direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D37, C37⋊C10, C185⋊2C2, SmallGroup(370,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C5×D37 |
Generators and relations for C5×D37
G = < a,b,c | a5=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D37
►On 185 points
Generators in S185
(1 180 125 109 47)(2 181 126 110 48)(3 182 127 111 49)(4 183 128 75 50)(5 184 129 76 51)(6 185 130 77 52)(7 149 131 78 53)(8 150 132 79 54)(9 151 133 80 55)(10 152 134 81 56)(11 153 135 82 57)(12 154 136 83 58)(13 155 137 84 59)(14 156 138 85 60)(15 157 139 86 61)(16 158 140 87 62)(17 159 141 88 63)(18 160 142 89 64)(19 161 143 90 65)(20 162 144 91 66)(21 163 145 92 67)(22 164 146 93 68)(23 165 147 94 69)(24 166 148 95 70)(25 167 112 96 71)(26 168 113 97 72)(27 169 114 98 73)(28 170 115 99 74)(29 171 116 100 38)(30 172 117 101 39)(31 173 118 102 40)(32 174 119 103 41)(33 175 120 104 42)(34 176 121 105 43)(35 177 122 106 44)(36 178 123 107 45)(37 179 124 108 46)
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(106 111)(107 110)(108 109)(112 137)(113 136)(114 135)(115 134)(116 133)(117 132)(118 131)(119 130)(120 129)(121 128)(122 127)(123 126)(124 125)(138 148)(139 147)(140 146)(141 145)(142 144)(149 173)(150 172)(151 171)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)
G:=sub<Sym(185)| (1,180,125,109,47)(2,181,126,110,48)(3,182,127,111,49)(4,183,128,75,50)(5,184,129,76,51)(6,185,130,77,52)(7,149,131,78,53)(8,150,132,79,54)(9,151,133,80,55)(10,152,134,81,56)(11,153,135,82,57)(12,154,136,83,58)(13,155,137,84,59)(14,156,138,85,60)(15,157,139,86,61)(16,158,140,87,62)(17,159,141,88,63)(18,160,142,89,64)(19,161,143,90,65)(20,162,144,91,66)(21,163,145,92,67)(22,164,146,93,68)(23,165,147,94,69)(24,166,148,95,70)(25,167,112,96,71)(26,168,113,97,72)(27,169,114,98,73)(28,170,115,99,74)(29,171,116,100,38)(30,172,117,101,39)(31,173,118,102,40)(32,174,119,103,41)(33,175,120,104,42)(34,176,121,105,43)(35,177,122,106,44)(36,178,123,107,45)(37,179,124,108,46), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(106,111)(107,110)(108,109)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(138,148)(139,147)(140,146)(141,145)(142,144)(149,173)(150,172)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)>;
G:=Group( (1,180,125,109,47)(2,181,126,110,48)(3,182,127,111,49)(4,183,128,75,50)(5,184,129,76,51)(6,185,130,77,52)(7,149,131,78,53)(8,150,132,79,54)(9,151,133,80,55)(10,152,134,81,56)(11,153,135,82,57)(12,154,136,83,58)(13,155,137,84,59)(14,156,138,85,60)(15,157,139,86,61)(16,158,140,87,62)(17,159,141,88,63)(18,160,142,89,64)(19,161,143,90,65)(20,162,144,91,66)(21,163,145,92,67)(22,164,146,93,68)(23,165,147,94,69)(24,166,148,95,70)(25,167,112,96,71)(26,168,113,97,72)(27,169,114,98,73)(28,170,115,99,74)(29,171,116,100,38)(30,172,117,101,39)(31,173,118,102,40)(32,174,119,103,41)(33,175,120,104,42)(34,176,121,105,43)(35,177,122,106,44)(36,178,123,107,45)(37,179,124,108,46), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(106,111)(107,110)(108,109)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(138,148)(139,147)(140,146)(141,145)(142,144)(149,173)(150,172)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180) );
G=PermutationGroup([[(1,180,125,109,47),(2,181,126,110,48),(3,182,127,111,49),(4,183,128,75,50),(5,184,129,76,51),(6,185,130,77,52),(7,149,131,78,53),(8,150,132,79,54),(9,151,133,80,55),(10,152,134,81,56),(11,153,135,82,57),(12,154,136,83,58),(13,155,137,84,59),(14,156,138,85,60),(15,157,139,86,61),(16,158,140,87,62),(17,159,141,88,63),(18,160,142,89,64),(19,161,143,90,65),(20,162,144,91,66),(21,163,145,92,67),(22,164,146,93,68),(23,165,147,94,69),(24,166,148,95,70),(25,167,112,96,71),(26,168,113,97,72),(27,169,114,98,73),(28,170,115,99,74),(29,171,116,100,38),(30,172,117,101,39),(31,173,118,102,40),(32,174,119,103,41),(33,175,120,104,42),(34,176,121,105,43),(35,177,122,106,44),(36,178,123,107,45),(37,179,124,108,46)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(56,74),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(75,105),(76,104),(77,103),(78,102),(79,101),(80,100),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(106,111),(107,110),(108,109),(112,137),(113,136),(114,135),(115,134),(116,133),(117,132),(118,131),(119,130),(120,129),(121,128),(122,127),(123,126),(124,125),(138,148),(139,147),(140,146),(141,145),(142,144),(149,173),(150,172),(151,171),(152,170),(153,169),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(174,185),(175,184),(176,183),(177,182),(178,181),(179,180)]])
100 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 37A | ··· | 37R | 185A | ··· | 185BT |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 37 | ··· | 37 | 185 | ··· | 185 |
| size | 1 | 37 | 1 | 1 | 1 | 1 | 37 | 37 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D37 | C5×D37 |
| kernel | C5×D37 | C185 | D37 | C37 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 18 | 72 |
Matrix representation of C5×D37 ►in GL2(𝔽1481) generated by
| 136 | 0 |
| 0 | 136 |
| 719 | 1264 |
| 1480 | 1092 |
| 489 | 914 |
| 568 | 992 |
G:=sub<GL(2,GF(1481))| [136,0,0,136],[719,1480,1264,1092],[489,568,914,992] >;
C5×D37 in GAP, Magma, Sage, TeX
C_5\times D_{37} % in TeX
G:=Group("C5xD37"); // GroupNames label
G:=SmallGroup(370,2);
// by ID
G=gap.SmallGroup(370,2);
# by ID
G:=PCGroup([3,-2,-5,-37,3242]);
// Polycyclic
G:=Group<a,b,c|a^5=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export