metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C57⋊1D4, D6⋊1D19, D38⋊1S3, C6.4D38, C38.4D6, Dic57⋊4C2, C114.4C22, (S3×C38)⋊1C2, (C6×D19)⋊1C2, C19⋊2(C3⋊D4), C3⋊2(C19⋊D4), C2.4(S3×D19), SmallGroup(456,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C57⋊D4
G = < a,b,c | a57=b4=c2=1, bab-1=a-1, cac=a37, cbc=b-1 >
Smallest permutation representation of C57⋊D4
►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)
(1 155 98 176)(2 154 99 175)(3 153 100 174)(4 152 101 173)(5 151 102 172)(6 150 103 228)(7 149 104 227)(8 148 105 226)(9 147 106 225)(10 146 107 224)(11 145 108 223)(12 144 109 222)(13 143 110 221)(14 142 111 220)(15 141 112 219)(16 140 113 218)(17 139 114 217)(18 138 58 216)(19 137 59 215)(20 136 60 214)(21 135 61 213)(22 134 62 212)(23 133 63 211)(24 132 64 210)(25 131 65 209)(26 130 66 208)(27 129 67 207)(28 128 68 206)(29 127 69 205)(30 126 70 204)(31 125 71 203)(32 124 72 202)(33 123 73 201)(34 122 74 200)(35 121 75 199)(36 120 76 198)(37 119 77 197)(38 118 78 196)(39 117 79 195)(40 116 80 194)(41 115 81 193)(42 171 82 192)(43 170 83 191)(44 169 84 190)(45 168 85 189)(46 167 86 188)(47 166 87 187)(48 165 88 186)(49 164 89 185)(50 163 90 184)(51 162 91 183)(52 161 92 182)(53 160 93 181)(54 159 94 180)(55 158 95 179)(56 157 96 178)(57 156 97 177)
(2 38)(3 18)(4 55)(5 35)(6 15)(7 52)(8 32)(9 12)(10 49)(11 29)(13 46)(14 26)(16 43)(17 23)(19 40)(21 57)(22 37)(24 54)(25 34)(27 51)(28 31)(30 48)(33 45)(36 42)(41 56)(44 53)(47 50)(58 100)(59 80)(61 97)(62 77)(63 114)(64 94)(65 74)(66 111)(67 91)(68 71)(69 108)(70 88)(72 105)(73 85)(75 102)(76 82)(78 99)(81 96)(83 113)(84 93)(86 110)(87 90)(89 107)(92 104)(95 101)(103 112)(106 109)(115 178)(116 215)(117 195)(118 175)(119 212)(120 192)(121 172)(122 209)(123 189)(124 226)(125 206)(126 186)(127 223)(128 203)(129 183)(130 220)(131 200)(132 180)(133 217)(134 197)(135 177)(136 214)(137 194)(138 174)(139 211)(140 191)(141 228)(142 208)(143 188)(144 225)(145 205)(146 185)(147 222)(148 202)(149 182)(150 219)(151 199)(152 179)(153 216)(154 196)(155 176)(156 213)(157 193)(158 173)(159 210)(160 190)(161 227)(162 207)(163 187)(164 224)(165 204)(166 184)(167 221)(168 201)(169 181)(170 218)(171 198)
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), (1,155,98,176)(2,154,99,175)(3,153,100,174)(4,152,101,173)(5,151,102,172)(6,150,103,228)(7,149,104,227)(8,148,105,226)(9,147,106,225)(10,146,107,224)(11,145,108,223)(12,144,109,222)(13,143,110,221)(14,142,111,220)(15,141,112,219)(16,140,113,218)(17,139,114,217)(18,138,58,216)(19,137,59,215)(20,136,60,214)(21,135,61,213)(22,134,62,212)(23,133,63,211)(24,132,64,210)(25,131,65,209)(26,130,66,208)(27,129,67,207)(28,128,68,206)(29,127,69,205)(30,126,70,204)(31,125,71,203)(32,124,72,202)(33,123,73,201)(34,122,74,200)(35,121,75,199)(36,120,76,198)(37,119,77,197)(38,118,78,196)(39,117,79,195)(40,116,80,194)(41,115,81,193)(42,171,82,192)(43,170,83,191)(44,169,84,190)(45,168,85,189)(46,167,86,188)(47,166,87,187)(48,165,88,186)(49,164,89,185)(50,163,90,184)(51,162,91,183)(52,161,92,182)(53,160,93,181)(54,159,94,180)(55,158,95,179)(56,157,96,178)(57,156,97,177), (2,38)(3,18)(4,55)(5,35)(6,15)(7,52)(8,32)(9,12)(10,49)(11,29)(13,46)(14,26)(16,43)(17,23)(19,40)(21,57)(22,37)(24,54)(25,34)(27,51)(28,31)(30,48)(33,45)(36,42)(41,56)(44,53)(47,50)(58,100)(59,80)(61,97)(62,77)(63,114)(64,94)(65,74)(66,111)(67,91)(68,71)(69,108)(70,88)(72,105)(73,85)(75,102)(76,82)(78,99)(81,96)(83,113)(84,93)(86,110)(87,90)(89,107)(92,104)(95,101)(103,112)(106,109)(115,178)(116,215)(117,195)(118,175)(119,212)(120,192)(121,172)(122,209)(123,189)(124,226)(125,206)(126,186)(127,223)(128,203)(129,183)(130,220)(131,200)(132,180)(133,217)(134,197)(135,177)(136,214)(137,194)(138,174)(139,211)(140,191)(141,228)(142,208)(143,188)(144,225)(145,205)(146,185)(147,222)(148,202)(149,182)(150,219)(151,199)(152,179)(153,216)(154,196)(155,176)(156,213)(157,193)(158,173)(159,210)(160,190)(161,227)(162,207)(163,187)(164,224)(165,204)(166,184)(167,221)(168,201)(169,181)(170,218)(171,198)>;
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), (1,155,98,176)(2,154,99,175)(3,153,100,174)(4,152,101,173)(5,151,102,172)(6,150,103,228)(7,149,104,227)(8,148,105,226)(9,147,106,225)(10,146,107,224)(11,145,108,223)(12,144,109,222)(13,143,110,221)(14,142,111,220)(15,141,112,219)(16,140,113,218)(17,139,114,217)(18,138,58,216)(19,137,59,215)(20,136,60,214)(21,135,61,213)(22,134,62,212)(23,133,63,211)(24,132,64,210)(25,131,65,209)(26,130,66,208)(27,129,67,207)(28,128,68,206)(29,127,69,205)(30,126,70,204)(31,125,71,203)(32,124,72,202)(33,123,73,201)(34,122,74,200)(35,121,75,199)(36,120,76,198)(37,119,77,197)(38,118,78,196)(39,117,79,195)(40,116,80,194)(41,115,81,193)(42,171,82,192)(43,170,83,191)(44,169,84,190)(45,168,85,189)(46,167,86,188)(47,166,87,187)(48,165,88,186)(49,164,89,185)(50,163,90,184)(51,162,91,183)(52,161,92,182)(53,160,93,181)(54,159,94,180)(55,158,95,179)(56,157,96,178)(57,156,97,177), (2,38)(3,18)(4,55)(5,35)(6,15)(7,52)(8,32)(9,12)(10,49)(11,29)(13,46)(14,26)(16,43)(17,23)(19,40)(21,57)(22,37)(24,54)(25,34)(27,51)(28,31)(30,48)(33,45)(36,42)(41,56)(44,53)(47,50)(58,100)(59,80)(61,97)(62,77)(63,114)(64,94)(65,74)(66,111)(67,91)(68,71)(69,108)(70,88)(72,105)(73,85)(75,102)(76,82)(78,99)(81,96)(83,113)(84,93)(86,110)(87,90)(89,107)(92,104)(95,101)(103,112)(106,109)(115,178)(116,215)(117,195)(118,175)(119,212)(120,192)(121,172)(122,209)(123,189)(124,226)(125,206)(126,186)(127,223)(128,203)(129,183)(130,220)(131,200)(132,180)(133,217)(134,197)(135,177)(136,214)(137,194)(138,174)(139,211)(140,191)(141,228)(142,208)(143,188)(144,225)(145,205)(146,185)(147,222)(148,202)(149,182)(150,219)(151,199)(152,179)(153,216)(154,196)(155,176)(156,213)(157,193)(158,173)(159,210)(160,190)(161,227)(162,207)(163,187)(164,224)(165,204)(166,184)(167,221)(168,201)(169,181)(170,218)(171,198) );
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)], [(1,155,98,176),(2,154,99,175),(3,153,100,174),(4,152,101,173),(5,151,102,172),(6,150,103,228),(7,149,104,227),(8,148,105,226),(9,147,106,225),(10,146,107,224),(11,145,108,223),(12,144,109,222),(13,143,110,221),(14,142,111,220),(15,141,112,219),(16,140,113,218),(17,139,114,217),(18,138,58,216),(19,137,59,215),(20,136,60,214),(21,135,61,213),(22,134,62,212),(23,133,63,211),(24,132,64,210),(25,131,65,209),(26,130,66,208),(27,129,67,207),(28,128,68,206),(29,127,69,205),(30,126,70,204),(31,125,71,203),(32,124,72,202),(33,123,73,201),(34,122,74,200),(35,121,75,199),(36,120,76,198),(37,119,77,197),(38,118,78,196),(39,117,79,195),(40,116,80,194),(41,115,81,193),(42,171,82,192),(43,170,83,191),(44,169,84,190),(45,168,85,189),(46,167,86,188),(47,166,87,187),(48,165,88,186),(49,164,89,185),(50,163,90,184),(51,162,91,183),(52,161,92,182),(53,160,93,181),(54,159,94,180),(55,158,95,179),(56,157,96,178),(57,156,97,177)], [(2,38),(3,18),(4,55),(5,35),(6,15),(7,52),(8,32),(9,12),(10,49),(11,29),(13,46),(14,26),(16,43),(17,23),(19,40),(21,57),(22,37),(24,54),(25,34),(27,51),(28,31),(30,48),(33,45),(36,42),(41,56),(44,53),(47,50),(58,100),(59,80),(61,97),(62,77),(63,114),(64,94),(65,74),(66,111),(67,91),(68,71),(69,108),(70,88),(72,105),(73,85),(75,102),(76,82),(78,99),(81,96),(83,113),(84,93),(86,110),(87,90),(89,107),(92,104),(95,101),(103,112),(106,109),(115,178),(116,215),(117,195),(118,175),(119,212),(120,192),(121,172),(122,209),(123,189),(124,226),(125,206),(126,186),(127,223),(128,203),(129,183),(130,220),(131,200),(132,180),(133,217),(134,197),(135,177),(136,214),(137,194),(138,174),(139,211),(140,191),(141,228),(142,208),(143,188),(144,225),(145,205),(146,185),(147,222),(148,202),(149,182),(150,219),(151,199),(152,179),(153,216),(154,196),(155,176),(156,213),(157,193),(158,173),(159,210),(160,190),(161,227),(162,207),(163,187),(164,224),(165,204),(166,184),(167,221),(168,201),(169,181),(170,218),(171,198)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 57A | ··· | 57I | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 6 | 38 | 2 | 114 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D19 | D38 | C19⋊D4 | S3×D19 | C57⋊D4 |
| kernel | C57⋊D4 | Dic57 | C6×D19 | S3×C38 | D38 | C57 | C38 | C19 | D6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of C57⋊D4 ►in GL4(𝔽229) generated by
| 57 | 217 | 0 | 0 |
| 20 | 40 | 0 | 0 |
| 0 | 0 | 94 | 0 |
| 0 | 0 | 17 | 134 |
| 117 | 25 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 18 | 204 |
| 0 | 0 | 13 | 211 |
| 117 | 25 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 148 | 228 |
G:=sub<GL(4,GF(229))| [57,20,0,0,217,40,0,0,0,0,94,17,0,0,0,134],[117,112,0,0,25,112,0,0,0,0,18,13,0,0,204,211],[117,112,0,0,25,112,0,0,0,0,1,148,0,0,0,228] >;
C57⋊D4 in GAP, Magma, Sage, TeX
C_{57}\rtimes D_4 % in TeX
G:=Group("C57:D4"); // GroupNames label
G:=SmallGroup(456,15);
// by ID
G=gap.SmallGroup(456,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,61,168,10804]);
// Polycyclic
G:=Group<a,b,c|a^57=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^37,c*b*c=b^-1>;
// generators/relations
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