metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C57⋊7D4, D114⋊2C2, C6.12D38, C38.12D6, C2.5D114, C22⋊2D57, Dic57⋊1C2, C114.12C22, (C2×C38)⋊4S3, (C2×C6)⋊2D19, (C2×C114)⋊2C2, C3⋊3(C19⋊D4), C19⋊3(C3⋊D4), SmallGroup(456,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C57⋊7D4
G = < a,b,c | a57=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C57⋊7D4
►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)
(1 129 98 186)(2 128 99 185)(3 127 100 184)(4 126 101 183)(5 125 102 182)(6 124 103 181)(7 123 104 180)(8 122 105 179)(9 121 106 178)(10 120 107 177)(11 119 108 176)(12 118 109 175)(13 117 110 174)(14 116 111 173)(15 115 112 172)(16 171 113 228)(17 170 114 227)(18 169 58 226)(19 168 59 225)(20 167 60 224)(21 166 61 223)(22 165 62 222)(23 164 63 221)(24 163 64 220)(25 162 65 219)(26 161 66 218)(27 160 67 217)(28 159 68 216)(29 158 69 215)(30 157 70 214)(31 156 71 213)(32 155 72 212)(33 154 73 211)(34 153 74 210)(35 152 75 209)(36 151 76 208)(37 150 77 207)(38 149 78 206)(39 148 79 205)(40 147 80 204)(41 146 81 203)(42 145 82 202)(43 144 83 201)(44 143 84 200)(45 142 85 199)(46 141 86 198)(47 140 87 197)(48 139 88 196)(49 138 89 195)(50 137 90 194)(51 136 91 193)(52 135 92 192)(53 134 93 191)(54 133 94 190)(55 132 95 189)(56 131 96 188)(57 130 97 187)
(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(82 114)(83 113)(84 112)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(115 200)(116 199)(117 198)(118 197)(119 196)(120 195)(121 194)(122 193)(123 192)(124 191)(125 190)(126 189)(127 188)(128 187)(129 186)(130 185)(131 184)(132 183)(133 182)(134 181)(135 180)(136 179)(137 178)(138 177)(139 176)(140 175)(141 174)(142 173)(143 172)(144 228)(145 227)(146 226)(147 225)(148 224)(149 223)(150 222)(151 221)(152 220)(153 219)(154 218)(155 217)(156 216)(157 215)(158 214)(159 213)(160 212)(161 211)(162 210)(163 209)(164 208)(165 207)(166 206)(167 205)(168 204)(169 203)(170 202)(171 201)
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), (1,129,98,186)(2,128,99,185)(3,127,100,184)(4,126,101,183)(5,125,102,182)(6,124,103,181)(7,123,104,180)(8,122,105,179)(9,121,106,178)(10,120,107,177)(11,119,108,176)(12,118,109,175)(13,117,110,174)(14,116,111,173)(15,115,112,172)(16,171,113,228)(17,170,114,227)(18,169,58,226)(19,168,59,225)(20,167,60,224)(21,166,61,223)(22,165,62,222)(23,164,63,221)(24,163,64,220)(25,162,65,219)(26,161,66,218)(27,160,67,217)(28,159,68,216)(29,158,69,215)(30,157,70,214)(31,156,71,213)(32,155,72,212)(33,154,73,211)(34,153,74,210)(35,152,75,209)(36,151,76,208)(37,150,77,207)(38,149,78,206)(39,148,79,205)(40,147,80,204)(41,146,81,203)(42,145,82,202)(43,144,83,201)(44,143,84,200)(45,142,85,199)(46,141,86,198)(47,140,87,197)(48,139,88,196)(49,138,89,195)(50,137,90,194)(51,136,91,193)(52,135,92,192)(53,134,93,191)(54,133,94,190)(55,132,95,189)(56,131,96,188)(57,130,97,187), (2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(82,114)(83,113)(84,112)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,194)(122,193)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(133,182)(134,181)(135,180)(136,179)(137,178)(138,177)(139,176)(140,175)(141,174)(142,173)(143,172)(144,228)(145,227)(146,226)(147,225)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)(171,201)>;
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), (1,129,98,186)(2,128,99,185)(3,127,100,184)(4,126,101,183)(5,125,102,182)(6,124,103,181)(7,123,104,180)(8,122,105,179)(9,121,106,178)(10,120,107,177)(11,119,108,176)(12,118,109,175)(13,117,110,174)(14,116,111,173)(15,115,112,172)(16,171,113,228)(17,170,114,227)(18,169,58,226)(19,168,59,225)(20,167,60,224)(21,166,61,223)(22,165,62,222)(23,164,63,221)(24,163,64,220)(25,162,65,219)(26,161,66,218)(27,160,67,217)(28,159,68,216)(29,158,69,215)(30,157,70,214)(31,156,71,213)(32,155,72,212)(33,154,73,211)(34,153,74,210)(35,152,75,209)(36,151,76,208)(37,150,77,207)(38,149,78,206)(39,148,79,205)(40,147,80,204)(41,146,81,203)(42,145,82,202)(43,144,83,201)(44,143,84,200)(45,142,85,199)(46,141,86,198)(47,140,87,197)(48,139,88,196)(49,138,89,195)(50,137,90,194)(51,136,91,193)(52,135,92,192)(53,134,93,191)(54,133,94,190)(55,132,95,189)(56,131,96,188)(57,130,97,187), (2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(82,114)(83,113)(84,112)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,194)(122,193)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(133,182)(134,181)(135,180)(136,179)(137,178)(138,177)(139,176)(140,175)(141,174)(142,173)(143,172)(144,228)(145,227)(146,226)(147,225)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)(171,201) );
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)], [(1,129,98,186),(2,128,99,185),(3,127,100,184),(4,126,101,183),(5,125,102,182),(6,124,103,181),(7,123,104,180),(8,122,105,179),(9,121,106,178),(10,120,107,177),(11,119,108,176),(12,118,109,175),(13,117,110,174),(14,116,111,173),(15,115,112,172),(16,171,113,228),(17,170,114,227),(18,169,58,226),(19,168,59,225),(20,167,60,224),(21,166,61,223),(22,165,62,222),(23,164,63,221),(24,163,64,220),(25,162,65,219),(26,161,66,218),(27,160,67,217),(28,159,68,216),(29,158,69,215),(30,157,70,214),(31,156,71,213),(32,155,72,212),(33,154,73,211),(34,153,74,210),(35,152,75,209),(36,151,76,208),(37,150,77,207),(38,149,78,206),(39,148,79,205),(40,147,80,204),(41,146,81,203),(42,145,82,202),(43,144,83,201),(44,143,84,200),(45,142,85,199),(46,141,86,198),(47,140,87,197),(48,139,88,196),(49,138,89,195),(50,137,90,194),(51,136,91,193),(52,135,92,192),(53,134,93,191),(54,133,94,190),(55,132,95,189),(56,131,96,188),(57,130,97,187)], [(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(82,114),(83,113),(84,112),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(115,200),(116,199),(117,198),(118,197),(119,196),(120,195),(121,194),(122,193),(123,192),(124,191),(125,190),(126,189),(127,188),(128,187),(129,186),(130,185),(131,184),(132,183),(133,182),(134,181),(135,180),(136,179),(137,178),(138,177),(139,176),(140,175),(141,174),(142,173),(143,172),(144,228),(145,227),(146,226),(147,225),(148,224),(149,223),(150,222),(151,221),(152,220),(153,219),(154,218),(155,217),(156,216),(157,215),(158,214),(159,213),(160,212),(161,211),(162,210),(163,209),(164,208),(165,207),(166,206),(167,205),(168,204),(169,203),(170,202),(171,201)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38AA | 57A | ··· | 57R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 2 | 114 | 2 | 114 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D19 | D38 | D57 | C19⋊D4 | D114 | C57⋊7D4 |
| kernel | C57⋊7D4 | Dic57 | D114 | C2×C114 | C2×C38 | C57 | C38 | C19 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C57⋊7D4 ►in GL2(𝔽229) generated by
| 156 | 180 |
| 49 | 196 |
| 82 | 178 |
| 69 | 147 |
| 1 | 0 |
| 102 | 228 |
G:=sub<GL(2,GF(229))| [156,49,180,196],[82,69,178,147],[1,102,0,228] >;
C57⋊7D4 in GAP, Magma, Sage, TeX
C_{57}\rtimes_7D_4 % in TeX
G:=Group("C57:7D4"); // GroupNames label
G:=SmallGroup(456,38);
// by ID
G=gap.SmallGroup(456,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,61,323,10804]);
// Polycyclic
G:=Group<a,b,c|a^57=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export