metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C45⋊2D4, D10⋊1D9, D18⋊2D5, C30.6D6, C10.6D18, C18.6D10, Dic45⋊4C2, C90.6C22, C9⋊2(C5⋊D4), C5⋊2(C9⋊D4), C2.6(D5×D9), (D5×C18)⋊1C2, (C10×D9)⋊2C2, C3.(C15⋊D4), (C6×D5).2S3, C6.13(S3×D5), C15.1(C3⋊D4), SmallGroup(360,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C45⋊D4
G = < a,b,c,d | a10=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=c-1 >
Smallest permutation representation of C45⋊D4
►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)
(1 102)(2 101)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 141)(12 150)(13 149)(14 148)(15 147)(16 146)(17 145)(18 144)(19 143)(20 142)(21 153)(22 152)(23 151)(24 160)(25 159)(26 158)(27 157)(28 156)(29 155)(30 154)(31 123)(32 122)(33 121)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 93)(42 92)(43 91)(44 100)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 115)(52 114)(53 113)(54 112)(55 111)(56 120)(57 119)(58 118)(59 117)(60 116)(61 133)(62 132)(63 131)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 164)(72 163)(73 162)(74 161)(75 170)(76 169)(77 168)(78 167)(79 166)(80 165)(81 178)(82 177)(83 176)(84 175)(85 174)(86 173)(87 172)(88 171)(89 180)(90 179)
(1 147 27 33 118 68 83 74 100)(2 148 28 34 119 69 84 75 91)(3 149 29 35 120 70 85 76 92)(4 150 30 36 111 61 86 77 93)(5 141 21 37 112 62 87 78 94)(6 142 22 38 113 63 88 79 95)(7 143 23 39 114 64 89 80 96)(8 144 24 40 115 65 90 71 97)(9 145 25 31 116 66 81 72 98)(10 146 26 32 117 67 82 73 99)(11 153 127 54 132 172 167 50 108)(12 154 128 55 133 173 168 41 109)(13 155 129 56 134 174 169 42 110)(14 156 130 57 135 175 170 43 101)(15 157 121 58 136 176 161 44 102)(16 158 122 59 137 177 162 45 103)(17 159 123 60 138 178 163 46 104)(18 160 124 51 139 179 164 47 105)(19 151 125 52 140 180 165 48 106)(20 152 126 53 131 171 166 49 107)
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 161)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 101)(49 102)(50 103)(51 56)(52 57)(53 58)(54 59)(55 60)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 141)(79 142)(80 143)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(151 175)(152 176)(153 177)(154 178)(155 179)(156 180)(157 171)(158 172)(159 173)(160 174)
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), (1,102)(2,101)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,141)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,153)(22,152)(23,151)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,123)(32,122)(33,121)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,93)(42,92)(43,91)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,115)(52,114)(53,113)(54,112)(55,111)(56,120)(57,119)(58,118)(59,117)(60,116)(61,133)(62,132)(63,131)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,164)(72,163)(73,162)(74,161)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,180)(90,179), (1,147,27,33,118,68,83,74,100)(2,148,28,34,119,69,84,75,91)(3,149,29,35,120,70,85,76,92)(4,150,30,36,111,61,86,77,93)(5,141,21,37,112,62,87,78,94)(6,142,22,38,113,63,88,79,95)(7,143,23,39,114,64,89,80,96)(8,144,24,40,115,65,90,71,97)(9,145,25,31,116,66,81,72,98)(10,146,26,32,117,67,82,73,99)(11,153,127,54,132,172,167,50,108)(12,154,128,55,133,173,168,41,109)(13,155,129,56,134,174,169,42,110)(14,156,130,57,135,175,170,43,101)(15,157,121,58,136,176,161,44,102)(16,158,122,59,137,177,162,45,103)(17,159,123,60,138,178,163,46,104)(18,160,124,51,139,179,164,47,105)(19,151,125,52,140,180,165,48,106)(20,152,126,53,131,171,166,49,107), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,161)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,56)(52,57)(53,58)(54,59)(55,60)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,141)(79,142)(80,143)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,171)(158,172)(159,173)(160,174)>;
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), (1,102)(2,101)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,141)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,153)(22,152)(23,151)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,123)(32,122)(33,121)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,93)(42,92)(43,91)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,115)(52,114)(53,113)(54,112)(55,111)(56,120)(57,119)(58,118)(59,117)(60,116)(61,133)(62,132)(63,131)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,164)(72,163)(73,162)(74,161)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,180)(90,179), (1,147,27,33,118,68,83,74,100)(2,148,28,34,119,69,84,75,91)(3,149,29,35,120,70,85,76,92)(4,150,30,36,111,61,86,77,93)(5,141,21,37,112,62,87,78,94)(6,142,22,38,113,63,88,79,95)(7,143,23,39,114,64,89,80,96)(8,144,24,40,115,65,90,71,97)(9,145,25,31,116,66,81,72,98)(10,146,26,32,117,67,82,73,99)(11,153,127,54,132,172,167,50,108)(12,154,128,55,133,173,168,41,109)(13,155,129,56,134,174,169,42,110)(14,156,130,57,135,175,170,43,101)(15,157,121,58,136,176,161,44,102)(16,158,122,59,137,177,162,45,103)(17,159,123,60,138,178,163,46,104)(18,160,124,51,139,179,164,47,105)(19,151,125,52,140,180,165,48,106)(20,152,126,53,131,171,166,49,107), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,161)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,56)(52,57)(53,58)(54,59)(55,60)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,141)(79,142)(80,143)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,171)(158,172)(159,173)(160,174) );
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)], [(1,102),(2,101),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,141),(12,150),(13,149),(14,148),(15,147),(16,146),(17,145),(18,144),(19,143),(20,142),(21,153),(22,152),(23,151),(24,160),(25,159),(26,158),(27,157),(28,156),(29,155),(30,154),(31,123),(32,122),(33,121),(34,130),(35,129),(36,128),(37,127),(38,126),(39,125),(40,124),(41,93),(42,92),(43,91),(44,100),(45,99),(46,98),(47,97),(48,96),(49,95),(50,94),(51,115),(52,114),(53,113),(54,112),(55,111),(56,120),(57,119),(58,118),(59,117),(60,116),(61,133),(62,132),(63,131),(64,140),(65,139),(66,138),(67,137),(68,136),(69,135),(70,134),(71,164),(72,163),(73,162),(74,161),(75,170),(76,169),(77,168),(78,167),(79,166),(80,165),(81,178),(82,177),(83,176),(84,175),(85,174),(86,173),(87,172),(88,171),(89,180),(90,179)], [(1,147,27,33,118,68,83,74,100),(2,148,28,34,119,69,84,75,91),(3,149,29,35,120,70,85,76,92),(4,150,30,36,111,61,86,77,93),(5,141,21,37,112,62,87,78,94),(6,142,22,38,113,63,88,79,95),(7,143,23,39,114,64,89,80,96),(8,144,24,40,115,65,90,71,97),(9,145,25,31,116,66,81,72,98),(10,146,26,32,117,67,82,73,99),(11,153,127,54,132,172,167,50,108),(12,154,128,55,133,173,168,41,109),(13,155,129,56,134,174,169,42,110),(14,156,130,57,135,175,170,43,101),(15,157,121,58,136,176,161,44,102),(16,158,122,59,137,177,162,45,103),(17,159,123,60,138,178,163,46,104),(18,160,124,51,139,179,164,47,105),(19,151,125,52,140,180,165,48,106),(20,152,126,53,131,171,166,49,107)], [(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,169),(19,170),(20,161),(21,87),(22,88),(23,89),(24,90),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,101),(49,102),(50,103),(51,56),(52,57),(53,58),(54,59),(55,60),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,141),(79,142),(80,143),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(151,175),(152,176),(153,177),(154,178),(155,179),(156,180),(157,171),(158,172),(159,173),(160,174)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 18A | 18B | 18C | 18D | ··· | 18I | 30A | 30B | 45A | ··· | 45F | 90A | ··· | 90F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 18 | 18 | 18 | 18 | ··· | 18 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 10 | 18 | 2 | 90 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 4 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 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 | C5⋊D4 | C9⋊D4 | S3×D5 | C15⋊D4 | D5×D9 | C45⋊D4 |
| kernel | C45⋊D4 | Dic45 | D5×C18 | C10×D9 | C6×D5 | C45 | D18 | 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 C45⋊D4 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 122 | 122 |
| 180 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 37 | 118 |
| 0 | 0 | 151 | 144 |
| 50 | 177 | 0 | 0 |
| 4 | 54 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 131 | 0 | 0 |
| 127 | 177 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 73 | 180 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,0,0,0,0,46,122,0,0,0,122],[180,0,0,0,0,180,0,0,0,0,37,151,0,0,118,144],[50,4,0,0,177,54,0,0,0,0,1,0,0,0,0,1],[4,127,0,0,131,177,0,0,0,0,1,73,0,0,0,180] >;
C45⋊D4 in GAP, Magma, Sage, TeX
C_{45}\rtimes D_4 % in TeX
G:=Group("C45:D4"); // GroupNames label
G:=SmallGroup(360,12);
// by ID
G=gap.SmallGroup(360,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,73,1641,741,2884,4331]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations
Export