direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic9, C9⋊C20, C45⋊4C4, C18.C10, C90.2C2, C30.5S3, C10.2D9, C15.3Dic3, C2.(C5×D9), C6.1(C5×S3), C3.(C5×Dic3), SmallGroup(180,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C5×Dic9 |
Generators and relations for C5×Dic9
G = < a,b,c | a5=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic9
►Regular action on 180 points
Generators in S180
(1 80 58 44 29)(2 81 59 45 30)(3 82 60 46 31)(4 83 61 47 32)(5 84 62 48 33)(6 85 63 49 34)(7 86 64 50 35)(8 87 65 51 36)(9 88 66 52 19)(10 89 67 53 20)(11 90 68 54 21)(12 73 69 37 22)(13 74 70 38 23)(14 75 71 39 24)(15 76 72 40 25)(16 77 55 41 26)(17 78 56 42 27)(18 79 57 43 28)(91 176 148 139 109)(92 177 149 140 110)(93 178 150 141 111)(94 179 151 142 112)(95 180 152 143 113)(96 163 153 144 114)(97 164 154 127 115)(98 165 155 128 116)(99 166 156 129 117)(100 167 157 130 118)(101 168 158 131 119)(102 169 159 132 120)(103 170 160 133 121)(104 171 161 134 122)(105 172 162 135 123)(106 173 145 136 124)(107 174 146 137 125)(108 175 147 138 126)
(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 95 10 104)(2 94 11 103)(3 93 12 102)(4 92 13 101)(5 91 14 100)(6 108 15 99)(7 107 16 98)(8 106 17 97)(9 105 18 96)(19 123 28 114)(20 122 29 113)(21 121 30 112)(22 120 31 111)(23 119 32 110)(24 118 33 109)(25 117 34 126)(26 116 35 125)(27 115 36 124)(37 132 46 141)(38 131 47 140)(39 130 48 139)(40 129 49 138)(41 128 50 137)(42 127 51 136)(43 144 52 135)(44 143 53 134)(45 142 54 133)(55 155 64 146)(56 154 65 145)(57 153 66 162)(58 152 67 161)(59 151 68 160)(60 150 69 159)(61 149 70 158)(62 148 71 157)(63 147 72 156)(73 169 82 178)(74 168 83 177)(75 167 84 176)(76 166 85 175)(77 165 86 174)(78 164 87 173)(79 163 88 172)(80 180 89 171)(81 179 90 170)
G:=sub<Sym(180)| (1,80,58,44,29)(2,81,59,45,30)(3,82,60,46,31)(4,83,61,47,32)(5,84,62,48,33)(6,85,63,49,34)(7,86,64,50,35)(8,87,65,51,36)(9,88,66,52,19)(10,89,67,53,20)(11,90,68,54,21)(12,73,69,37,22)(13,74,70,38,23)(14,75,71,39,24)(15,76,72,40,25)(16,77,55,41,26)(17,78,56,42,27)(18,79,57,43,28)(91,176,148,139,109)(92,177,149,140,110)(93,178,150,141,111)(94,179,151,142,112)(95,180,152,143,113)(96,163,153,144,114)(97,164,154,127,115)(98,165,155,128,116)(99,166,156,129,117)(100,167,157,130,118)(101,168,158,131,119)(102,169,159,132,120)(103,170,160,133,121)(104,171,161,134,122)(105,172,162,135,123)(106,173,145,136,124)(107,174,146,137,125)(108,175,147,138,126), (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), (1,95,10,104)(2,94,11,103)(3,93,12,102)(4,92,13,101)(5,91,14,100)(6,108,15,99)(7,107,16,98)(8,106,17,97)(9,105,18,96)(19,123,28,114)(20,122,29,113)(21,121,30,112)(22,120,31,111)(23,119,32,110)(24,118,33,109)(25,117,34,126)(26,116,35,125)(27,115,36,124)(37,132,46,141)(38,131,47,140)(39,130,48,139)(40,129,49,138)(41,128,50,137)(42,127,51,136)(43,144,52,135)(44,143,53,134)(45,142,54,133)(55,155,64,146)(56,154,65,145)(57,153,66,162)(58,152,67,161)(59,151,68,160)(60,150,69,159)(61,149,70,158)(62,148,71,157)(63,147,72,156)(73,169,82,178)(74,168,83,177)(75,167,84,176)(76,166,85,175)(77,165,86,174)(78,164,87,173)(79,163,88,172)(80,180,89,171)(81,179,90,170)>;
G:=Group( (1,80,58,44,29)(2,81,59,45,30)(3,82,60,46,31)(4,83,61,47,32)(5,84,62,48,33)(6,85,63,49,34)(7,86,64,50,35)(8,87,65,51,36)(9,88,66,52,19)(10,89,67,53,20)(11,90,68,54,21)(12,73,69,37,22)(13,74,70,38,23)(14,75,71,39,24)(15,76,72,40,25)(16,77,55,41,26)(17,78,56,42,27)(18,79,57,43,28)(91,176,148,139,109)(92,177,149,140,110)(93,178,150,141,111)(94,179,151,142,112)(95,180,152,143,113)(96,163,153,144,114)(97,164,154,127,115)(98,165,155,128,116)(99,166,156,129,117)(100,167,157,130,118)(101,168,158,131,119)(102,169,159,132,120)(103,170,160,133,121)(104,171,161,134,122)(105,172,162,135,123)(106,173,145,136,124)(107,174,146,137,125)(108,175,147,138,126), (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), (1,95,10,104)(2,94,11,103)(3,93,12,102)(4,92,13,101)(5,91,14,100)(6,108,15,99)(7,107,16,98)(8,106,17,97)(9,105,18,96)(19,123,28,114)(20,122,29,113)(21,121,30,112)(22,120,31,111)(23,119,32,110)(24,118,33,109)(25,117,34,126)(26,116,35,125)(27,115,36,124)(37,132,46,141)(38,131,47,140)(39,130,48,139)(40,129,49,138)(41,128,50,137)(42,127,51,136)(43,144,52,135)(44,143,53,134)(45,142,54,133)(55,155,64,146)(56,154,65,145)(57,153,66,162)(58,152,67,161)(59,151,68,160)(60,150,69,159)(61,149,70,158)(62,148,71,157)(63,147,72,156)(73,169,82,178)(74,168,83,177)(75,167,84,176)(76,166,85,175)(77,165,86,174)(78,164,87,173)(79,163,88,172)(80,180,89,171)(81,179,90,170) );
G=PermutationGroup([[(1,80,58,44,29),(2,81,59,45,30),(3,82,60,46,31),(4,83,61,47,32),(5,84,62,48,33),(6,85,63,49,34),(7,86,64,50,35),(8,87,65,51,36),(9,88,66,52,19),(10,89,67,53,20),(11,90,68,54,21),(12,73,69,37,22),(13,74,70,38,23),(14,75,71,39,24),(15,76,72,40,25),(16,77,55,41,26),(17,78,56,42,27),(18,79,57,43,28),(91,176,148,139,109),(92,177,149,140,110),(93,178,150,141,111),(94,179,151,142,112),(95,180,152,143,113),(96,163,153,144,114),(97,164,154,127,115),(98,165,155,128,116),(99,166,156,129,117),(100,167,157,130,118),(101,168,158,131,119),(102,169,159,132,120),(103,170,160,133,121),(104,171,161,134,122),(105,172,162,135,123),(106,173,145,136,124),(107,174,146,137,125),(108,175,147,138,126)], [(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)], [(1,95,10,104),(2,94,11,103),(3,93,12,102),(4,92,13,101),(5,91,14,100),(6,108,15,99),(7,107,16,98),(8,106,17,97),(9,105,18,96),(19,123,28,114),(20,122,29,113),(21,121,30,112),(22,120,31,111),(23,119,32,110),(24,118,33,109),(25,117,34,126),(26,116,35,125),(27,115,36,124),(37,132,46,141),(38,131,47,140),(39,130,48,139),(40,129,49,138),(41,128,50,137),(42,127,51,136),(43,144,52,135),(44,143,53,134),(45,142,54,133),(55,155,64,146),(56,154,65,145),(57,153,66,162),(58,152,67,161),(59,151,68,160),(60,150,69,159),(61,149,70,158),(62,148,71,157),(63,147,72,156),(73,169,82,178),(74,168,83,177),(75,167,84,176),(76,166,85,175),(77,165,86,174),(78,164,87,173),(79,163,88,172),(80,180,89,171),(81,179,90,170)]])
C5×Dic9 is a maximal subgroup of
C45⋊Q8 D90.C2 C9⋊D20 D9×C20
60 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 20A | ··· | 20H | 30A | 30B | 30C | 30D | 45A | ··· | 45L | 90A | ··· | 90L |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | D9 | C5×S3 | Dic9 | C5×Dic3 | C5×D9 | C5×Dic9 |
| kernel | C5×Dic9 | C90 | C45 | Dic9 | C18 | C9 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 3 | 4 | 3 | 4 | 12 | 12 |
Matrix representation of C5×Dic9 ►in GL3(𝔽181) generated by
| 59 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 180 | 0 | 0 |
| 0 | 4 | 54 |
| 0 | 127 | 131 |
| 19 | 0 | 0 |
| 0 | 131 | 4 |
| 0 | 54 | 50 |
G:=sub<GL(3,GF(181))| [59,0,0,0,42,0,0,0,42],[180,0,0,0,4,127,0,54,131],[19,0,0,0,131,54,0,4,50] >;
C5×Dic9 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_9 % in TeX
G:=Group("C5xDic9"); // GroupNames label
G:=SmallGroup(180,1);
// by ID
G=gap.SmallGroup(180,1);
# by ID
G:=PCGroup([5,-2,-5,-2,-3,-3,50,2003,138,3004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export