metabelian, supersoluble, monomial, A-group
Aliases: C5⋊D45, C45⋊1D5, C52⋊3D9, C15.3D15, C9⋊(C5⋊D5), (C5×C45)⋊1C2, C3.(C5⋊D15), (C5×C15).2S3, SmallGroup(450,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C45 — C5⋊D45 |
Generators and relations for C5⋊D45
G = < a,b,c | a5=b45=c2=1, ab=ba, cac=a-1, cbc=b-1 >
225C2
75S3
45D5
45D5
45D5
45D5
45D5
45D5
25D9
15D15
15D15
15D15
15D15
15D15
15D15
5D45
5D45
5D45
5D45
5D45
5D45
Smallest permutation representation of C5⋊D45
►On 225 points
Generators in S225
(1 187 75 173 126)(2 188 76 174 127)(3 189 77 175 128)(4 190 78 176 129)(5 191 79 177 130)(6 192 80 178 131)(7 193 81 179 132)(8 194 82 180 133)(9 195 83 136 134)(10 196 84 137 135)(11 197 85 138 91)(12 198 86 139 92)(13 199 87 140 93)(14 200 88 141 94)(15 201 89 142 95)(16 202 90 143 96)(17 203 46 144 97)(18 204 47 145 98)(19 205 48 146 99)(20 206 49 147 100)(21 207 50 148 101)(22 208 51 149 102)(23 209 52 150 103)(24 210 53 151 104)(25 211 54 152 105)(26 212 55 153 106)(27 213 56 154 107)(28 214 57 155 108)(29 215 58 156 109)(30 216 59 157 110)(31 217 60 158 111)(32 218 61 159 112)(33 219 62 160 113)(34 220 63 161 114)(35 221 64 162 115)(36 222 65 163 116)(37 223 66 164 117)(38 224 67 165 118)(39 225 68 166 119)(40 181 69 167 120)(41 182 70 168 121)(42 183 71 169 122)(43 184 72 170 123)(44 185 73 171 124)(45 186 74 172 125)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)
(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)(46 157)(47 156)(48 155)(49 154)(50 153)(51 152)(52 151)(53 150)(54 149)(55 148)(56 147)(57 146)(58 145)(59 144)(60 143)(61 142)(62 141)(63 140)(64 139)(65 138)(66 137)(67 136)(68 180)(69 179)(70 178)(71 177)(72 176)(73 175)(74 174)(75 173)(76 172)(77 171)(78 170)(79 169)(80 168)(81 167)(82 166)(83 165)(84 164)(85 163)(86 162)(87 161)(88 160)(89 159)(90 158)(91 222)(92 221)(93 220)(94 219)(95 218)(96 217)(97 216)(98 215)(99 214)(100 213)(101 212)(102 211)(103 210)(104 209)(105 208)(106 207)(107 206)(108 205)(109 204)(110 203)(111 202)(112 201)(113 200)(114 199)(115 198)(116 197)(117 196)(118 195)(119 194)(120 193)(121 192)(122 191)(123 190)(124 189)(125 188)(126 187)(127 186)(128 185)(129 184)(130 183)(131 182)(132 181)(133 225)(134 224)(135 223)
G:=sub<Sym(225)| (1,187,75,173,126)(2,188,76,174,127)(3,189,77,175,128)(4,190,78,176,129)(5,191,79,177,130)(6,192,80,178,131)(7,193,81,179,132)(8,194,82,180,133)(9,195,83,136,134)(10,196,84,137,135)(11,197,85,138,91)(12,198,86,139,92)(13,199,87,140,93)(14,200,88,141,94)(15,201,89,142,95)(16,202,90,143,96)(17,203,46,144,97)(18,204,47,145,98)(19,205,48,146,99)(20,206,49,147,100)(21,207,50,148,101)(22,208,51,149,102)(23,209,52,150,103)(24,210,53,151,104)(25,211,54,152,105)(26,212,55,153,106)(27,213,56,154,107)(28,214,57,155,108)(29,215,58,156,109)(30,216,59,157,110)(31,217,60,158,111)(32,218,61,159,112)(33,219,62,160,113)(34,220,63,161,114)(35,221,64,162,115)(36,222,65,163,116)(37,223,66,164,117)(38,224,67,165,118)(39,225,68,166,119)(40,181,69,167,120)(41,182,70,168,121)(42,183,71,169,122)(43,184,72,170,123)(44,185,73,171,124)(45,186,74,172,125), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (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)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,166)(83,165)(84,164)(85,163)(86,162)(87,161)(88,160)(89,159)(90,158)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,225)(134,224)(135,223)>;
G:=Group( (1,187,75,173,126)(2,188,76,174,127)(3,189,77,175,128)(4,190,78,176,129)(5,191,79,177,130)(6,192,80,178,131)(7,193,81,179,132)(8,194,82,180,133)(9,195,83,136,134)(10,196,84,137,135)(11,197,85,138,91)(12,198,86,139,92)(13,199,87,140,93)(14,200,88,141,94)(15,201,89,142,95)(16,202,90,143,96)(17,203,46,144,97)(18,204,47,145,98)(19,205,48,146,99)(20,206,49,147,100)(21,207,50,148,101)(22,208,51,149,102)(23,209,52,150,103)(24,210,53,151,104)(25,211,54,152,105)(26,212,55,153,106)(27,213,56,154,107)(28,214,57,155,108)(29,215,58,156,109)(30,216,59,157,110)(31,217,60,158,111)(32,218,61,159,112)(33,219,62,160,113)(34,220,63,161,114)(35,221,64,162,115)(36,222,65,163,116)(37,223,66,164,117)(38,224,67,165,118)(39,225,68,166,119)(40,181,69,167,120)(41,182,70,168,121)(42,183,71,169,122)(43,184,72,170,123)(44,185,73,171,124)(45,186,74,172,125), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225), (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)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,166)(83,165)(84,164)(85,163)(86,162)(87,161)(88,160)(89,159)(90,158)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,225)(134,224)(135,223) );
G=PermutationGroup([[(1,187,75,173,126),(2,188,76,174,127),(3,189,77,175,128),(4,190,78,176,129),(5,191,79,177,130),(6,192,80,178,131),(7,193,81,179,132),(8,194,82,180,133),(9,195,83,136,134),(10,196,84,137,135),(11,197,85,138,91),(12,198,86,139,92),(13,199,87,140,93),(14,200,88,141,94),(15,201,89,142,95),(16,202,90,143,96),(17,203,46,144,97),(18,204,47,145,98),(19,205,48,146,99),(20,206,49,147,100),(21,207,50,148,101),(22,208,51,149,102),(23,209,52,150,103),(24,210,53,151,104),(25,211,54,152,105),(26,212,55,153,106),(27,213,56,154,107),(28,214,57,155,108),(29,215,58,156,109),(30,216,59,157,110),(31,217,60,158,111),(32,218,61,159,112),(33,219,62,160,113),(34,220,63,161,114),(35,221,64,162,115),(36,222,65,163,116),(37,223,66,164,117),(38,224,67,165,118),(39,225,68,166,119),(40,181,69,167,120),(41,182,70,168,121),(42,183,71,169,122),(43,184,72,170,123),(44,185,73,171,124),(45,186,74,172,125)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)], [(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),(46,157),(47,156),(48,155),(49,154),(50,153),(51,152),(52,151),(53,150),(54,149),(55,148),(56,147),(57,146),(58,145),(59,144),(60,143),(61,142),(62,141),(63,140),(64,139),(65,138),(66,137),(67,136),(68,180),(69,179),(70,178),(71,177),(72,176),(73,175),(74,174),(75,173),(76,172),(77,171),(78,170),(79,169),(80,168),(81,167),(82,166),(83,165),(84,164),(85,163),(86,162),(87,161),(88,160),(89,159),(90,158),(91,222),(92,221),(93,220),(94,219),(95,218),(96,217),(97,216),(98,215),(99,214),(100,213),(101,212),(102,211),(103,210),(104,209),(105,208),(106,207),(107,206),(108,205),(109,204),(110,203),(111,202),(112,201),(113,200),(114,199),(115,198),(116,197),(117,196),(118,195),(119,194),(120,193),(121,192),(122,191),(123,190),(124,189),(125,188),(126,187),(127,186),(128,185),(129,184),(130,183),(131,182),(132,181),(133,225),(134,224),(135,223)]])
114 conjugacy classes
| class | 1 | 2 | 3 | 5A | ··· | 5L | 9A | 9B | 9C | 15A | ··· | 15X | 45A | ··· | 45BT |
| order | 1 | 2 | 3 | 5 | ··· | 5 | 9 | 9 | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 225 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D5 | D9 | D15 | D45 |
| kernel | C5⋊D45 | C5×C45 | C5×C15 | C45 | C52 | C15 | C5 |
| # reps | 1 | 1 | 1 | 12 | 3 | 24 | 72 |
Matrix representation of C5⋊D45 ►in GL4(𝔽181) generated by
| 180 | 180 | 0 | 0 |
| 15 | 14 | 0 | 0 |
| 0 | 0 | 119 | 137 |
| 0 | 0 | 44 | 75 |
| 167 | 168 | 0 | 0 |
| 14 | 0 | 0 | 0 |
| 0 | 0 | 63 | 90 |
| 0 | 0 | 91 | 153 |
| 167 | 168 | 0 | 0 |
| 15 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(181))| [180,15,0,0,180,14,0,0,0,0,119,44,0,0,137,75],[167,14,0,0,168,0,0,0,0,0,63,91,0,0,90,153],[167,15,0,0,168,14,0,0,0,0,0,1,0,0,1,0] >;
C5⋊D45 in GAP, Magma, Sage, TeX
C_5\rtimes D_{45} % in TeX
G:=Group("C5:D45"); // GroupNames label
G:=SmallGroup(450,18);
// by ID
G=gap.SmallGroup(450,18);
# by ID
G:=PCGroup([5,-2,-3,-5,-5,-3,1541,1506,362,2403,7504]);
// Polycyclic
G:=Group<a,b,c|a^5=b^45=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export