metabelian, soluble, monomial, A-group
Aliases: C39.A4, (C2×C26)⋊2C9, C13⋊(C3.A4), C3.(C13⋊A4), C22⋊(C13⋊C9), (C2×C78).2C3, (C2×C6).(C13⋊C3), SmallGroup(468,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C26 — C39.A4 |
Generators and relations for C39.A4
G = < a,b,c,d | a39=b2=c2=1, d3=a13, ab=ba, ac=ca, dad-1=a22, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C39.A4
►On 234 points
Generators in S234
(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)(196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)
(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)(118 218)(119 219)(120 220)(121 221)(122 222)(123 223)(124 224)(125 225)(126 226)(127 227)(128 228)(129 229)(130 230)(131 231)(132 232)(133 233)(134 234)(135 196)(136 197)(137 198)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 205)(145 206)(146 207)(147 208)(148 209)(149 210)(150 211)(151 212)(152 213)(153 214)(154 215)(155 216)(156 217)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 40)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)
(1 132 174 14 145 187 27 119 161)(2 148 157 15 122 170 28 135 183)(3 125 179 16 138 192 29 151 166)(4 141 162 17 154 175 30 128 188)(5 118 184 18 131 158 31 144 171)(6 134 167 19 147 180 32 121 193)(7 150 189 20 124 163 33 137 176)(8 127 172 21 140 185 34 153 159)(9 143 194 22 156 168 35 130 181)(10 120 177 23 133 190 36 146 164)(11 136 160 24 149 173 37 123 186)(12 152 182 25 126 195 38 139 169)(13 129 165 26 142 178 39 155 191)(40 221 115 53 234 89 66 208 102)(41 198 98 54 211 111 67 224 85)(42 214 81 55 227 94 68 201 107)(43 230 103 56 204 116 69 217 90)(44 207 86 57 220 99 70 233 112)(45 223 108 58 197 82 71 210 95)(46 200 91 59 213 104 72 226 117)(47 216 113 60 229 87 73 203 100)(48 232 96 61 206 109 74 219 83)(49 209 79 62 222 92 75 196 105)(50 225 101 63 199 114 76 212 88)(51 202 84 64 215 97 77 228 110)(52 218 106 65 231 80 78 205 93)
G:=sub<Sym(234)| (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)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,218)(119,219)(120,220)(121,221)(122,222)(123,223)(124,224)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(133,233)(134,234)(135,196)(136,197)(137,198)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195), (1,132,174,14,145,187,27,119,161)(2,148,157,15,122,170,28,135,183)(3,125,179,16,138,192,29,151,166)(4,141,162,17,154,175,30,128,188)(5,118,184,18,131,158,31,144,171)(6,134,167,19,147,180,32,121,193)(7,150,189,20,124,163,33,137,176)(8,127,172,21,140,185,34,153,159)(9,143,194,22,156,168,35,130,181)(10,120,177,23,133,190,36,146,164)(11,136,160,24,149,173,37,123,186)(12,152,182,25,126,195,38,139,169)(13,129,165,26,142,178,39,155,191)(40,221,115,53,234,89,66,208,102)(41,198,98,54,211,111,67,224,85)(42,214,81,55,227,94,68,201,107)(43,230,103,56,204,116,69,217,90)(44,207,86,57,220,99,70,233,112)(45,223,108,58,197,82,71,210,95)(46,200,91,59,213,104,72,226,117)(47,216,113,60,229,87,73,203,100)(48,232,96,61,206,109,74,219,83)(49,209,79,62,222,92,75,196,105)(50,225,101,63,199,114,76,212,88)(51,202,84,64,215,97,77,228,110)(52,218,106,65,231,80,78,205,93)>;
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)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,218)(119,219)(120,220)(121,221)(122,222)(123,223)(124,224)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(133,233)(134,234)(135,196)(136,197)(137,198)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195), (1,132,174,14,145,187,27,119,161)(2,148,157,15,122,170,28,135,183)(3,125,179,16,138,192,29,151,166)(4,141,162,17,154,175,30,128,188)(5,118,184,18,131,158,31,144,171)(6,134,167,19,147,180,32,121,193)(7,150,189,20,124,163,33,137,176)(8,127,172,21,140,185,34,153,159)(9,143,194,22,156,168,35,130,181)(10,120,177,23,133,190,36,146,164)(11,136,160,24,149,173,37,123,186)(12,152,182,25,126,195,38,139,169)(13,129,165,26,142,178,39,155,191)(40,221,115,53,234,89,66,208,102)(41,198,98,54,211,111,67,224,85)(42,214,81,55,227,94,68,201,107)(43,230,103,56,204,116,69,217,90)(44,207,86,57,220,99,70,233,112)(45,223,108,58,197,82,71,210,95)(46,200,91,59,213,104,72,226,117)(47,216,113,60,229,87,73,203,100)(48,232,96,61,206,109,74,219,83)(49,209,79,62,222,92,75,196,105)(50,225,101,63,199,114,76,212,88)(51,202,84,64,215,97,77,228,110)(52,218,106,65,231,80,78,205,93) );
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),(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)], [(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,169),(92,170),(93,171),(94,172),(95,173),(96,174),(97,175),(98,176),(99,177),(100,178),(101,179),(102,180),(103,181),(104,182),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(113,191),(114,192),(115,193),(116,194),(117,195),(118,218),(119,219),(120,220),(121,221),(122,222),(123,223),(124,224),(125,225),(126,226),(127,227),(128,228),(129,229),(130,230),(131,231),(132,232),(133,233),(134,234),(135,196),(136,197),(137,198),(138,199),(139,200),(140,201),(141,202),(142,203),(143,204),(144,205),(145,206),(146,207),(147,208),(148,209),(149,210),(150,211),(151,212),(152,213),(153,214),(154,215),(155,216),(156,217)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,40),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,169),(92,170),(93,171),(94,172),(95,173),(96,174),(97,175),(98,176),(99,177),(100,178),(101,179),(102,180),(103,181),(104,182),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(113,191),(114,192),(115,193),(116,194),(117,195)], [(1,132,174,14,145,187,27,119,161),(2,148,157,15,122,170,28,135,183),(3,125,179,16,138,192,29,151,166),(4,141,162,17,154,175,30,128,188),(5,118,184,18,131,158,31,144,171),(6,134,167,19,147,180,32,121,193),(7,150,189,20,124,163,33,137,176),(8,127,172,21,140,185,34,153,159),(9,143,194,22,156,168,35,130,181),(10,120,177,23,133,190,36,146,164),(11,136,160,24,149,173,37,123,186),(12,152,182,25,126,195,38,139,169),(13,129,165,26,142,178,39,155,191),(40,221,115,53,234,89,66,208,102),(41,198,98,54,211,111,67,224,85),(42,214,81,55,227,94,68,201,107),(43,230,103,56,204,116,69,217,90),(44,207,86,57,220,99,70,233,112),(45,223,108,58,197,82,71,210,95),(46,200,91,59,213,104,72,226,117),(47,216,113,60,229,87,73,203,100),(48,232,96,61,206,109,74,219,83),(49,209,79,62,222,92,75,196,105),(50,225,101,63,199,114,76,212,88),(51,202,84,64,215,97,77,228,110),(52,218,106,65,231,80,78,205,93)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 13A | 13B | 13C | 13D | 26A | ··· | 26L | 39A | ··· | 39H | 78A | ··· | 78X |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 52 | ··· | 52 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C9 | A4 | C3.A4 | C13⋊C3 | C13⋊C9 | C13⋊A4 | C39.A4 |
| kernel | C39.A4 | C2×C78 | C2×C26 | C39 | C13 | C2×C6 | C22 | C3 | C1 |
| # reps | 1 | 2 | 6 | 1 | 2 | 4 | 8 | 12 | 24 |
Matrix representation of C39.A4 ►in GL3(𝔽937) generated by
| 571 | 0 | 0 |
| 0 | 146 | 0 |
| 0 | 0 | 347 |
| 1 | 0 | 0 |
| 0 | 936 | 0 |
| 0 | 0 | 936 |
| 936 | 0 | 0 |
| 0 | 936 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 322 | 0 | 0 |
G:=sub<GL(3,GF(937))| [571,0,0,0,146,0,0,0,347],[1,0,0,0,936,0,0,0,936],[936,0,0,0,936,0,0,0,1],[0,0,322,1,0,0,0,1,0] >;
C39.A4 in GAP, Magma, Sage, TeX
C_{39}.A_4 % in TeX
G:=Group("C39.A4"); // GroupNames label
G:=SmallGroup(468,14);
// by ID
G=gap.SmallGroup(468,14);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-13,15,272,543,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^39=b^2=c^2=1,d^3=a^13,a*b=b*a,a*c=c*a,d*a*d^-1=a^22,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export