direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×C2×C10, C45⋊3C23, C90⋊3C22, C30.61D6, C18⋊(C2×C10), C9⋊(C22×C10), (C2×C90)⋊5C2, (C2×C18)⋊3C10, C6.11(S3×C10), (C2×C30).10S3, C15.3(C22×S3), C3.(S3×C2×C10), (C2×C6).4(C5×S3), SmallGroup(360,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C2×C10 |
Generators and relations for D9×C2×C10
G = < a,b,c,d | a2=b10=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 316 in 96 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C9, C10, C10, D6, C2×C6, C15, D9, C18, C2×C10, C2×C10, C22×S3, C5×S3, C30, D18, C2×C18, C22×C10, C45, S3×C10, C2×C30, C22×D9, C5×D9, C90, S3×C2×C10, C10×D9, C2×C90, D9×C2×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, D9, C2×C10, C22×S3, C5×S3, D18, C22×C10, S3×C10, C22×D9, C5×D9, S3×C2×C10, C10×D9, D9×C2×C10
►On 180 points
Generators in S180
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 171)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 151)(18 152)(19 153)(20 154)(21 165)(22 166)(23 167)(24 168)(25 169)(26 170)(27 161)(28 162)(29 163)(30 164)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 91)(40 92)(41 140)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 150)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 101)(68 102)(69 103)(70 104)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 111)(78 112)(79 113)(80 114)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 121)(88 122)(89 123)(90 124)
(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 98 141 157 118 133 122 101 167)(2 99 142 158 119 134 123 102 168)(3 100 143 159 120 135 124 103 169)(4 91 144 160 111 136 125 104 170)(5 92 145 151 112 137 126 105 161)(6 93 146 152 113 138 127 106 162)(7 94 147 153 114 139 128 107 163)(8 95 148 154 115 140 129 108 164)(9 96 149 155 116 131 130 109 165)(10 97 150 156 117 132 121 110 166)(11 72 42 86 65 21 180 34 60)(12 73 43 87 66 22 171 35 51)(13 74 44 88 67 23 172 36 52)(14 75 45 89 68 24 173 37 53)(15 76 46 90 69 25 174 38 54)(16 77 47 81 70 26 175 39 55)(17 78 48 82 61 27 176 40 56)(18 79 49 83 62 28 177 31 57)(19 80 50 84 63 29 178 32 58)(20 71 41 85 64 30 179 33 59)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 136)(12 137)(13 138)(14 139)(15 140)(16 131)(17 132)(18 133)(19 134)(20 135)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 159)(42 160)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 97)(62 98)(63 99)(64 100)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 120)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 149)(82 150)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)
G:=sub<Sym(180)| (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,171)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,151)(18,152)(19,153)(20,154)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,161)(28,162)(29,163)(30,164)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,91)(40,92)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,150)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,101)(68,102)(69,103)(70,104)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,111)(78,112)(79,113)(80,114)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124), (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), (1,98,141,157,118,133,122,101,167)(2,99,142,158,119,134,123,102,168)(3,100,143,159,120,135,124,103,169)(4,91,144,160,111,136,125,104,170)(5,92,145,151,112,137,126,105,161)(6,93,146,152,113,138,127,106,162)(7,94,147,153,114,139,128,107,163)(8,95,148,154,115,140,129,108,164)(9,96,149,155,116,131,130,109,165)(10,97,150,156,117,132,121,110,166)(11,72,42,86,65,21,180,34,60)(12,73,43,87,66,22,171,35,51)(13,74,44,88,67,23,172,36,52)(14,75,45,89,68,24,173,37,53)(15,76,46,90,69,25,174,38,54)(16,77,47,81,70,26,175,39,55)(17,78,48,82,61,27,176,40,56)(18,79,49,83,62,28,177,31,57)(19,80,50,84,63,29,178,32,58)(20,71,41,85,64,30,179,33,59), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,136)(12,137)(13,138)(14,139)(15,140)(16,131)(17,132)(18,133)(19,134)(20,135)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,159)(42,160)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,126)(52,127)(53,128)(54,129)(55,130)(56,121)(57,122)(58,123)(59,124)(60,125)(61,97)(62,98)(63,99)(64,100)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,120)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,149)(82,150)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)>;
G:=Group( (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,171)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,151)(18,152)(19,153)(20,154)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,161)(28,162)(29,163)(30,164)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,91)(40,92)(41,140)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,150)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,101)(68,102)(69,103)(70,104)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,111)(78,112)(79,113)(80,114)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,121)(88,122)(89,123)(90,124), (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), (1,98,141,157,118,133,122,101,167)(2,99,142,158,119,134,123,102,168)(3,100,143,159,120,135,124,103,169)(4,91,144,160,111,136,125,104,170)(5,92,145,151,112,137,126,105,161)(6,93,146,152,113,138,127,106,162)(7,94,147,153,114,139,128,107,163)(8,95,148,154,115,140,129,108,164)(9,96,149,155,116,131,130,109,165)(10,97,150,156,117,132,121,110,166)(11,72,42,86,65,21,180,34,60)(12,73,43,87,66,22,171,35,51)(13,74,44,88,67,23,172,36,52)(14,75,45,89,68,24,173,37,53)(15,76,46,90,69,25,174,38,54)(16,77,47,81,70,26,175,39,55)(17,78,48,82,61,27,176,40,56)(18,79,49,83,62,28,177,31,57)(19,80,50,84,63,29,178,32,58)(20,71,41,85,64,30,179,33,59), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,136)(12,137)(13,138)(14,139)(15,140)(16,131)(17,132)(18,133)(19,134)(20,135)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,159)(42,160)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,126)(52,127)(53,128)(54,129)(55,130)(56,121)(57,122)(58,123)(59,124)(60,125)(61,97)(62,98)(63,99)(64,100)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,120)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,149)(82,150)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180) );
G=PermutationGroup([[(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,171),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,151),(18,152),(19,153),(20,154),(21,165),(22,166),(23,167),(24,168),(25,169),(26,170),(27,161),(28,162),(29,163),(30,164),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,91),(40,92),(41,140),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,150),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,101),(68,102),(69,103),(70,104),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,111),(78,112),(79,113),(80,114),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,121),(88,122),(89,123),(90,124)], [(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)], [(1,98,141,157,118,133,122,101,167),(2,99,142,158,119,134,123,102,168),(3,100,143,159,120,135,124,103,169),(4,91,144,160,111,136,125,104,170),(5,92,145,151,112,137,126,105,161),(6,93,146,152,113,138,127,106,162),(7,94,147,153,114,139,128,107,163),(8,95,148,154,115,140,129,108,164),(9,96,149,155,116,131,130,109,165),(10,97,150,156,117,132,121,110,166),(11,72,42,86,65,21,180,34,60),(12,73,43,87,66,22,171,35,51),(13,74,44,88,67,23,172,36,52),(14,75,45,89,68,24,173,37,53),(15,76,46,90,69,25,174,38,54),(16,77,47,81,70,26,175,39,55),(17,78,48,82,61,27,176,40,56),(18,79,49,83,62,28,177,31,57),(19,80,50,84,63,29,178,32,58),(20,71,41,85,64,30,179,33,59)], [(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,136),(12,137),(13,138),(14,139),(15,140),(16,131),(17,132),(18,133),(19,134),(20,135),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,159),(42,160),(43,151),(44,152),(45,153),(46,154),(47,155),(48,156),(49,157),(50,158),(51,126),(52,127),(53,128),(54,129),(55,130),(56,121),(57,122),(58,123),(59,124),(60,125),(61,97),(62,98),(63,99),(64,100),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,120),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,149),(82,150),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178),(169,179),(170,180)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 9A | 9B | 9C | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | 15B | 15C | 15D | 18A | ··· | 18I | 30A | ··· | 30L | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | D9 | C5×S3 | D18 | S3×C10 | C5×D9 | C10×D9 |
| kernel | D9×C2×C10 | C10×D9 | C2×C90 | C22×D9 | D18 | C2×C18 | C2×C30 | C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 4 | 24 | 4 | 1 | 3 | 3 | 4 | 9 | 12 | 12 | 36 |
Matrix representation of D9×C2×C10 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 180 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 0 |
| 0 | 0 | 0 | 59 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 131 | 54 |
| 0 | 0 | 127 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 50 |
| 0 | 0 | 54 | 177 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,180,0,0,0,0,1,0,0,0,0,1],[180,0,0,0,0,1,0,0,0,0,59,0,0,0,0,59],[1,0,0,0,0,1,0,0,0,0,131,127,0,0,54,4],[1,0,0,0,0,1,0,0,0,0,4,54,0,0,50,177] >;
D9×C2×C10 in GAP, Magma, Sage, TeX
D_9\times C_2\times C_{10} % in TeX
G:=Group("D9xC2xC10"); // GroupNames label
G:=SmallGroup(360,48);
// by ID
G=gap.SmallGroup(360,48);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-3,-3,6004,208,8645]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations