direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D57, C76⋊2S3, C228⋊2C2, C12⋊2D19, C6.9D38, C38.9D6, C2.1D114, Dic57⋊5C2, D114.2C2, C114.9C22, C19⋊2(C4×S3), C57⋊4(C2×C4), C3⋊2(C4×D19), SmallGroup(456,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — C4×D57 |
Generators and relations for C4×D57
G = < a,b,c | a4=b57=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D57
►On 228 points
Generators in S228
(1 173 96 167)(2 174 97 168)(3 175 98 169)(4 176 99 170)(5 177 100 171)(6 178 101 115)(7 179 102 116)(8 180 103 117)(9 181 104 118)(10 182 105 119)(11 183 106 120)(12 184 107 121)(13 185 108 122)(14 186 109 123)(15 187 110 124)(16 188 111 125)(17 189 112 126)(18 190 113 127)(19 191 114 128)(20 192 58 129)(21 193 59 130)(22 194 60 131)(23 195 61 132)(24 196 62 133)(25 197 63 134)(26 198 64 135)(27 199 65 136)(28 200 66 137)(29 201 67 138)(30 202 68 139)(31 203 69 140)(32 204 70 141)(33 205 71 142)(34 206 72 143)(35 207 73 144)(36 208 74 145)(37 209 75 146)(38 210 76 147)(39 211 77 148)(40 212 78 149)(41 213 79 150)(42 214 80 151)(43 215 81 152)(44 216 82 153)(45 217 83 154)(46 218 84 155)(47 219 85 156)(48 220 86 157)(49 221 87 158)(50 222 88 159)(51 223 89 160)(52 224 90 161)(53 225 91 162)(54 226 92 163)(55 227 93 164)(56 228 94 165)(57 172 95 166)
(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)(172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)
(1 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 84)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 96)(115 224)(116 223)(117 222)(118 221)(119 220)(120 219)(121 218)(122 217)(123 216)(124 215)(125 214)(126 213)(127 212)(128 211)(129 210)(130 209)(131 208)(132 207)(133 206)(134 205)(135 204)(136 203)(137 202)(138 201)(139 200)(140 199)(141 198)(142 197)(143 196)(144 195)(145 194)(146 193)(147 192)(148 191)(149 190)(150 189)(151 188)(152 187)(153 186)(154 185)(155 184)(156 183)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 228)(169 227)(170 226)(171 225)
G:=sub<Sym(228)| (1,173,96,167)(2,174,97,168)(3,175,98,169)(4,176,99,170)(5,177,100,171)(6,178,101,115)(7,179,102,116)(8,180,103,117)(9,181,104,118)(10,182,105,119)(11,183,106,120)(12,184,107,121)(13,185,108,122)(14,186,109,123)(15,187,110,124)(16,188,111,125)(17,189,112,126)(18,190,113,127)(19,191,114,128)(20,192,58,129)(21,193,59,130)(22,194,60,131)(23,195,61,132)(24,196,62,133)(25,197,63,134)(26,198,64,135)(27,199,65,136)(28,200,66,137)(29,201,67,138)(30,202,68,139)(31,203,69,140)(32,204,70,141)(33,205,71,142)(34,206,72,143)(35,207,73,144)(36,208,74,145)(37,209,75,146)(38,210,76,147)(39,211,77,148)(40,212,78,149)(41,213,79,150)(42,214,80,151)(43,215,81,152)(44,216,82,153)(45,217,83,154)(46,218,84,155)(47,219,85,156)(48,220,86,157)(49,221,87,158)(50,222,88,159)(51,223,89,160)(52,224,90,161)(53,225,91,162)(54,226,92,163)(55,227,93,164)(56,228,94,165)(57,172,95,166), (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)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(127,212)(128,211)(129,210)(130,209)(131,208)(132,207)(133,206)(134,205)(135,204)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,189)(151,188)(152,187)(153,186)(154,185)(155,184)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,228)(169,227)(170,226)(171,225)>;
G:=Group( (1,173,96,167)(2,174,97,168)(3,175,98,169)(4,176,99,170)(5,177,100,171)(6,178,101,115)(7,179,102,116)(8,180,103,117)(9,181,104,118)(10,182,105,119)(11,183,106,120)(12,184,107,121)(13,185,108,122)(14,186,109,123)(15,187,110,124)(16,188,111,125)(17,189,112,126)(18,190,113,127)(19,191,114,128)(20,192,58,129)(21,193,59,130)(22,194,60,131)(23,195,61,132)(24,196,62,133)(25,197,63,134)(26,198,64,135)(27,199,65,136)(28,200,66,137)(29,201,67,138)(30,202,68,139)(31,203,69,140)(32,204,70,141)(33,205,71,142)(34,206,72,143)(35,207,73,144)(36,208,74,145)(37,209,75,146)(38,210,76,147)(39,211,77,148)(40,212,78,149)(41,213,79,150)(42,214,80,151)(43,215,81,152)(44,216,82,153)(45,217,83,154)(46,218,84,155)(47,219,85,156)(48,220,86,157)(49,221,87,158)(50,222,88,159)(51,223,89,160)(52,224,90,161)(53,225,91,162)(54,226,92,163)(55,227,93,164)(56,228,94,165)(57,172,95,166), (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)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(127,212)(128,211)(129,210)(130,209)(131,208)(132,207)(133,206)(134,205)(135,204)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,189)(151,188)(152,187)(153,186)(154,185)(155,184)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,228)(169,227)(170,226)(171,225) );
G=PermutationGroup([[(1,173,96,167),(2,174,97,168),(3,175,98,169),(4,176,99,170),(5,177,100,171),(6,178,101,115),(7,179,102,116),(8,180,103,117),(9,181,104,118),(10,182,105,119),(11,183,106,120),(12,184,107,121),(13,185,108,122),(14,186,109,123),(15,187,110,124),(16,188,111,125),(17,189,112,126),(18,190,113,127),(19,191,114,128),(20,192,58,129),(21,193,59,130),(22,194,60,131),(23,195,61,132),(24,196,62,133),(25,197,63,134),(26,198,64,135),(27,199,65,136),(28,200,66,137),(29,201,67,138),(30,202,68,139),(31,203,69,140),(32,204,70,141),(33,205,71,142),(34,206,72,143),(35,207,73,144),(36,208,74,145),(37,209,75,146),(38,210,76,147),(39,211,77,148),(40,212,78,149),(41,213,79,150),(42,214,80,151),(43,215,81,152),(44,216,82,153),(45,217,83,154),(46,218,84,155),(47,219,85,156),(48,220,86,157),(49,221,87,158),(50,222,88,159),(51,223,89,160),(52,224,90,161),(53,225,91,162),(54,226,92,163),(55,227,93,164),(56,228,94,165),(57,172,95,166)], [(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),(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,90),(7,89),(8,88),(9,87),(10,86),(11,85),(12,84),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,114),(40,113),(41,112),(42,111),(43,110),(44,109),(45,108),(46,107),(47,106),(48,105),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,96),(115,224),(116,223),(117,222),(118,221),(119,220),(120,219),(121,218),(122,217),(123,216),(124,215),(125,214),(126,213),(127,212),(128,211),(129,210),(130,209),(131,208),(132,207),(133,206),(134,205),(135,204),(136,203),(137,202),(138,201),(139,200),(140,199),(141,198),(142,197),(143,196),(144,195),(145,194),(146,193),(147,192),(148,191),(149,190),(150,189),(151,188),(152,187),(153,186),(154,185),(155,184),(156,183),(157,182),(158,181),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,228),(169,227),(170,226),(171,225)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 57 | 57 | 2 | 1 | 1 | 57 | 57 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 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 | D19 | D38 | D57 | C4×D19 | D114 | C4×D57 |
| kernel | C4×D57 | Dic57 | C228 | D114 | D57 | C76 | C38 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C4×D57 ►in GL3(𝔽229) generated by
| 122 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 140 | 155 |
| 0 | 74 | 149 |
| 1 | 0 | 0 |
| 0 | 140 | 155 |
| 0 | 8 | 89 |
G:=sub<GL(3,GF(229))| [122,0,0,0,1,0,0,0,1],[1,0,0,0,140,74,0,155,149],[1,0,0,0,140,8,0,155,89] >;
C4×D57 in GAP, Magma, Sage, TeX
C_4\times D_{57} % in TeX
G:=Group("C4xD57"); // GroupNames label
G:=SmallGroup(456,35);
// by ID
G=gap.SmallGroup(456,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,26,323,10804]);
// Polycyclic
G:=Group<a,b,c|a^4=b^57=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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