metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D57⋊C4, D114.C2, C6.3D38, C38.3D6, Dic19⋊2S3, Dic3⋊2D19, C114.3C22, C19⋊1(C4×S3), C57⋊3(C2×C4), C3⋊1(C4×D19), C2.3(S3×D19), (C3×Dic19)⋊2C2, (Dic3×C19)⋊2C2, SmallGroup(456,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — D57⋊C4 |
Generators and relations for D57⋊C4
G = < a,b,c | a57=b2=c4=1, bab=a-1, cac-1=a20, cbc-1=a19b >
Smallest permutation representation of D57⋊C4
►On 228 points
Generators in S228
(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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(115 202)(116 201)(117 200)(118 199)(119 198)(120 197)(121 196)(122 195)(123 194)(124 193)(125 192)(126 191)(127 190)(128 189)(129 188)(130 187)(131 186)(132 185)(133 184)(134 183)(135 182)(136 181)(137 180)(138 179)(139 178)(140 177)(141 176)(142 175)(143 174)(144 173)(145 172)(146 228)(147 227)(148 226)(149 225)(150 224)(151 223)(152 222)(153 221)(154 220)(155 219)(156 218)(157 217)(158 216)(159 215)(160 214)(161 213)(162 212)(163 211)(164 210)(165 209)(166 208)(167 207)(168 206)(169 205)(170 204)(171 203)
(1 188 88 130)(2 208 89 150)(3 228 90 170)(4 191 91 133)(5 211 92 153)(6 174 93 116)(7 194 94 136)(8 214 95 156)(9 177 96 119)(10 197 97 139)(11 217 98 159)(12 180 99 122)(13 200 100 142)(14 220 101 162)(15 183 102 125)(16 203 103 145)(17 223 104 165)(18 186 105 128)(19 206 106 148)(20 226 107 168)(21 189 108 131)(22 209 109 151)(23 172 110 171)(24 192 111 134)(25 212 112 154)(26 175 113 117)(27 195 114 137)(28 215 58 157)(29 178 59 120)(30 198 60 140)(31 218 61 160)(32 181 62 123)(33 201 63 143)(34 221 64 163)(35 184 65 126)(36 204 66 146)(37 224 67 166)(38 187 68 129)(39 207 69 149)(40 227 70 169)(41 190 71 132)(42 210 72 152)(43 173 73 115)(44 193 74 135)(45 213 75 155)(46 176 76 118)(47 196 77 138)(48 216 78 158)(49 179 79 121)(50 199 80 141)(51 219 81 161)(52 182 82 124)(53 202 83 144)(54 222 84 164)(55 185 85 127)(56 205 86 147)(57 225 87 167)
G:=sub<Sym(228)| (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), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,196)(122,195)(123,194)(124,193)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,228)(147,227)(148,226)(149,225)(150,224)(151,223)(152,222)(153,221)(154,220)(155,219)(156,218)(157,217)(158,216)(159,215)(160,214)(161,213)(162,212)(163,211)(164,210)(165,209)(166,208)(167,207)(168,206)(169,205)(170,204)(171,203), (1,188,88,130)(2,208,89,150)(3,228,90,170)(4,191,91,133)(5,211,92,153)(6,174,93,116)(7,194,94,136)(8,214,95,156)(9,177,96,119)(10,197,97,139)(11,217,98,159)(12,180,99,122)(13,200,100,142)(14,220,101,162)(15,183,102,125)(16,203,103,145)(17,223,104,165)(18,186,105,128)(19,206,106,148)(20,226,107,168)(21,189,108,131)(22,209,109,151)(23,172,110,171)(24,192,111,134)(25,212,112,154)(26,175,113,117)(27,195,114,137)(28,215,58,157)(29,178,59,120)(30,198,60,140)(31,218,61,160)(32,181,62,123)(33,201,63,143)(34,221,64,163)(35,184,65,126)(36,204,66,146)(37,224,67,166)(38,187,68,129)(39,207,69,149)(40,227,70,169)(41,190,71,132)(42,210,72,152)(43,173,73,115)(44,193,74,135)(45,213,75,155)(46,176,76,118)(47,196,77,138)(48,216,78,158)(49,179,79,121)(50,199,80,141)(51,219,81,161)(52,182,82,124)(53,202,83,144)(54,222,84,164)(55,185,85,127)(56,205,86,147)(57,225,87,167)>;
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)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,196)(122,195)(123,194)(124,193)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,228)(147,227)(148,226)(149,225)(150,224)(151,223)(152,222)(153,221)(154,220)(155,219)(156,218)(157,217)(158,216)(159,215)(160,214)(161,213)(162,212)(163,211)(164,210)(165,209)(166,208)(167,207)(168,206)(169,205)(170,204)(171,203), (1,188,88,130)(2,208,89,150)(3,228,90,170)(4,191,91,133)(5,211,92,153)(6,174,93,116)(7,194,94,136)(8,214,95,156)(9,177,96,119)(10,197,97,139)(11,217,98,159)(12,180,99,122)(13,200,100,142)(14,220,101,162)(15,183,102,125)(16,203,103,145)(17,223,104,165)(18,186,105,128)(19,206,106,148)(20,226,107,168)(21,189,108,131)(22,209,109,151)(23,172,110,171)(24,192,111,134)(25,212,112,154)(26,175,113,117)(27,195,114,137)(28,215,58,157)(29,178,59,120)(30,198,60,140)(31,218,61,160)(32,181,62,123)(33,201,63,143)(34,221,64,163)(35,184,65,126)(36,204,66,146)(37,224,67,166)(38,187,68,129)(39,207,69,149)(40,227,70,169)(41,190,71,132)(42,210,72,152)(43,173,73,115)(44,193,74,135)(45,213,75,155)(46,176,76,118)(47,196,77,138)(48,216,78,158)(49,179,79,121)(50,199,80,141)(51,219,81,161)(52,182,82,124)(53,202,83,144)(54,222,84,164)(55,185,85,127)(56,205,86,147)(57,225,87,167) );
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),(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)], [(1,87),(2,86),(3,85),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,70),(19,69),(20,68),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,114),(32,113),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(115,202),(116,201),(117,200),(118,199),(119,198),(120,197),(121,196),(122,195),(123,194),(124,193),(125,192),(126,191),(127,190),(128,189),(129,188),(130,187),(131,186),(132,185),(133,184),(134,183),(135,182),(136,181),(137,180),(138,179),(139,178),(140,177),(141,176),(142,175),(143,174),(144,173),(145,172),(146,228),(147,227),(148,226),(149,225),(150,224),(151,223),(152,222),(153,221),(154,220),(155,219),(156,218),(157,217),(158,216),(159,215),(160,214),(161,213),(162,212),(163,211),(164,210),(165,209),(166,208),(167,207),(168,206),(169,205),(170,204),(171,203)], [(1,188,88,130),(2,208,89,150),(3,228,90,170),(4,191,91,133),(5,211,92,153),(6,174,93,116),(7,194,94,136),(8,214,95,156),(9,177,96,119),(10,197,97,139),(11,217,98,159),(12,180,99,122),(13,200,100,142),(14,220,101,162),(15,183,102,125),(16,203,103,145),(17,223,104,165),(18,186,105,128),(19,206,106,148),(20,226,107,168),(21,189,108,131),(22,209,109,151),(23,172,110,171),(24,192,111,134),(25,212,112,154),(26,175,113,117),(27,195,114,137),(28,215,58,157),(29,178,59,120),(30,198,60,140),(31,218,61,160),(32,181,62,123),(33,201,63,143),(34,221,64,163),(35,184,65,126),(36,204,66,146),(37,224,67,166),(38,187,68,129),(39,207,69,149),(40,227,70,169),(41,190,71,132),(42,210,72,152),(43,173,73,115),(44,193,74,135),(45,213,75,155),(46,176,76,118),(47,196,77,138),(48,216,78,158),(49,179,79,121),(50,199,80,141),(51,219,81,161),(52,182,82,124),(53,202,83,144),(54,222,84,164),(55,185,85,127),(56,205,86,147),(57,225,87,167)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57I | 76A | ··· | 76R | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 57 | 57 | 2 | 3 | 3 | 19 | 19 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D19 | D38 | C4×D19 | S3×D19 | D57⋊C4 |
| kernel | D57⋊C4 | Dic3×C19 | C3×Dic19 | D114 | D57 | Dic19 | C38 | C19 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of D57⋊C4 ►in GL4(𝔽229) generated by
| 172 | 209 | 0 | 0 |
| 36 | 101 | 0 | 0 |
| 0 | 0 | 1 | 130 |
| 0 | 0 | 118 | 227 |
| 54 | 228 | 0 | 0 |
| 167 | 175 | 0 | 0 |
| 0 | 0 | 228 | 99 |
| 0 | 0 | 0 | 1 |
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 122 | 0 |
| 0 | 0 | 198 | 107 |
G:=sub<GL(4,GF(229))| [172,36,0,0,209,101,0,0,0,0,1,118,0,0,130,227],[54,167,0,0,228,175,0,0,0,0,228,0,0,0,99,1],[228,0,0,0,0,228,0,0,0,0,122,198,0,0,0,107] >;
D57⋊C4 in GAP, Magma, Sage, TeX
D_{57}\rtimes C_4 % in TeX
G:=Group("D57:C4"); // GroupNames label
G:=SmallGroup(456,14);
// by ID
G=gap.SmallGroup(456,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,26,168,10804]);
// Polycyclic
G:=Group<a,b,c|a^57=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^20,c*b*c^-1=a^19*b>;
// generators/relations
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