direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C38, C10⋊C38, C190⋊3C2, C95⋊4C22, C5⋊(C2×C38), SmallGroup(380,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C38 |
Generators and relations for D5×C38
G = < a,b,c | a38=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C38
►On 190 points
Generators in S190
(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)
(1 135 71 163 101)(2 136 72 164 102)(3 137 73 165 103)(4 138 74 166 104)(5 139 75 167 105)(6 140 76 168 106)(7 141 39 169 107)(8 142 40 170 108)(9 143 41 171 109)(10 144 42 172 110)(11 145 43 173 111)(12 146 44 174 112)(13 147 45 175 113)(14 148 46 176 114)(15 149 47 177 77)(16 150 48 178 78)(17 151 49 179 79)(18 152 50 180 80)(19 115 51 181 81)(20 116 52 182 82)(21 117 53 183 83)(22 118 54 184 84)(23 119 55 185 85)(24 120 56 186 86)(25 121 57 187 87)(26 122 58 188 88)(27 123 59 189 89)(28 124 60 190 90)(29 125 61 153 91)(30 126 62 154 92)(31 127 63 155 93)(32 128 64 156 94)(33 129 65 157 95)(34 130 66 158 96)(35 131 67 159 97)(36 132 68 160 98)(37 133 69 161 99)(38 134 70 162 100)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 77)(35 78)(36 79)(37 80)(38 81)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 188)(142 189)(143 190)(144 153)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)(151 160)(152 161)
G:=sub<Sym(190)| (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), (1,135,71,163,101)(2,136,72,164,102)(3,137,73,165,103)(4,138,74,166,104)(5,139,75,167,105)(6,140,76,168,106)(7,141,39,169,107)(8,142,40,170,108)(9,143,41,171,109)(10,144,42,172,110)(11,145,43,173,111)(12,146,44,174,112)(13,147,45,175,113)(14,148,46,176,114)(15,149,47,177,77)(16,150,48,178,78)(17,151,49,179,79)(18,152,50,180,80)(19,115,51,181,81)(20,116,52,182,82)(21,117,53,183,83)(22,118,54,184,84)(23,119,55,185,85)(24,120,56,186,86)(25,121,57,187,87)(26,122,58,188,88)(27,123,59,189,89)(28,124,60,190,90)(29,125,61,153,91)(30,126,62,154,92)(31,127,63,155,93)(32,128,64,156,94)(33,129,65,157,95)(34,130,66,158,96)(35,131,67,159,97)(36,132,68,160,98)(37,133,69,161,99)(38,134,70,162,100), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,77)(35,78)(36,79)(37,80)(38,81)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,188)(142,189)(143,190)(144,153)(145,154)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)>;
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), (1,135,71,163,101)(2,136,72,164,102)(3,137,73,165,103)(4,138,74,166,104)(5,139,75,167,105)(6,140,76,168,106)(7,141,39,169,107)(8,142,40,170,108)(9,143,41,171,109)(10,144,42,172,110)(11,145,43,173,111)(12,146,44,174,112)(13,147,45,175,113)(14,148,46,176,114)(15,149,47,177,77)(16,150,48,178,78)(17,151,49,179,79)(18,152,50,180,80)(19,115,51,181,81)(20,116,52,182,82)(21,117,53,183,83)(22,118,54,184,84)(23,119,55,185,85)(24,120,56,186,86)(25,121,57,187,87)(26,122,58,188,88)(27,123,59,189,89)(28,124,60,190,90)(29,125,61,153,91)(30,126,62,154,92)(31,127,63,155,93)(32,128,64,156,94)(33,129,65,157,95)(34,130,66,158,96)(35,131,67,159,97)(36,132,68,160,98)(37,133,69,161,99)(38,134,70,162,100), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,77)(35,78)(36,79)(37,80)(38,81)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,188)(142,189)(143,190)(144,153)(145,154)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161) );
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)], [(1,135,71,163,101),(2,136,72,164,102),(3,137,73,165,103),(4,138,74,166,104),(5,139,75,167,105),(6,140,76,168,106),(7,141,39,169,107),(8,142,40,170,108),(9,143,41,171,109),(10,144,42,172,110),(11,145,43,173,111),(12,146,44,174,112),(13,147,45,175,113),(14,148,46,176,114),(15,149,47,177,77),(16,150,48,178,78),(17,151,49,179,79),(18,152,50,180,80),(19,115,51,181,81),(20,116,52,182,82),(21,117,53,183,83),(22,118,54,184,84),(23,119,55,185,85),(24,120,56,186,86),(25,121,57,187,87),(26,122,58,188,88),(27,123,59,189,89),(28,124,60,190,90),(29,125,61,153,91),(30,126,62,154,92),(31,127,63,155,93),(32,128,64,156,94),(33,129,65,157,95),(34,130,66,158,96),(35,131,67,159,97),(36,132,68,160,98),(37,133,69,161,99),(38,134,70,162,100)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,77),(35,78),(36,79),(37,80),(38,81),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167),(121,168),(122,169),(123,170),(124,171),(125,172),(126,173),(127,174),(128,175),(129,176),(130,177),(131,178),(132,179),(133,180),(134,181),(135,182),(136,183),(137,184),(138,185),(139,186),(140,187),(141,188),(142,189),(143,190),(144,153),(145,154),(146,155),(147,156),(148,157),(149,158),(150,159),(151,160),(152,161)]])
152 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BB | 95A | ··· | 95AJ | 190A | ··· | 190AJ |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 95 | ··· | 95 | 190 | ··· | 190 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
152 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | D5 | D10 | D5×C19 | D5×C38 |
| kernel | D5×C38 | D5×C19 | C190 | D10 | D5 | C10 | C38 | C19 | C2 | C1 |
| # reps | 1 | 2 | 1 | 18 | 36 | 18 | 2 | 2 | 36 | 36 |
Matrix representation of D5×C38 ►in GL2(𝔽191) generated by
| 139 | 0 |
| 0 | 139 |
| 0 | 1 |
| 190 | 88 |
| 0 | 190 |
| 190 | 0 |
G:=sub<GL(2,GF(191))| [139,0,0,139],[0,190,1,88],[0,190,190,0] >;
D5×C38 in GAP, Magma, Sage, TeX
D_5\times C_{38} % in TeX
G:=Group("D5xC38"); // GroupNames label
G:=SmallGroup(380,9);
// by ID
G=gap.SmallGroup(380,9);
# by ID
G:=PCGroup([4,-2,-2,-19,-5,4867]);
// Polycyclic
G:=Group<a,b,c|a^38=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export