metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C45⋊3D4, C9⋊2D20, Dic9⋊D5, D90⋊4C2, D10⋊2D9, C30.7D6, C10.7D18, C18.7D10, C90.7C22, C5⋊1(C9⋊D4), C2.7(D5×D9), (D5×C18)⋊2C2, C3.(C3⋊D20), (C6×D5).3S3, C6.14(S3×D5), (C5×Dic9)⋊3C2, C15.2(C3⋊D4), SmallGroup(360,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊D20
G = < a,b,c | a9=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
10C2
90C2
5C22
9C4
45C22
10C6
30S3
2D5
18D5
45D4
3Dic3
15D6
10C18
10D9
9C20
9D10
6D15
15C3⋊D4
5D18
9D20
3D30
2D45
Smallest permutation representation of C9⋊D20
►On 180 points
Generators in S180
(1 87 71 32 166 104 122 142 60)(2 41 143 123 105 167 33 72 88)(3 89 73 34 168 106 124 144 42)(4 43 145 125 107 169 35 74 90)(5 91 75 36 170 108 126 146 44)(6 45 147 127 109 171 37 76 92)(7 93 77 38 172 110 128 148 46)(8 47 149 129 111 173 39 78 94)(9 95 79 40 174 112 130 150 48)(10 49 151 131 113 175 21 80 96)(11 97 61 22 176 114 132 152 50)(12 51 153 133 115 177 23 62 98)(13 99 63 24 178 116 134 154 52)(14 53 155 135 117 179 25 64 100)(15 81 65 26 180 118 136 156 54)(16 55 157 137 119 161 27 66 82)(17 83 67 28 162 120 138 158 56)(18 57 159 139 101 163 29 68 84)(19 85 69 30 164 102 140 160 58)(20 59 141 121 103 165 31 70 86)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 127)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 133)(36 132)(37 131)(38 130)(39 129)(40 128)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 146)(62 145)(63 144)(64 143)(65 142)(66 141)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)(101 163)(102 162)(103 161)(104 180)(105 179)(106 178)(107 177)(108 176)(109 175)(110 174)(111 173)(112 172)(113 171)(114 170)(115 169)(116 168)(117 167)(118 166)(119 165)(120 164)
G:=sub<Sym(180)| (1,87,71,32,166,104,122,142,60)(2,41,143,123,105,167,33,72,88)(3,89,73,34,168,106,124,144,42)(4,43,145,125,107,169,35,74,90)(5,91,75,36,170,108,126,146,44)(6,45,147,127,109,171,37,76,92)(7,93,77,38,172,110,128,148,46)(8,47,149,129,111,173,39,78,94)(9,95,79,40,174,112,130,150,48)(10,49,151,131,113,175,21,80,96)(11,97,61,22,176,114,132,152,50)(12,51,153,133,115,177,23,62,98)(13,99,63,24,178,116,134,154,52)(14,53,155,135,117,179,25,64,100)(15,81,65,26,180,118,136,156,54)(16,55,157,137,119,161,27,66,82)(17,83,67,28,162,120,138,158,56)(18,57,159,139,101,163,29,68,84)(19,85,69,30,164,102,140,160,58)(20,59,141,121,103,165,31,70,86), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(101,163)(102,162)(103,161)(104,180)(105,179)(106,178)(107,177)(108,176)(109,175)(110,174)(111,173)(112,172)(113,171)(114,170)(115,169)(116,168)(117,167)(118,166)(119,165)(120,164)>;
G:=Group( (1,87,71,32,166,104,122,142,60)(2,41,143,123,105,167,33,72,88)(3,89,73,34,168,106,124,144,42)(4,43,145,125,107,169,35,74,90)(5,91,75,36,170,108,126,146,44)(6,45,147,127,109,171,37,76,92)(7,93,77,38,172,110,128,148,46)(8,47,149,129,111,173,39,78,94)(9,95,79,40,174,112,130,150,48)(10,49,151,131,113,175,21,80,96)(11,97,61,22,176,114,132,152,50)(12,51,153,133,115,177,23,62,98)(13,99,63,24,178,116,134,154,52)(14,53,155,135,117,179,25,64,100)(15,81,65,26,180,118,136,156,54)(16,55,157,137,119,161,27,66,82)(17,83,67,28,162,120,138,158,56)(18,57,159,139,101,163,29,68,84)(19,85,69,30,164,102,140,160,58)(20,59,141,121,103,165,31,70,86), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(101,163)(102,162)(103,161)(104,180)(105,179)(106,178)(107,177)(108,176)(109,175)(110,174)(111,173)(112,172)(113,171)(114,170)(115,169)(116,168)(117,167)(118,166)(119,165)(120,164) );
G=PermutationGroup([[(1,87,71,32,166,104,122,142,60),(2,41,143,123,105,167,33,72,88),(3,89,73,34,168,106,124,144,42),(4,43,145,125,107,169,35,74,90),(5,91,75,36,170,108,126,146,44),(6,45,147,127,109,171,37,76,92),(7,93,77,38,172,110,128,148,46),(8,47,149,129,111,173,39,78,94),(9,95,79,40,174,112,130,150,48),(10,49,151,131,113,175,21,80,96),(11,97,61,22,176,114,132,152,50),(12,51,153,133,115,177,23,62,98),(13,99,63,24,178,116,134,154,52),(14,53,155,135,117,179,25,64,100),(15,81,65,26,180,118,136,156,54),(16,55,157,137,119,161,27,66,82),(17,83,67,28,162,120,138,158,56),(18,57,159,139,101,163,29,68,84),(19,85,69,30,164,102,140,160,58),(20,59,141,121,103,165,31,70,86)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,127),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,140),(29,139),(30,138),(31,137),(32,136),(33,135),(34,134),(35,133),(36,132),(37,131),(38,130),(39,129),(40,128),(41,100),(42,99),(43,98),(44,97),(45,96),(46,95),(47,94),(48,93),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(61,146),(62,145),(63,144),(64,143),(65,142),(66,141),(67,160),(68,159),(69,158),(70,157),(71,156),(72,155),(73,154),(74,153),(75,152),(76,151),(77,150),(78,149),(79,148),(80,147),(101,163),(102,162),(103,161),(104,180),(105,179),(106,178),(107,177),(108,176),(109,175),(110,174),(111,173),(112,172),(113,171),(114,170),(115,169),(116,168),(117,167),(118,166),(119,165),(120,164)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | 10B | 15A | 15B | 18A | 18B | 18C | 18D | ··· | 18I | 20A | 20B | 20C | 20D | 30A | 30B | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | 10 | 15 | 15 | 18 | 18 | 18 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 10 | 90 | 2 | 18 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 18 | 18 | 18 | 18 | 4 | 4 | 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 | C3⋊D4 | D18 | D20 | C9⋊D4 | S3×D5 | C3⋊D20 | D5×D9 | C9⋊D20 |
| kernel | C9⋊D20 | C5×Dic9 | D5×C18 | D90 | C6×D5 | C45 | Dic9 | C30 | D10 | 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 C9⋊D20 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 131 | 54 |
| 0 | 0 | 127 | 4 |
| 180 | 178 | 0 | 0 |
| 117 | 169 | 0 | 0 |
| 0 | 0 | 176 | 21 |
| 0 | 0 | 16 | 5 |
| 1 | 3 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 50 | 127 |
| 0 | 0 | 177 | 131 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,0,0,0,0,131,127,0,0,54,4],[180,117,0,0,178,169,0,0,0,0,176,16,0,0,21,5],[1,0,0,0,3,180,0,0,0,0,50,177,0,0,127,131] >;
C9⋊D20 in GAP, Magma, Sage, TeX
C_9\rtimes D_{20} % in TeX
G:=Group("C9:D20"); // GroupNames label
G:=SmallGroup(360,13);
// by ID
G=gap.SmallGroup(360,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,73,31,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c|a^9=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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