metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D36, C45⋊1D4, Dic5⋊D9, C15.D12, D18⋊1D5, D90⋊3C2, C30.4D6, C10.4D18, C18.4D10, C90.4C22, C9⋊1(C5⋊D4), C2.5(D5×D9), (C10×D9)⋊1C2, C3.(C5⋊D12), C6.11(S3×D5), (C9×Dic5)⋊3C2, (C3×Dic5).4S3, SmallGroup(360,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D36
G = < a,b,c | a45=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >
18C2
90C2
5C4
9C22
45C22
6S3
30S3
18D5
18C10
45D4
3D6
5C12
15D6
2D9
10D9
9D10
6D15
15D12
5C36
5D18
3D30
2D45
5D36
Smallest permutation representation of C5⋊D36
►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)
(1 115 56 177)(2 134 57 151)(3 108 58 170)(4 127 59 144)(5 101 60 163)(6 120 61 137)(7 94 62 156)(8 113 63 175)(9 132 64 149)(10 106 65 168)(11 125 66 142)(12 99 67 161)(13 118 68 180)(14 92 69 154)(15 111 70 173)(16 130 71 147)(17 104 72 166)(18 123 73 140)(19 97 74 159)(20 116 75 178)(21 135 76 152)(22 109 77 171)(23 128 78 145)(24 102 79 164)(25 121 80 138)(26 95 81 157)(27 114 82 176)(28 133 83 150)(29 107 84 169)(30 126 85 143)(31 100 86 162)(32 119 87 136)(33 93 88 155)(34 112 89 174)(35 131 90 148)(36 105 46 167)(37 124 47 141)(38 98 48 160)(39 117 49 179)(40 91 50 153)(41 110 51 172)(42 129 52 146)(43 103 53 165)(44 122 54 139)(45 96 55 158)
(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(91 156)(92 155)(93 154)(94 153)(95 152)(96 151)(97 150)(98 149)(99 148)(100 147)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 180)(113 179)(114 178)(115 177)(116 176)(117 175)(118 174)(119 173)(120 172)(121 171)(122 170)(123 169)(124 168)(125 167)(126 166)(127 165)(128 164)(129 163)(130 162)(131 161)(132 160)(133 159)(134 158)(135 157)
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), (1,115,56,177)(2,134,57,151)(3,108,58,170)(4,127,59,144)(5,101,60,163)(6,120,61,137)(7,94,62,156)(8,113,63,175)(9,132,64,149)(10,106,65,168)(11,125,66,142)(12,99,67,161)(13,118,68,180)(14,92,69,154)(15,111,70,173)(16,130,71,147)(17,104,72,166)(18,123,73,140)(19,97,74,159)(20,116,75,178)(21,135,76,152)(22,109,77,171)(23,128,78,145)(24,102,79,164)(25,121,80,138)(26,95,81,157)(27,114,82,176)(28,133,83,150)(29,107,84,169)(30,126,85,143)(31,100,86,162)(32,119,87,136)(33,93,88,155)(34,112,89,174)(35,131,90,148)(36,105,46,167)(37,124,47,141)(38,98,48,160)(39,117,49,179)(40,91,50,153)(41,110,51,172)(42,129,52,146)(43,103,53,165)(44,122,54,139)(45,96,55,158), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)>;
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), (1,115,56,177)(2,134,57,151)(3,108,58,170)(4,127,59,144)(5,101,60,163)(6,120,61,137)(7,94,62,156)(8,113,63,175)(9,132,64,149)(10,106,65,168)(11,125,66,142)(12,99,67,161)(13,118,68,180)(14,92,69,154)(15,111,70,173)(16,130,71,147)(17,104,72,166)(18,123,73,140)(19,97,74,159)(20,116,75,178)(21,135,76,152)(22,109,77,171)(23,128,78,145)(24,102,79,164)(25,121,80,138)(26,95,81,157)(27,114,82,176)(28,133,83,150)(29,107,84,169)(30,126,85,143)(31,100,86,162)(32,119,87,136)(33,93,88,155)(34,112,89,174)(35,131,90,148)(36,105,46,167)(37,124,47,141)(38,98,48,160)(39,117,49,179)(40,91,50,153)(41,110,51,172)(42,129,52,146)(43,103,53,165)(44,122,54,139)(45,96,55,158), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157) );
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)], [(1,115,56,177),(2,134,57,151),(3,108,58,170),(4,127,59,144),(5,101,60,163),(6,120,61,137),(7,94,62,156),(8,113,63,175),(9,132,64,149),(10,106,65,168),(11,125,66,142),(12,99,67,161),(13,118,68,180),(14,92,69,154),(15,111,70,173),(16,130,71,147),(17,104,72,166),(18,123,73,140),(19,97,74,159),(20,116,75,178),(21,135,76,152),(22,109,77,171),(23,128,78,145),(24,102,79,164),(25,121,80,138),(26,95,81,157),(27,114,82,176),(28,133,83,150),(29,107,84,169),(30,126,85,143),(31,100,86,162),(32,119,87,136),(33,93,88,155),(34,112,89,174),(35,131,90,148),(36,105,46,167),(37,124,47,141),(38,98,48,160),(39,117,49,179),(40,91,50,153),(41,110,51,172),(42,129,52,146),(43,103,53,165),(44,122,54,139),(45,96,55,158)], [(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(91,156),(92,155),(93,154),(94,153),(95,152),(96,151),(97,150),(98,149),(99,148),(100,147),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,140),(108,139),(109,138),(110,137),(111,136),(112,180),(113,179),(114,178),(115,177),(116,176),(117,175),(118,174),(119,173),(120,172),(121,171),(122,170),(123,169),(124,168),(125,167),(126,166),(127,165),(128,164),(129,163),(130,162),(131,161),(132,160),(133,159),(134,158),(135,157)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6 | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 18A | 18B | 18C | 30A | 30B | 36A | ··· | 36F | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 18 | 18 | 18 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 18 | 90 | 2 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D9 | D10 | D12 | D18 | C5⋊D4 | D36 | S3×D5 | C5⋊D12 | D5×D9 | C5⋊D36 |
| kernel | C5⋊D36 | C9×Dic5 | C10×D9 | D90 | C3×Dic5 | C45 | D18 | C30 | Dic5 | C18 | C15 | C10 | C9 | C5 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 2 | 2 | 6 | 6 |
Matrix representation of C5⋊D36 ►in GL4(𝔽181) generated by
| 13 | 180 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 177 | 131 |
| 0 | 0 | 50 | 127 |
| 74 | 100 | 0 | 0 |
| 157 | 107 | 0 | 0 |
| 0 | 0 | 180 | 0 |
| 0 | 0 | 0 | 180 |
| 1 | 0 | 0 | 0 |
| 13 | 180 | 0 | 0 |
| 0 | 0 | 54 | 177 |
| 0 | 0 | 50 | 127 |
G:=sub<GL(4,GF(181))| [13,1,0,0,180,0,0,0,0,0,177,50,0,0,131,127],[74,157,0,0,100,107,0,0,0,0,180,0,0,0,0,180],[1,13,0,0,0,180,0,0,0,0,54,50,0,0,177,127] >;
C5⋊D36 in GAP, Magma, Sage, TeX
C_5\rtimes D_{36} % in TeX
G:=Group("C5:D36"); // GroupNames label
G:=SmallGroup(360,10);
// by ID
G=gap.SmallGroup(360,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,73,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c|a^45=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export