direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C49, C4⋊C98, C22⋊C98, C196⋊3C2, C28.3C14, C98.6C22, C7.(C7×D4), (C7×D4).C7, (C2×C98)⋊1C2, C2.1(C2×C98), (C2×C14).1C14, C14.6(C2×C14), SmallGroup(392,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C49
G = < a,b,c | a49=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C49
►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)
(1 137 161 92)(2 138 162 93)(3 139 163 94)(4 140 164 95)(5 141 165 96)(6 142 166 97)(7 143 167 98)(8 144 168 50)(9 145 169 51)(10 146 170 52)(11 147 171 53)(12 99 172 54)(13 100 173 55)(14 101 174 56)(15 102 175 57)(16 103 176 58)(17 104 177 59)(18 105 178 60)(19 106 179 61)(20 107 180 62)(21 108 181 63)(22 109 182 64)(23 110 183 65)(24 111 184 66)(25 112 185 67)(26 113 186 68)(27 114 187 69)(28 115 188 70)(29 116 189 71)(30 117 190 72)(31 118 191 73)(32 119 192 74)(33 120 193 75)(34 121 194 76)(35 122 195 77)(36 123 196 78)(37 124 148 79)(38 125 149 80)(39 126 150 81)(40 127 151 82)(41 128 152 83)(42 129 153 84)(43 130 154 85)(44 131 155 86)(45 132 156 87)(46 133 157 88)(47 134 158 89)(48 135 159 90)(49 136 160 91)
(50 144)(51 145)(52 146)(53 147)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)
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), (1,137,161,92)(2,138,162,93)(3,139,163,94)(4,140,164,95)(5,141,165,96)(6,142,166,97)(7,143,167,98)(8,144,168,50)(9,145,169,51)(10,146,170,52)(11,147,171,53)(12,99,172,54)(13,100,173,55)(14,101,174,56)(15,102,175,57)(16,103,176,58)(17,104,177,59)(18,105,178,60)(19,106,179,61)(20,107,180,62)(21,108,181,63)(22,109,182,64)(23,110,183,65)(24,111,184,66)(25,112,185,67)(26,113,186,68)(27,114,187,69)(28,115,188,70)(29,116,189,71)(30,117,190,72)(31,118,191,73)(32,119,192,74)(33,120,193,75)(34,121,194,76)(35,122,195,77)(36,123,196,78)(37,124,148,79)(38,125,149,80)(39,126,150,81)(40,127,151,82)(41,128,152,83)(42,129,153,84)(43,130,154,85)(44,131,155,86)(45,132,156,87)(46,133,157,88)(47,134,158,89)(48,135,159,90)(49,136,160,91), (50,144)(51,145)(52,146)(53,147)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,142)(98,143)>;
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), (1,137,161,92)(2,138,162,93)(3,139,163,94)(4,140,164,95)(5,141,165,96)(6,142,166,97)(7,143,167,98)(8,144,168,50)(9,145,169,51)(10,146,170,52)(11,147,171,53)(12,99,172,54)(13,100,173,55)(14,101,174,56)(15,102,175,57)(16,103,176,58)(17,104,177,59)(18,105,178,60)(19,106,179,61)(20,107,180,62)(21,108,181,63)(22,109,182,64)(23,110,183,65)(24,111,184,66)(25,112,185,67)(26,113,186,68)(27,114,187,69)(28,115,188,70)(29,116,189,71)(30,117,190,72)(31,118,191,73)(32,119,192,74)(33,120,193,75)(34,121,194,76)(35,122,195,77)(36,123,196,78)(37,124,148,79)(38,125,149,80)(39,126,150,81)(40,127,151,82)(41,128,152,83)(42,129,153,84)(43,130,154,85)(44,131,155,86)(45,132,156,87)(46,133,157,88)(47,134,158,89)(48,135,159,90)(49,136,160,91), (50,144)(51,145)(52,146)(53,147)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,142)(98,143) );
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)], [(1,137,161,92),(2,138,162,93),(3,139,163,94),(4,140,164,95),(5,141,165,96),(6,142,166,97),(7,143,167,98),(8,144,168,50),(9,145,169,51),(10,146,170,52),(11,147,171,53),(12,99,172,54),(13,100,173,55),(14,101,174,56),(15,102,175,57),(16,103,176,58),(17,104,177,59),(18,105,178,60),(19,106,179,61),(20,107,180,62),(21,108,181,63),(22,109,182,64),(23,110,183,65),(24,111,184,66),(25,112,185,67),(26,113,186,68),(27,114,187,69),(28,115,188,70),(29,116,189,71),(30,117,190,72),(31,118,191,73),(32,119,192,74),(33,120,193,75),(34,121,194,76),(35,122,195,77),(36,123,196,78),(37,124,148,79),(38,125,149,80),(39,126,150,81),(40,127,151,82),(41,128,152,83),(42,129,153,84),(43,130,154,85),(44,131,155,86),(45,132,156,87),(46,133,157,88),(47,134,158,89),(48,135,159,90),(49,136,160,91)], [(50,144),(51,145),(52,146),(53,147),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137),(93,138),(94,139),(95,140),(96,141),(97,142),(98,143)]])
245 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14R | 28A | ··· | 28F | 49A | ··· | 49AP | 98A | ··· | 98AP | 98AQ | ··· | 98DV | 196A | ··· | 196AP |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
245 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | C49 | C98 | C98 | D4 | C7×D4 | D4×C49 |
| kernel | D4×C49 | C196 | C2×C98 | C7×D4 | C28 | C2×C14 | D4 | C4 | C22 | C49 | C7 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 42 | 42 | 84 | 1 | 6 | 42 |
Matrix representation of D4×C49 ►in GL2(𝔽197) generated by
| 171 | 0 |
| 0 | 171 |
| 98 | 172 |
| 69 | 99 |
| 1 | 20 |
| 0 | 196 |
G:=sub<GL(2,GF(197))| [171,0,0,171],[98,69,172,99],[1,0,20,196] >;
D4×C49 in GAP, Magma, Sage, TeX
D_4\times C_{49} % in TeX
G:=Group("D4xC49"); // GroupNames label
G:=SmallGroup(392,9);
// by ID
G=gap.SmallGroup(392,9);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-7,301,222]);
// Polycyclic
G:=Group<a,b,c|a^49=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export