direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D51, C68⋊2S3, C204⋊2C2, C12⋊2D17, C6.9D34, C34.9D6, C2.1D102, Dic51⋊5C2, D102.2C2, C102.9C22, C17⋊3(C4×S3), C51⋊7(C2×C4), C3⋊2(C4×D17), SmallGroup(408,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C4×D51 |
Generators and relations for C4×D51
G = < a,b,c | a4=b51=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D51
►On 204 points
Generators in S204
(1 167 68 108)(2 168 69 109)(3 169 70 110)(4 170 71 111)(5 171 72 112)(6 172 73 113)(7 173 74 114)(8 174 75 115)(9 175 76 116)(10 176 77 117)(11 177 78 118)(12 178 79 119)(13 179 80 120)(14 180 81 121)(15 181 82 122)(16 182 83 123)(17 183 84 124)(18 184 85 125)(19 185 86 126)(20 186 87 127)(21 187 88 128)(22 188 89 129)(23 189 90 130)(24 190 91 131)(25 191 92 132)(26 192 93 133)(27 193 94 134)(28 194 95 135)(29 195 96 136)(30 196 97 137)(31 197 98 138)(32 198 99 139)(33 199 100 140)(34 200 101 141)(35 201 102 142)(36 202 52 143)(37 203 53 144)(38 204 54 145)(39 154 55 146)(40 155 56 147)(41 156 57 148)(42 157 58 149)(43 158 59 150)(44 159 60 151)(45 160 61 152)(46 161 62 153)(47 162 63 103)(48 163 64 104)(49 164 65 105)(50 165 66 106)(51 166 67 107)
(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)(154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 102)(18 101)(19 100)(20 99)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(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)(121 204)(122 203)(123 202)(124 201)(125 200)(126 199)(127 198)(128 197)(129 196)(130 195)(131 194)(132 193)(133 192)(134 191)(135 190)(136 189)(137 188)(138 187)(139 186)(140 185)(141 184)(142 183)(143 182)(144 181)(145 180)(146 179)(147 178)(148 177)(149 176)(150 175)(151 174)(152 173)(153 172)
G:=sub<Sym(204)| (1,167,68,108)(2,168,69,109)(3,169,70,110)(4,170,71,111)(5,171,72,112)(6,172,73,113)(7,173,74,114)(8,174,75,115)(9,175,76,116)(10,176,77,117)(11,177,78,118)(12,178,79,119)(13,179,80,120)(14,180,81,121)(15,181,82,122)(16,182,83,123)(17,183,84,124)(18,184,85,125)(19,185,86,126)(20,186,87,127)(21,187,88,128)(22,188,89,129)(23,189,90,130)(24,190,91,131)(25,191,92,132)(26,192,93,133)(27,193,94,134)(28,194,95,135)(29,195,96,136)(30,196,97,137)(31,197,98,138)(32,198,99,139)(33,199,100,140)(34,200,101,141)(35,201,102,142)(36,202,52,143)(37,203,53,144)(38,204,54,145)(39,154,55,146)(40,155,56,147)(41,156,57,148)(42,157,58,149)(43,158,59,150)(44,159,60,151)(45,160,61,152)(46,161,62,153)(47,162,63,103)(48,163,64,104)(49,164,65,105)(50,165,66,106)(51,166,67,107), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,102)(18,101)(19,100)(20,99)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(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)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187)(139,186)(140,185)(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)>;
G:=Group( (1,167,68,108)(2,168,69,109)(3,169,70,110)(4,170,71,111)(5,171,72,112)(6,172,73,113)(7,173,74,114)(8,174,75,115)(9,175,76,116)(10,176,77,117)(11,177,78,118)(12,178,79,119)(13,179,80,120)(14,180,81,121)(15,181,82,122)(16,182,83,123)(17,183,84,124)(18,184,85,125)(19,185,86,126)(20,186,87,127)(21,187,88,128)(22,188,89,129)(23,189,90,130)(24,190,91,131)(25,191,92,132)(26,192,93,133)(27,193,94,134)(28,194,95,135)(29,195,96,136)(30,196,97,137)(31,197,98,138)(32,198,99,139)(33,199,100,140)(34,200,101,141)(35,201,102,142)(36,202,52,143)(37,203,53,144)(38,204,54,145)(39,154,55,146)(40,155,56,147)(41,156,57,148)(42,157,58,149)(43,158,59,150)(44,159,60,151)(45,160,61,152)(46,161,62,153)(47,162,63,103)(48,163,64,104)(49,164,65,105)(50,165,66,106)(51,166,67,107), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,102)(18,101)(19,100)(20,99)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(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)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187)(139,186)(140,185)(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172) );
G=PermutationGroup([[(1,167,68,108),(2,168,69,109),(3,169,70,110),(4,170,71,111),(5,171,72,112),(6,172,73,113),(7,173,74,114),(8,174,75,115),(9,175,76,116),(10,176,77,117),(11,177,78,118),(12,178,79,119),(13,179,80,120),(14,180,81,121),(15,181,82,122),(16,182,83,123),(17,183,84,124),(18,184,85,125),(19,185,86,126),(20,186,87,127),(21,187,88,128),(22,188,89,129),(23,189,90,130),(24,190,91,131),(25,191,92,132),(26,192,93,133),(27,193,94,134),(28,194,95,135),(29,195,96,136),(30,196,97,137),(31,197,98,138),(32,198,99,139),(33,199,100,140),(34,200,101,141),(35,201,102,142),(36,202,52,143),(37,203,53,144),(38,204,54,145),(39,154,55,146),(40,155,56,147),(41,156,57,148),(42,157,58,149),(43,158,59,150),(44,159,60,151),(45,160,61,152),(46,161,62,153),(47,162,63,103),(48,163,64,104),(49,164,65,105),(50,165,66,106),(51,166,67,107)], [(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),(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,102),(18,101),(19,100),(20,99),(21,98),(22,97),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,90),(30,89),(31,88),(32,87),(33,86),(34,85),(35,84),(36,83),(37,82),(38,81),(39,80),(40,79),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,72),(48,71),(49,70),(50,69),(51,68),(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),(121,204),(122,203),(123,202),(124,201),(125,200),(126,199),(127,198),(128,197),(129,196),(130,195),(131,194),(132,193),(133,192),(134,191),(135,190),(136,189),(137,188),(138,187),(139,186),(140,185),(141,184),(142,183),(143,182),(144,181),(145,180),(146,179),(147,178),(148,177),(149,176),(150,175),(151,174),(152,173),(153,172)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 51 | 51 | 2 | 1 | 1 | 51 | 51 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D17 | D34 | D51 | C4×D17 | D102 | C4×D51 |
| kernel | C4×D51 | Dic51 | C204 | D102 | D51 | C68 | C34 | C17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C4×D51 ►in GL2(𝔽409) generated by
| 266 | 0 |
| 0 | 266 |
| 19 | 252 |
| 157 | 145 |
| 390 | 157 |
| 352 | 19 |
G:=sub<GL(2,GF(409))| [266,0,0,266],[19,157,252,145],[390,352,157,19] >;
C4×D51 in GAP, Magma, Sage, TeX
C_4\times D_{51} % in TeX
G:=Group("C4xD51"); // GroupNames label
G:=SmallGroup(408,26);
// by ID
G=gap.SmallGroup(408,26);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,26,323,9604]);
// Polycyclic
G:=Group<a,b,c|a^4=b^51=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export