direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D36, C45⋊5D4, C20⋊3D9, C36⋊1C10, C180⋊3C2, C60.9S3, D18⋊1C10, C30.58D6, C15.3D12, C10.15D18, C90.15C22, C4⋊(C5×D9), C9⋊1(C5×D4), C3.(C5×D12), (C10×D9)⋊4C2, C6.8(S3×C10), C12.2(C5×S3), C2.4(C10×D9), C18.3(C2×C10), SmallGroup(360,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D36
G = < a,b,c | a5=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D36
►On 180 points
Generators in S180
(1 109 97 62 157)(2 110 98 63 158)(3 111 99 64 159)(4 112 100 65 160)(5 113 101 66 161)(6 114 102 67 162)(7 115 103 68 163)(8 116 104 69 164)(9 117 105 70 165)(10 118 106 71 166)(11 119 107 72 167)(12 120 108 37 168)(13 121 73 38 169)(14 122 74 39 170)(15 123 75 40 171)(16 124 76 41 172)(17 125 77 42 173)(18 126 78 43 174)(19 127 79 44 175)(20 128 80 45 176)(21 129 81 46 177)(22 130 82 47 178)(23 131 83 48 179)(24 132 84 49 180)(25 133 85 50 145)(26 134 86 51 146)(27 135 87 52 147)(28 136 88 53 148)(29 137 89 54 149)(30 138 90 55 150)(31 139 91 56 151)(32 140 92 57 152)(33 141 93 58 153)(34 142 94 59 154)(35 143 95 60 155)(36 144 96 61 156)
(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(109 144)(110 143)(111 142)(112 141)(113 140)(114 139)(115 138)(116 137)(117 136)(118 135)(119 134)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(145 168)(146 167)(147 166)(148 165)(149 164)(150 163)(151 162)(152 161)(153 160)(154 159)(155 158)(156 157)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)
G:=sub<Sym(180)| (1,109,97,62,157)(2,110,98,63,158)(3,111,99,64,159)(4,112,100,65,160)(5,113,101,66,161)(6,114,102,67,162)(7,115,103,68,163)(8,116,104,69,164)(9,117,105,70,165)(10,118,106,71,166)(11,119,107,72,167)(12,120,108,37,168)(13,121,73,38,169)(14,122,74,39,170)(15,123,75,40,171)(16,124,76,41,172)(17,125,77,42,173)(18,126,78,43,174)(19,127,79,44,175)(20,128,80,45,176)(21,129,81,46,177)(22,130,82,47,178)(23,131,83,48,179)(24,132,84,49,180)(25,133,85,50,145)(26,134,86,51,146)(27,135,87,52,147)(28,136,88,53,148)(29,137,89,54,149)(30,138,90,55,150)(31,139,91,56,151)(32,140,92,57,152)(33,141,93,58,153)(34,142,94,59,154)(35,143,95,60,155)(36,144,96,61,156), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)>;
G:=Group( (1,109,97,62,157)(2,110,98,63,158)(3,111,99,64,159)(4,112,100,65,160)(5,113,101,66,161)(6,114,102,67,162)(7,115,103,68,163)(8,116,104,69,164)(9,117,105,70,165)(10,118,106,71,166)(11,119,107,72,167)(12,120,108,37,168)(13,121,73,38,169)(14,122,74,39,170)(15,123,75,40,171)(16,124,76,41,172)(17,125,77,42,173)(18,126,78,43,174)(19,127,79,44,175)(20,128,80,45,176)(21,129,81,46,177)(22,130,82,47,178)(23,131,83,48,179)(24,132,84,49,180)(25,133,85,50,145)(26,134,86,51,146)(27,135,87,52,147)(28,136,88,53,148)(29,137,89,54,149)(30,138,90,55,150)(31,139,91,56,151)(32,140,92,57,152)(33,141,93,58,153)(34,142,94,59,154)(35,143,95,60,155)(36,144,96,61,156), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,144)(110,143)(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175) );
G=PermutationGroup([[(1,109,97,62,157),(2,110,98,63,158),(3,111,99,64,159),(4,112,100,65,160),(5,113,101,66,161),(6,114,102,67,162),(7,115,103,68,163),(8,116,104,69,164),(9,117,105,70,165),(10,118,106,71,166),(11,119,107,72,167),(12,120,108,37,168),(13,121,73,38,169),(14,122,74,39,170),(15,123,75,40,171),(16,124,76,41,172),(17,125,77,42,173),(18,126,78,43,174),(19,127,79,44,175),(20,128,80,45,176),(21,129,81,46,177),(22,130,82,47,178),(23,131,83,48,179),(24,132,84,49,180),(25,133,85,50,145),(26,134,86,51,146),(27,135,87,52,147),(28,136,88,53,148),(29,137,89,54,149),(30,138,90,55,150),(31,139,91,56,151),(32,140,92,57,152),(33,141,93,58,153),(34,142,94,59,154),(35,143,95,60,155),(36,144,96,61,156)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(109,144),(110,143),(111,142),(112,141),(113,140),(114,139),(115,138),(116,137),(117,136),(118,135),(119,134),(120,133),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(145,168),(146,167),(147,166),(148,165),(149,164),(150,163),(151,162),(152,161),(153,160),(154,159),(155,158),(156,157),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 5C | 5D | 6 | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 36A | ··· | 36F | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 18 | 18 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | D9 | D12 | C5×S3 | D18 | C5×D4 | S3×C10 | D36 | C5×D9 | C5×D12 | C10×D9 | C5×D36 |
| kernel | C5×D36 | C180 | C10×D9 | D36 | C36 | D18 | C60 | C45 | C30 | C20 | C15 | C12 | C10 | C9 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 3 | 2 | 4 | 3 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C5×D36 ►in GL2(𝔽181) generated by
| 42 | 0 |
| 0 | 42 |
| 37 | 123 |
| 58 | 95 |
| 159 | 170 |
| 11 | 22 |
G:=sub<GL(2,GF(181))| [42,0,0,42],[37,58,123,95],[159,11,170,22] >;
C5×D36 in GAP, Magma, Sage, TeX
C_5\times D_{36} % in TeX
G:=Group("C5xD36"); // GroupNames label
G:=SmallGroup(360,22);
// by ID
G=gap.SmallGroup(360,22);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,6004,208,8645]);
// Polycyclic
G:=Group<a,b,c|a^5=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export