direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D61, C61⋊3C6, C183⋊2C2, SmallGroup(366,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C3×D61 |
Generators and relations for C3×D61
G = < a,b,c | a3=b61=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D61
►On 183 points
Generators in S183
(1 160 64)(2 161 65)(3 162 66)(4 163 67)(5 164 68)(6 165 69)(7 166 70)(8 167 71)(9 168 72)(10 169 73)(11 170 74)(12 171 75)(13 172 76)(14 173 77)(15 174 78)(16 175 79)(17 176 80)(18 177 81)(19 178 82)(20 179 83)(21 180 84)(22 181 85)(23 182 86)(24 183 87)(25 123 88)(26 124 89)(27 125 90)(28 126 91)(29 127 92)(30 128 93)(31 129 94)(32 130 95)(33 131 96)(34 132 97)(35 133 98)(36 134 99)(37 135 100)(38 136 101)(39 137 102)(40 138 103)(41 139 104)(42 140 105)(43 141 106)(44 142 107)(45 143 108)(46 144 109)(47 145 110)(48 146 111)(49 147 112)(50 148 113)(51 149 114)(52 150 115)(53 151 116)(54 152 117)(55 153 118)(56 154 119)(57 155 120)(58 156 121)(59 157 122)(60 158 62)(61 159 63)
(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)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(62 65)(63 64)(66 122)(67 121)(68 120)(69 119)(70 118)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(136 183)(137 182)(138 181)(139 180)(140 179)(141 178)(142 177)(143 176)(144 175)(145 174)(146 173)(147 172)(148 171)(149 170)(150 169)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)
G:=sub<Sym(183)| (1,160,64)(2,161,65)(3,162,66)(4,163,67)(5,164,68)(6,165,69)(7,166,70)(8,167,71)(9,168,72)(10,169,73)(11,170,74)(12,171,75)(13,172,76)(14,173,77)(15,174,78)(16,175,79)(17,176,80)(18,177,81)(19,178,82)(20,179,83)(21,180,84)(22,181,85)(23,182,86)(24,183,87)(25,123,88)(26,124,89)(27,125,90)(28,126,91)(29,127,92)(30,128,93)(31,129,94)(32,130,95)(33,131,96)(34,132,97)(35,133,98)(36,134,99)(37,135,100)(38,136,101)(39,137,102)(40,138,103)(41,139,104)(42,140,105)(43,141,106)(44,142,107)(45,143,108)(46,144,109)(47,145,110)(48,146,111)(49,147,112)(50,148,113)(51,149,114)(52,150,115)(53,151,116)(54,152,117)(55,153,118)(56,154,119)(57,155,120)(58,156,121)(59,157,122)(60,158,62)(61,159,63), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(62,65)(63,64)(66,122)(67,121)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)>;
G:=Group( (1,160,64)(2,161,65)(3,162,66)(4,163,67)(5,164,68)(6,165,69)(7,166,70)(8,167,71)(9,168,72)(10,169,73)(11,170,74)(12,171,75)(13,172,76)(14,173,77)(15,174,78)(16,175,79)(17,176,80)(18,177,81)(19,178,82)(20,179,83)(21,180,84)(22,181,85)(23,182,86)(24,183,87)(25,123,88)(26,124,89)(27,125,90)(28,126,91)(29,127,92)(30,128,93)(31,129,94)(32,130,95)(33,131,96)(34,132,97)(35,133,98)(36,134,99)(37,135,100)(38,136,101)(39,137,102)(40,138,103)(41,139,104)(42,140,105)(43,141,106)(44,142,107)(45,143,108)(46,144,109)(47,145,110)(48,146,111)(49,147,112)(50,148,113)(51,149,114)(52,150,115)(53,151,116)(54,152,117)(55,153,118)(56,154,119)(57,155,120)(58,156,121)(59,157,122)(60,158,62)(61,159,63), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(62,65)(63,64)(66,122)(67,121)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160) );
G=PermutationGroup([[(1,160,64),(2,161,65),(3,162,66),(4,163,67),(5,164,68),(6,165,69),(7,166,70),(8,167,71),(9,168,72),(10,169,73),(11,170,74),(12,171,75),(13,172,76),(14,173,77),(15,174,78),(16,175,79),(17,176,80),(18,177,81),(19,178,82),(20,179,83),(21,180,84),(22,181,85),(23,182,86),(24,183,87),(25,123,88),(26,124,89),(27,125,90),(28,126,91),(29,127,92),(30,128,93),(31,129,94),(32,130,95),(33,131,96),(34,132,97),(35,133,98),(36,134,99),(37,135,100),(38,136,101),(39,137,102),(40,138,103),(41,139,104),(42,140,105),(43,141,106),(44,142,107),(45,143,108),(46,144,109),(47,145,110),(48,146,111),(49,147,112),(50,148,113),(51,149,114),(52,150,115),(53,151,116),(54,152,117),(55,153,118),(56,154,119),(57,155,120),(58,156,121),(59,157,122),(60,158,62),(61,159,63)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(62,65),(63,64),(66,122),(67,121),(68,120),(69,119),(70,118),(71,117),(72,116),(73,115),(74,114),(75,113),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(136,183),(137,182),(138,181),(139,180),(140,179),(141,178),(142,177),(143,176),(144,175),(145,174),(146,173),(147,172),(148,171),(149,170),(150,169),(151,168),(152,167),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160)]])
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 61A | ··· | 61AD | 183A | ··· | 183BH |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 61 | ··· | 61 | 183 | ··· | 183 |
| size | 1 | 61 | 1 | 1 | 61 | 61 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D61 | C3×D61 |
| kernel | C3×D61 | C183 | D61 | C61 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 30 | 60 |
Matrix representation of C3×D61 ►in GL2(𝔽367) generated by
| 283 | 0 |
| 0 | 283 |
| 8 | 1 |
| 126 | 337 |
| 337 | 366 |
| 165 | 30 |
G:=sub<GL(2,GF(367))| [283,0,0,283],[8,126,1,337],[337,165,366,30] >;
C3×D61 in GAP, Magma, Sage, TeX
C_3\times D_{61} % in TeX
G:=Group("C3xD61"); // GroupNames label
G:=SmallGroup(366,4);
// by ID
G=gap.SmallGroup(366,4);
# by ID
G:=PCGroup([3,-2,-3,-61,3242]);
// Polycyclic
G:=Group<a,b,c|a^3=b^61=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export