direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C36, C20⋊2C18, C180⋊6C2, C60.8C6, Dic5⋊2C18, D10.2C18, C18.14D10, C90.19C22, C5⋊2(C2×C36), C45⋊9(C2×C4), C3.(D5×C12), (D5×C12).C3, (C6×D5).7C6, C2.1(D5×C18), C12.7(C3×D5), C6.14(C6×D5), C30.14(C2×C6), C10.2(C2×C18), C15.2(C2×C12), (C9×Dic5)⋊5C2, (C3×D5).2C12, (D5×C18).4C2, (C3×Dic5).8C6, SmallGroup(360,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C36 |
Generators and relations for D5×C36
G = < a,b,c | a36=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C36
►On 180 points
Generators in S180
(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)
(1 135 153 40 83)(2 136 154 41 84)(3 137 155 42 85)(4 138 156 43 86)(5 139 157 44 87)(6 140 158 45 88)(7 141 159 46 89)(8 142 160 47 90)(9 143 161 48 91)(10 144 162 49 92)(11 109 163 50 93)(12 110 164 51 94)(13 111 165 52 95)(14 112 166 53 96)(15 113 167 54 97)(16 114 168 55 98)(17 115 169 56 99)(18 116 170 57 100)(19 117 171 58 101)(20 118 172 59 102)(21 119 173 60 103)(22 120 174 61 104)(23 121 175 62 105)(24 122 176 63 106)(25 123 177 64 107)(26 124 178 65 108)(27 125 179 66 73)(28 126 180 67 74)(29 127 145 68 75)(30 128 146 69 76)(31 129 147 70 77)(32 130 148 71 78)(33 131 149 72 79)(34 132 150 37 80)(35 133 151 38 81)(36 134 152 39 82)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 132)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)
G:=sub<Sym(180)| (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), (1,135,153,40,83)(2,136,154,41,84)(3,137,155,42,85)(4,138,156,43,86)(5,139,157,44,87)(6,140,158,45,88)(7,141,159,46,89)(8,142,160,47,90)(9,143,161,48,91)(10,144,162,49,92)(11,109,163,50,93)(12,110,164,51,94)(13,111,165,52,95)(14,112,166,53,96)(15,113,167,54,97)(16,114,168,55,98)(17,115,169,56,99)(18,116,170,57,100)(19,117,171,58,101)(20,118,172,59,102)(21,119,173,60,103)(22,120,174,61,104)(23,121,175,62,105)(24,122,176,63,106)(25,123,177,64,107)(26,124,178,65,108)(27,125,179,66,73)(28,126,180,67,74)(29,127,145,68,75)(30,128,146,69,76)(31,129,147,70,77)(32,130,148,71,78)(33,131,149,72,79)(34,132,150,37,80)(35,133,151,38,81)(36,134,152,39,82), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)>;
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), (1,135,153,40,83)(2,136,154,41,84)(3,137,155,42,85)(4,138,156,43,86)(5,139,157,44,87)(6,140,158,45,88)(7,141,159,46,89)(8,142,160,47,90)(9,143,161,48,91)(10,144,162,49,92)(11,109,163,50,93)(12,110,164,51,94)(13,111,165,52,95)(14,112,166,53,96)(15,113,167,54,97)(16,114,168,55,98)(17,115,169,56,99)(18,116,170,57,100)(19,117,171,58,101)(20,118,172,59,102)(21,119,173,60,103)(22,120,174,61,104)(23,121,175,62,105)(24,122,176,63,106)(25,123,177,64,107)(26,124,178,65,108)(27,125,179,66,73)(28,126,180,67,74)(29,127,145,68,75)(30,128,146,69,76)(31,129,147,70,77)(32,130,148,71,78)(33,131,149,72,79)(34,132,150,37,80)(35,133,151,38,81)(36,134,152,39,82), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131) );
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)], [(1,135,153,40,83),(2,136,154,41,84),(3,137,155,42,85),(4,138,156,43,86),(5,139,157,44,87),(6,140,158,45,88),(7,141,159,46,89),(8,142,160,47,90),(9,143,161,48,91),(10,144,162,49,92),(11,109,163,50,93),(12,110,164,51,94),(13,111,165,52,95),(14,112,166,53,96),(15,113,167,54,97),(16,114,168,55,98),(17,115,169,56,99),(18,116,170,57,100),(19,117,171,58,101),(20,118,172,59,102),(21,119,173,60,103),(22,120,174,61,104),(23,121,175,62,105),(24,122,176,63,106),(25,123,177,64,107),(26,124,178,65,108),(27,125,179,66,73),(28,126,180,67,74),(29,127,145,68,75),(30,128,146,69,76),(31,129,147,70,77),(32,130,148,71,78),(33,131,149,72,79),(34,132,150,37,80),(35,133,151,38,81),(36,134,152,39,82)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,132),(38,133),(39,134),(40,135),(41,136),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,144),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 18G | ··· | 18R | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 36A | ··· | 36L | 36M | ··· | 36X | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C36 | D5 | D10 | C3×D5 | C4×D5 | C6×D5 | C9×D5 | D5×C12 | D5×C18 | D5×C36 |
| kernel | D5×C36 | C9×Dic5 | C180 | D5×C18 | D5×C12 | C9×D5 | C3×Dic5 | C60 | C6×D5 | C4×D5 | C3×D5 | Dic5 | C20 | D10 | D5 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 24 | 2 | 2 | 4 | 4 | 4 | 12 | 8 | 12 | 24 |
Matrix representation of D5×C36 ►in GL3(𝔽181) generated by
| 142 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 167 | 1 |
| 0 | 180 | 0 |
| 180 | 0 | 0 |
| 0 | 1 | 167 |
| 0 | 0 | 180 |
G:=sub<GL(3,GF(181))| [142,0,0,0,7,0,0,0,7],[1,0,0,0,167,180,0,1,0],[180,0,0,0,1,0,0,167,180] >;
D5×C36 in GAP, Magma, Sage, TeX
D_5\times C_{36} % in TeX
G:=Group("D5xC36"); // GroupNames label
G:=SmallGroup(360,16);
// by ID
G=gap.SmallGroup(360,16);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,79,122,10373]);
// Polycyclic
G:=Group<a,b,c|a^36=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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