direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D31, C62⋊3C6, C186⋊2C2, C93⋊3C22, C31⋊3(C2×C6), SmallGroup(372,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C6×D31 |
Generators and relations for C6×D31
G = < a,b,c | a6=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D31
►On 186 points
Generators in S186
(1 150 80 97 42 171)(2 151 81 98 43 172)(3 152 82 99 44 173)(4 153 83 100 45 174)(5 154 84 101 46 175)(6 155 85 102 47 176)(7 125 86 103 48 177)(8 126 87 104 49 178)(9 127 88 105 50 179)(10 128 89 106 51 180)(11 129 90 107 52 181)(12 130 91 108 53 182)(13 131 92 109 54 183)(14 132 93 110 55 184)(15 133 63 111 56 185)(16 134 64 112 57 186)(17 135 65 113 58 156)(18 136 66 114 59 157)(19 137 67 115 60 158)(20 138 68 116 61 159)(21 139 69 117 62 160)(22 140 70 118 32 161)(23 141 71 119 33 162)(24 142 72 120 34 163)(25 143 73 121 35 164)(26 144 74 122 36 165)(27 145 75 123 37 166)(28 146 76 124 38 167)(29 147 77 94 39 168)(30 148 78 95 40 169)(31 149 79 96 41 170)
(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)
(1 96)(2 95)(3 94)(4 124)(5 123)(6 122)(7 121)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 99)(30 98)(31 97)(32 128)(33 127)(34 126)(35 125)(36 155)(37 154)(38 153)(39 152)(40 151)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 130)(62 129)(63 156)(64 186)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(79 171)(80 170)(81 169)(82 168)(83 167)(84 166)(85 165)(86 164)(87 163)(88 162)(89 161)(90 160)(91 159)(92 158)(93 157)
G:=sub<Sym(186)| (1,150,80,97,42,171)(2,151,81,98,43,172)(3,152,82,99,44,173)(4,153,83,100,45,174)(5,154,84,101,46,175)(6,155,85,102,47,176)(7,125,86,103,48,177)(8,126,87,104,49,178)(9,127,88,105,50,179)(10,128,89,106,51,180)(11,129,90,107,52,181)(12,130,91,108,53,182)(13,131,92,109,54,183)(14,132,93,110,55,184)(15,133,63,111,56,185)(16,134,64,112,57,186)(17,135,65,113,58,156)(18,136,66,114,59,157)(19,137,67,115,60,158)(20,138,68,116,61,159)(21,139,69,117,62,160)(22,140,70,118,32,161)(23,141,71,119,33,162)(24,142,72,120,34,163)(25,143,73,121,35,164)(26,144,74,122,36,165)(27,145,75,123,37,166)(28,146,76,124,38,167)(29,147,77,94,39,168)(30,148,78,95,40,169)(31,149,79,96,41,170), (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), (1,96)(2,95)(3,94)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,128)(33,127)(34,126)(35,125)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,156)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157)>;
G:=Group( (1,150,80,97,42,171)(2,151,81,98,43,172)(3,152,82,99,44,173)(4,153,83,100,45,174)(5,154,84,101,46,175)(6,155,85,102,47,176)(7,125,86,103,48,177)(8,126,87,104,49,178)(9,127,88,105,50,179)(10,128,89,106,51,180)(11,129,90,107,52,181)(12,130,91,108,53,182)(13,131,92,109,54,183)(14,132,93,110,55,184)(15,133,63,111,56,185)(16,134,64,112,57,186)(17,135,65,113,58,156)(18,136,66,114,59,157)(19,137,67,115,60,158)(20,138,68,116,61,159)(21,139,69,117,62,160)(22,140,70,118,32,161)(23,141,71,119,33,162)(24,142,72,120,34,163)(25,143,73,121,35,164)(26,144,74,122,36,165)(27,145,75,123,37,166)(28,146,76,124,38,167)(29,147,77,94,39,168)(30,148,78,95,40,169)(31,149,79,96,41,170), (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), (1,96)(2,95)(3,94)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,128)(33,127)(34,126)(35,125)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,156)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157) );
G=PermutationGroup([[(1,150,80,97,42,171),(2,151,81,98,43,172),(3,152,82,99,44,173),(4,153,83,100,45,174),(5,154,84,101,46,175),(6,155,85,102,47,176),(7,125,86,103,48,177),(8,126,87,104,49,178),(9,127,88,105,50,179),(10,128,89,106,51,180),(11,129,90,107,52,181),(12,130,91,108,53,182),(13,131,92,109,54,183),(14,132,93,110,55,184),(15,133,63,111,56,185),(16,134,64,112,57,186),(17,135,65,113,58,156),(18,136,66,114,59,157),(19,137,67,115,60,158),(20,138,68,116,61,159),(21,139,69,117,62,160),(22,140,70,118,32,161),(23,141,71,119,33,162),(24,142,72,120,34,163),(25,143,73,121,35,164),(26,144,74,122,36,165),(27,145,75,123,37,166),(28,146,76,124,38,167),(29,147,77,94,39,168),(30,148,78,95,40,169),(31,149,79,96,41,170)], [(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)], [(1,96),(2,95),(3,94),(4,124),(5,123),(6,122),(7,121),(8,120),(9,119),(10,118),(11,117),(12,116),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,100),(29,99),(30,98),(31,97),(32,128),(33,127),(34,126),(35,125),(36,155),(37,154),(38,153),(39,152),(40,151),(41,150),(42,149),(43,148),(44,147),(45,146),(46,145),(47,144),(48,143),(49,142),(50,141),(51,140),(52,139),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,130),(62,129),(63,156),(64,186),(65,185),(66,184),(67,183),(68,182),(69,181),(70,180),(71,179),(72,178),(73,177),(74,176),(75,175),(76,174),(77,173),(78,172),(79,171),(80,170),(81,169),(82,168),(83,167),(84,166),(85,165),(86,164),(87,163),(88,162),(89,161),(90,160),(91,159),(92,158),(93,157)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 31A | ··· | 31O | 62A | ··· | 62O | 93A | ··· | 93AD | 186A | ··· | 186AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 | 186 | ··· | 186 |
| size | 1 | 1 | 31 | 31 | 1 | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D31 | D62 | C3×D31 | C6×D31 |
| kernel | C6×D31 | C3×D31 | C186 | D62 | D31 | C62 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 15 | 15 | 30 | 30 |
Matrix representation of C6×D31 ►in GL2(𝔽373) generated by
| 89 | 0 |
| 0 | 89 |
| 346 | 1 |
| 2 | 290 |
| 253 | 83 |
| 213 | 120 |
G:=sub<GL(2,GF(373))| [89,0,0,89],[346,2,1,290],[253,213,83,120] >;
C6×D31 in GAP, Magma, Sage, TeX
C_6\times D_{31} % in TeX
G:=Group("C6xD31"); // GroupNames label
G:=SmallGroup(372,12);
// by ID
G=gap.SmallGroup(372,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-31,5763]);
// Polycyclic
G:=Group<a,b,c|a^6=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export