metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C49⋊2D4, C22⋊D49, D98⋊2C2, Dic49⋊C2, C2.5D98, C14.10D14, C98.5C22, (C2×C98)⋊2C2, C7.(C7⋊D4), (C2×C14).2D7, SmallGroup(392,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C49⋊D4
G = < a,b,c | a49=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C49⋊D4
►On 196 points
Generators in S196
(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)
(1 145 50 182)(2 144 51 181)(3 143 52 180)(4 142 53 179)(5 141 54 178)(6 140 55 177)(7 139 56 176)(8 138 57 175)(9 137 58 174)(10 136 59 173)(11 135 60 172)(12 134 61 171)(13 133 62 170)(14 132 63 169)(15 131 64 168)(16 130 65 167)(17 129 66 166)(18 128 67 165)(19 127 68 164)(20 126 69 163)(21 125 70 162)(22 124 71 161)(23 123 72 160)(24 122 73 159)(25 121 74 158)(26 120 75 157)(27 119 76 156)(28 118 77 155)(29 117 78 154)(30 116 79 153)(31 115 80 152)(32 114 81 151)(33 113 82 150)(34 112 83 149)(35 111 84 148)(36 110 85 196)(37 109 86 195)(38 108 87 194)(39 107 88 193)(40 106 89 192)(41 105 90 191)(42 104 91 190)(43 103 92 189)(44 102 93 188)(45 101 94 187)(46 100 95 186)(47 99 96 185)(48 147 97 184)(49 146 98 183)
(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(99 179)(100 178)(101 177)(102 176)(103 175)(104 174)(105 173)(106 172)(107 171)(108 170)(109 169)(110 168)(111 167)(112 166)(113 165)(114 164)(115 163)(116 162)(117 161)(118 160)(119 159)(120 158)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 196)(132 195)(133 194)(134 193)(135 192)(136 191)(137 190)(138 189)(139 188)(140 187)(141 186)(142 185)(143 184)(144 183)(145 182)(146 181)(147 180)
G:=sub<Sym(196)| (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), (1,145,50,182)(2,144,51,181)(3,143,52,180)(4,142,53,179)(5,141,54,178)(6,140,55,177)(7,139,56,176)(8,138,57,175)(9,137,58,174)(10,136,59,173)(11,135,60,172)(12,134,61,171)(13,133,62,170)(14,132,63,169)(15,131,64,168)(16,130,65,167)(17,129,66,166)(18,128,67,165)(19,127,68,164)(20,126,69,163)(21,125,70,162)(22,124,71,161)(23,123,72,160)(24,122,73,159)(25,121,74,158)(26,120,75,157)(27,119,76,156)(28,118,77,155)(29,117,78,154)(30,116,79,153)(31,115,80,152)(32,114,81,151)(33,113,82,150)(34,112,83,149)(35,111,84,148)(36,110,85,196)(37,109,86,195)(38,108,87,194)(39,107,88,193)(40,106,89,192)(41,105,90,191)(42,104,91,190)(43,103,92,189)(44,102,93,188)(45,101,94,187)(46,100,95,186)(47,99,96,185)(48,147,97,184)(49,146,98,183), (2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,196)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)(140,187)(141,186)(142,185)(143,184)(144,183)(145,182)(146,181)(147,180)>;
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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,145,50,182)(2,144,51,181)(3,143,52,180)(4,142,53,179)(5,141,54,178)(6,140,55,177)(7,139,56,176)(8,138,57,175)(9,137,58,174)(10,136,59,173)(11,135,60,172)(12,134,61,171)(13,133,62,170)(14,132,63,169)(15,131,64,168)(16,130,65,167)(17,129,66,166)(18,128,67,165)(19,127,68,164)(20,126,69,163)(21,125,70,162)(22,124,71,161)(23,123,72,160)(24,122,73,159)(25,121,74,158)(26,120,75,157)(27,119,76,156)(28,118,77,155)(29,117,78,154)(30,116,79,153)(31,115,80,152)(32,114,81,151)(33,113,82,150)(34,112,83,149)(35,111,84,148)(36,110,85,196)(37,109,86,195)(38,108,87,194)(39,107,88,193)(40,106,89,192)(41,105,90,191)(42,104,91,190)(43,103,92,189)(44,102,93,188)(45,101,94,187)(46,100,95,186)(47,99,96,185)(48,147,97,184)(49,146,98,183), (2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,196)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)(140,187)(141,186)(142,185)(143,184)(144,183)(145,182)(146,181)(147,180) );
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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)], [(1,145,50,182),(2,144,51,181),(3,143,52,180),(4,142,53,179),(5,141,54,178),(6,140,55,177),(7,139,56,176),(8,138,57,175),(9,137,58,174),(10,136,59,173),(11,135,60,172),(12,134,61,171),(13,133,62,170),(14,132,63,169),(15,131,64,168),(16,130,65,167),(17,129,66,166),(18,128,67,165),(19,127,68,164),(20,126,69,163),(21,125,70,162),(22,124,71,161),(23,123,72,160),(24,122,73,159),(25,121,74,158),(26,120,75,157),(27,119,76,156),(28,118,77,155),(29,117,78,154),(30,116,79,153),(31,115,80,152),(32,114,81,151),(33,113,82,150),(34,112,83,149),(35,111,84,148),(36,110,85,196),(37,109,86,195),(38,108,87,194),(39,107,88,193),(40,106,89,192),(41,105,90,191),(42,104,91,190),(43,103,92,189),(44,102,93,188),(45,101,94,187),(46,100,95,186),(47,99,96,185),(48,147,97,184),(49,146,98,183)], [(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(51,98),(52,97),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,90),(60,89),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(99,179),(100,178),(101,177),(102,176),(103,175),(104,174),(105,173),(106,172),(107,171),(108,170),(109,169),(110,168),(111,167),(112,166),(113,165),(114,164),(115,163),(116,162),(117,161),(118,160),(119,159),(120,158),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,196),(132,195),(133,194),(134,193),(135,192),(136,191),(137,190),(138,189),(139,188),(140,187),(141,186),(142,185),(143,184),(144,183),(145,182),(146,181),(147,180)]])
101 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 14A | ··· | 14I | 49A | ··· | 49U | 98A | ··· | 98BK |
| order | 1 | 2 | 2 | 2 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 49 | ··· | 49 | 98 | ··· | 98 |
| size | 1 | 1 | 2 | 98 | 98 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D14 | C7⋊D4 | D49 | D98 | C49⋊D4 |
| kernel | C49⋊D4 | Dic49 | D98 | C2×C98 | C49 | C2×C14 | C14 | C7 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 21 | 21 | 42 |
Matrix representation of C49⋊D4 ►in GL2(𝔽197) generated by
| 161 | 169 |
| 28 | 71 |
| 89 | 146 |
| 24 | 108 |
| 1 | 0 |
| 158 | 196 |
G:=sub<GL(2,GF(197))| [161,28,169,71],[89,24,146,108],[1,158,0,196] >;
C49⋊D4 in GAP, Magma, Sage, TeX
C_{49}\rtimes D_4 % in TeX
G:=Group("C49:D4"); // GroupNames label
G:=SmallGroup(392,7);
// by ID
G=gap.SmallGroup(392,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,61,2083,858,8404]);
// Polycyclic
G:=Group<a,b,c|a^49=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export