direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C9×C5⋊D4, C45⋊9D4, Dic5⋊C18, D10⋊2C18, C18.17D10, C90.22C22, C5⋊2(D4×C9), (C2×C90)⋊6C2, (C2×C18)⋊1D5, (C2×C10)⋊4C18, (D5×C18)⋊5C2, (C2×C30).6C6, (C6×D5).3C6, C6.17(C6×D5), C15.2(C3×D4), C2.5(D5×C18), C22⋊2(C9×D5), C30.17(C2×C6), C10.5(C2×C18), (C9×Dic5)⋊4C2, (C3×Dic5).3C6, C3.(C3×C5⋊D4), (C3×C5⋊D4).C3, (C2×C6).2(C3×D5), SmallGroup(360,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C5⋊D4
G = < a,b,c,d | a9=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C9×C5⋊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)
(1 161 134 76 88)(2 162 135 77 89)(3 154 127 78 90)(4 155 128 79 82)(5 156 129 80 83)(6 157 130 81 84)(7 158 131 73 85)(8 159 132 74 86)(9 160 133 75 87)(10 65 61 171 144)(11 66 62 163 136)(12 67 63 164 137)(13 68 55 165 138)(14 69 56 166 139)(15 70 57 167 140)(16 71 58 168 141)(17 72 59 169 142)(18 64 60 170 143)(19 95 117 101 46)(20 96 109 102 47)(21 97 110 103 48)(22 98 111 104 49)(23 99 112 105 50)(24 91 113 106 51)(25 92 114 107 52)(26 93 115 108 53)(27 94 116 100 54)(28 118 173 149 44)(29 119 174 150 45)(30 120 175 151 37)(31 121 176 152 38)(32 122 177 153 39)(33 123 178 145 40)(34 124 179 146 41)(35 125 180 147 42)(36 126 172 148 43)
(1 145 102 57)(2 146 103 58)(3 147 104 59)(4 148 105 60)(5 149 106 61)(6 150 107 62)(7 151 108 63)(8 152 100 55)(9 153 101 56)(10 129 118 24)(11 130 119 25)(12 131 120 26)(13 132 121 27)(14 133 122 19)(15 134 123 20)(16 135 124 21)(17 127 125 22)(18 128 126 23)(28 91 144 80)(29 92 136 81)(30 93 137 73)(31 94 138 74)(32 95 139 75)(33 96 140 76)(34 97 141 77)(35 98 142 78)(36 99 143 79)(37 115 164 85)(38 116 165 86)(39 117 166 87)(40 109 167 88)(41 110 168 89)(42 111 169 90)(43 112 170 82)(44 113 171 83)(45 114 163 84)(46 69 160 177)(47 70 161 178)(48 71 162 179)(49 72 154 180)(50 64 155 172)(51 65 156 173)(52 66 157 174)(53 67 158 175)(54 68 159 176)
(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 95)(20 96)(21 97)(22 98)(23 99)(24 91)(25 92)(26 93)(27 94)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 64)(44 65)(45 66)(46 117)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 152)(56 153)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(73 131)(74 132)(75 133)(76 134)(77 135)(78 127)(79 128)(80 129)(81 130)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 154)(118 144)(119 136)(120 137)(121 138)(122 139)(123 140)(124 141)(125 142)(126 143)(163 174)(164 175)(165 176)(166 177)(167 178)(168 179)(169 180)(170 172)(171 173)
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), (1,161,134,76,88)(2,162,135,77,89)(3,154,127,78,90)(4,155,128,79,82)(5,156,129,80,83)(6,157,130,81,84)(7,158,131,73,85)(8,159,132,74,86)(9,160,133,75,87)(10,65,61,171,144)(11,66,62,163,136)(12,67,63,164,137)(13,68,55,165,138)(14,69,56,166,139)(15,70,57,167,140)(16,71,58,168,141)(17,72,59,169,142)(18,64,60,170,143)(19,95,117,101,46)(20,96,109,102,47)(21,97,110,103,48)(22,98,111,104,49)(23,99,112,105,50)(24,91,113,106,51)(25,92,114,107,52)(26,93,115,108,53)(27,94,116,100,54)(28,118,173,149,44)(29,119,174,150,45)(30,120,175,151,37)(31,121,176,152,38)(32,122,177,153,39)(33,123,178,145,40)(34,124,179,146,41)(35,125,180,147,42)(36,126,172,148,43), (1,145,102,57)(2,146,103,58)(3,147,104,59)(4,148,105,60)(5,149,106,61)(6,150,107,62)(7,151,108,63)(8,152,100,55)(9,153,101,56)(10,129,118,24)(11,130,119,25)(12,131,120,26)(13,132,121,27)(14,133,122,19)(15,134,123,20)(16,135,124,21)(17,127,125,22)(18,128,126,23)(28,91,144,80)(29,92,136,81)(30,93,137,73)(31,94,138,74)(32,95,139,75)(33,96,140,76)(34,97,141,77)(35,98,142,78)(36,99,143,79)(37,115,164,85)(38,116,165,86)(39,117,166,87)(40,109,167,88)(41,110,168,89)(42,111,169,90)(43,112,170,82)(44,113,171,83)(45,114,163,84)(46,69,160,177)(47,70,161,178)(48,71,162,179)(49,72,154,180)(50,64,155,172)(51,65,156,173)(52,66,157,174)(53,67,158,175)(54,68,159,176), (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,95)(20,96)(21,97)(22,98)(23,99)(24,91)(25,92)(26,93)(27,94)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,64)(44,65)(45,66)(46,117)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,152)(56,153)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(73,131)(74,132)(75,133)(76,134)(77,135)(78,127)(79,128)(80,129)(81,130)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,154)(118,144)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179)(169,180)(170,172)(171,173)>;
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), (1,161,134,76,88)(2,162,135,77,89)(3,154,127,78,90)(4,155,128,79,82)(5,156,129,80,83)(6,157,130,81,84)(7,158,131,73,85)(8,159,132,74,86)(9,160,133,75,87)(10,65,61,171,144)(11,66,62,163,136)(12,67,63,164,137)(13,68,55,165,138)(14,69,56,166,139)(15,70,57,167,140)(16,71,58,168,141)(17,72,59,169,142)(18,64,60,170,143)(19,95,117,101,46)(20,96,109,102,47)(21,97,110,103,48)(22,98,111,104,49)(23,99,112,105,50)(24,91,113,106,51)(25,92,114,107,52)(26,93,115,108,53)(27,94,116,100,54)(28,118,173,149,44)(29,119,174,150,45)(30,120,175,151,37)(31,121,176,152,38)(32,122,177,153,39)(33,123,178,145,40)(34,124,179,146,41)(35,125,180,147,42)(36,126,172,148,43), (1,145,102,57)(2,146,103,58)(3,147,104,59)(4,148,105,60)(5,149,106,61)(6,150,107,62)(7,151,108,63)(8,152,100,55)(9,153,101,56)(10,129,118,24)(11,130,119,25)(12,131,120,26)(13,132,121,27)(14,133,122,19)(15,134,123,20)(16,135,124,21)(17,127,125,22)(18,128,126,23)(28,91,144,80)(29,92,136,81)(30,93,137,73)(31,94,138,74)(32,95,139,75)(33,96,140,76)(34,97,141,77)(35,98,142,78)(36,99,143,79)(37,115,164,85)(38,116,165,86)(39,117,166,87)(40,109,167,88)(41,110,168,89)(42,111,169,90)(43,112,170,82)(44,113,171,83)(45,114,163,84)(46,69,160,177)(47,70,161,178)(48,71,162,179)(49,72,154,180)(50,64,155,172)(51,65,156,173)(52,66,157,174)(53,67,158,175)(54,68,159,176), (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,95)(20,96)(21,97)(22,98)(23,99)(24,91)(25,92)(26,93)(27,94)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,64)(44,65)(45,66)(46,117)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,152)(56,153)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(73,131)(74,132)(75,133)(76,134)(77,135)(78,127)(79,128)(80,129)(81,130)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,154)(118,144)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179)(169,180)(170,172)(171,173) );
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)], [(1,161,134,76,88),(2,162,135,77,89),(3,154,127,78,90),(4,155,128,79,82),(5,156,129,80,83),(6,157,130,81,84),(7,158,131,73,85),(8,159,132,74,86),(9,160,133,75,87),(10,65,61,171,144),(11,66,62,163,136),(12,67,63,164,137),(13,68,55,165,138),(14,69,56,166,139),(15,70,57,167,140),(16,71,58,168,141),(17,72,59,169,142),(18,64,60,170,143),(19,95,117,101,46),(20,96,109,102,47),(21,97,110,103,48),(22,98,111,104,49),(23,99,112,105,50),(24,91,113,106,51),(25,92,114,107,52),(26,93,115,108,53),(27,94,116,100,54),(28,118,173,149,44),(29,119,174,150,45),(30,120,175,151,37),(31,121,176,152,38),(32,122,177,153,39),(33,123,178,145,40),(34,124,179,146,41),(35,125,180,147,42),(36,126,172,148,43)], [(1,145,102,57),(2,146,103,58),(3,147,104,59),(4,148,105,60),(5,149,106,61),(6,150,107,62),(7,151,108,63),(8,152,100,55),(9,153,101,56),(10,129,118,24),(11,130,119,25),(12,131,120,26),(13,132,121,27),(14,133,122,19),(15,134,123,20),(16,135,124,21),(17,127,125,22),(18,128,126,23),(28,91,144,80),(29,92,136,81),(30,93,137,73),(31,94,138,74),(32,95,139,75),(33,96,140,76),(34,97,141,77),(35,98,142,78),(36,99,143,79),(37,115,164,85),(38,116,165,86),(39,117,166,87),(40,109,167,88),(41,110,168,89),(42,111,169,90),(43,112,170,82),(44,113,171,83),(45,114,163,84),(46,69,160,177),(47,70,161,178),(48,71,162,179),(49,72,154,180),(50,64,155,172),(51,65,156,173),(52,66,157,174),(53,67,158,175),(54,68,159,176)], [(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,95),(20,96),(21,97),(22,98),(23,99),(24,91),(25,92),(26,93),(27,94),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,64),(44,65),(45,66),(46,117),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,152),(56,153),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(73,131),(74,132),(75,133),(76,134),(77,135),(78,127),(79,128),(80,129),(81,130),(82,155),(83,156),(84,157),(85,158),(86,159),(87,160),(88,161),(89,162),(90,154),(118,144),(119,136),(120,137),(121,138),(122,139),(123,140),(124,141),(125,142),(126,143),(163,174),(164,175),(165,176),(166,177),(167,178),(168,179),(169,180),(170,172),(171,173)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 30A | ··· | 30L | 36A | ··· | 36F | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 30 | ··· | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 10 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | D5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | D4×C9 | C9×D5 | C3×C5⋊D4 | D5×C18 | C9×C5⋊D4 |
| kernel | C9×C5⋊D4 | C9×Dic5 | D5×C18 | C2×C90 | C3×C5⋊D4 | C3×Dic5 | C6×D5 | C2×C30 | C5⋊D4 | Dic5 | D10 | C2×C10 | C45 | C2×C18 | C18 | C15 | C2×C6 | C9 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C9×C5⋊D4 ►in GL2(𝔽181) generated by
| 80 | 0 |
| 0 | 80 |
| 167 | 180 |
| 1 | 0 |
| 155 | 48 |
| 50 | 26 |
| 1 | 0 |
| 167 | 180 |
G:=sub<GL(2,GF(181))| [80,0,0,80],[167,1,180,0],[155,50,48,26],[1,167,0,180] >;
C9×C5⋊D4 in GAP, Magma, Sage, TeX
C_9\times C_5\rtimes D_4
% in TeX
G:=Group("C9xC5:D4"); // GroupNames label
G:=SmallGroup(360,19);
// by ID
G=gap.SmallGroup(360,19);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,122,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export