183 79 104)(8 184 80 105)(9 185 81 106)(10 186 82 107)(11 187 83 108)(12 188 84 109)(13 189 85 110)(14 190 86 111)(15 191 87 112)(16 192 88 113)(17 193 89 114)(18 194 90 115)(19 195 91 116)(20 196 92 117)(21 148 93 118)(22 149 94 119)(23 150 95 120)(24 151 96 121)(25 152 97 122)(26 153 98 123)(27 154 50 124)(28 155 51 125)(29 156 52 126)(30 157 53 127)(31 158 54 128)(32 159 55 129)(33 160 56 130)(34 161 57 131)(35 162 58 132)(36 163 59 133)(37 164 60 134)(38 165 61 135)(39 166 62 136)(40 167 63 137)(41 168 64 138)(42 169 65 139)(43 170 66 140)(44 171 67 141)(45 172 68 142)(46 173 69 143)(47 174 70 144)(48 175 71 145)(49 176 72 146)
(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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 168)(107 167)(108 166)(109 165)(110 164)(111 163)(112 162)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 182)(142 181)(143 180)(144 179)(145 178)(146 177)(147 176)
G:=sub<Sym(196)| (1,177,73,147)(2,178,74,99)(3,179,75,100)(4,180,76,101)(5,181,77,102)(6,182,78,103)(7,183,79,104)(8,184,80,105)(9,185,81,106)(10,186,82,107)(11,187,83,108)(12,188,84,109)(13,189,85,110)(14,190,86,111)(15,191,87,112)(16,192,88,113)(17,193,89,114)(18,194,90,115)(19,195,91,116)(20,196,92,117)(21,148,93,118)(22,149,94,119)(23,150,95,120)(24,151,96,121)(25,152,97,122)(26,153,98,123)(27,154,50,124)(28,155,51,125)(29,156,52,126)(30,157,53,127)(31,158,54,128)(32,159,55,129)(33,160,56,130)(34,161,57,131)(35,162,58,132)(36,163,59,133)(37,164,60,134)(38,165,61,135)(39,166,62,136)(40,167,63,137)(41,168,64,138)(42,169,65,139)(43,170,66,140)(44,171,67,141)(45,172,68,142)(46,173,69,143)(47,174,70,144)(48,175,71,145)(49,176,72,146), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,98)(25,97)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)>;
G:=Group( (1,177,73,147)(2,178,74,99)(3,179,75,100)(4,180,76,101)(5,181,77,102)(6,182,78,103)(7,183,79,104)(8,184,80,105)(9,185,81,106)(10,186,82,107)(11,187,83,108)(12,188,84,109)(13,189,85,110)(14,190,86,111)(15,191,87,112)(16,192,88,113)(17,193,89,114)(18,194,90,115)(19,195,91,116)(20,196,92,117)(21,148,93,118)(22,149,94,119)(23,150,95,120)(24,151,96,121)(25,152,97,122)(26,153,98,123)(27,154,50,124)(28,155,51,125)(29,156,52,126)(30,157,53,127)(31,158,54,128)(32,159,55,129)(33,160,56,130)(34,161,57,131)(35,162,58,132)(36,163,59,133)(37,164,60,134)(38,165,61,135)(39,166,62,136)(40,167,63,137)(41,168,64,138)(42,169,65,139)(43,170,66,140)(44,171,67,141)(45,172,68,142)(46,173,69,143)(47,174,70,144)(48,175,71,145)(49,176,72,146), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,98)(25,97)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176) );
G=PermutationGroup([[(1,177,73,147),(2,178,74,99),(3,179,75,100),(4,180,76,101),(5,181,77,102),(6,182,78,103),(7,183,79,104),(8,184,80,105),(9,185,81,106),(10,186,82,107),(11,187,83,108),(12,188,84,109),(13,189,85,110),(14,190,86,111),(15,191,87,112),(16,192,88,113),(17,193,89,114),(18,194,90,115),(19,195,91,116),(20,196,92,117),(21,148,93,118),(22,149,94,119),(23,150,95,120),(24,151,96,121),(25,152,97,122),(26,153,98,123),(27,154,50,124),(28,155,51,125),(29,156,52,126),(30,157,53,127),(31,158,54,128),(32,159,55,129),(33,160,56,130),(34,161,57,131),(35,162,58,132),(36,163,59,133),(37,164,60,134),(38,165,61,135),(39,166,62,136),(40,167,63,137),(41,168,64,138),(42,169,65,139),(43,170,66,140),(44,171,67,141),(45,172,68,142),(46,173,69,143),(47,174,70,144),(48,175,71,145),(49,176,72,146)], [(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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,51),(23,50),(24,98),(25,97),(26,96),(27,95),(28,94),(29,93),(30,92),(31,91),(32,90),(33,89),(34,88),(35,87),(36,86),(37,85),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(99,175),(100,174),(101,173),(102,172),(103,171),(104,170),(105,169),(106,168),(107,167),(108,166),(109,165),(110,164),(111,163),(112,162),(113,161),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,154),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(127,196),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,182),(142,181),(143,180),(144,179),(145,178),(146,177),(147,176)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 14A | 14B | 14C | 28A | ··· | 28F | 49A | ··· | 49U | 98A | ··· | 98U | 196A | ··· | 196AP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 49 | 49 | 1 | 1 | 49 | 49 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D7 | D14 | C4×D7 | D49 | D98 | C4×D49 |
| kernel | C4×D49 | Dic49 | C196 | D98 | D49 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 6 | 21 | 21 | 42 |
Matrix representation of C4×D49 ►in GL2(𝔽197) generated by
| 183 | 0 |
| 0 | 183 |
| 126 | 54 |
| 28 | 98 |
| 109 | 17 |
| 66 | 88 |
G:=sub<GL(2,GF(197))| [183,0,0,183],[126,28,54,98],[109,66,17,88] >;
C4×D49 in GAP, Magma, Sage, TeX
C_4\times D_{49} % in TeX
G:=Group("C4xD49"); // GroupNames label
G:=SmallGroup(392,4);
// by ID
G=gap.SmallGroup(392,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,26,2083,858,8404]);
// Polycyclic
G:=Group<a,b,c|a^4=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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