direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D45, C20⋊2D9, C36⋊2D5, C180⋊2C2, C60.5S3, C6.9D30, C2.1D90, D90.2C2, C12.5D15, C18.9D10, C10.9D18, C30.41D6, Dic45⋊5C2, C90.9C22, C5⋊3(C4×D9), C9⋊2(C4×D5), C45⋊7(C2×C4), C3.(C4×D15), C15.4(C4×S3), SmallGroup(360,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C4×D45 |
Generators and relations for C4×D45
G = < a,b,c | a4=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >
45C2
45C2
45C4
45C22
15S3
15S3
9D5
9D5
45C2×C4
15Dic3
15D6
5D9
5D9
9Dic5
9D10
3D15
3D15
15C4×S3
5D18
5Dic9
3D30
Smallest permutation representation of C4×D45
►On 180 points
Generators in S180
(1 141 78 131)(2 142 79 132)(3 143 80 133)(4 144 81 134)(5 145 82 135)(6 146 83 91)(7 147 84 92)(8 148 85 93)(9 149 86 94)(10 150 87 95)(11 151 88 96)(12 152 89 97)(13 153 90 98)(14 154 46 99)(15 155 47 100)(16 156 48 101)(17 157 49 102)(18 158 50 103)(19 159 51 104)(20 160 52 105)(21 161 53 106)(22 162 54 107)(23 163 55 108)(24 164 56 109)(25 165 57 110)(26 166 58 111)(27 167 59 112)(28 168 60 113)(29 169 61 114)(30 170 62 115)(31 171 63 116)(32 172 64 117)(33 173 65 118)(34 174 66 119)(35 175 67 120)(36 176 68 121)(37 177 69 122)(38 178 70 123)(39 179 71 124)(40 180 72 125)(41 136 73 126)(42 137 74 127)(43 138 75 128)(44 139 76 129)(45 140 77 130)
(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)
(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 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(127 135)(128 134)(129 133)(130 132)(136 146)(137 145)(138 144)(139 143)(140 142)(147 180)(148 179)(149 178)(150 177)(151 176)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)
G:=sub<Sym(180)| (1,141,78,131)(2,142,79,132)(3,143,80,133)(4,144,81,134)(5,145,82,135)(6,146,83,91)(7,147,84,92)(8,148,85,93)(9,149,86,94)(10,150,87,95)(11,151,88,96)(12,152,89,97)(13,153,90,98)(14,154,46,99)(15,155,47,100)(16,156,48,101)(17,157,49,102)(18,158,50,103)(19,159,51,104)(20,160,52,105)(21,161,53,106)(22,162,54,107)(23,163,55,108)(24,164,56,109)(25,165,57,110)(26,166,58,111)(27,167,59,112)(28,168,60,113)(29,169,61,114)(30,170,62,115)(31,171,63,116)(32,172,64,117)(33,173,65,118)(34,174,66,119)(35,175,67,120)(36,176,68,121)(37,177,69,122)(38,178,70,123)(39,179,71,124)(40,180,72,125)(41,136,73,126)(42,137,74,127)(43,138,75,128)(44,139,76,129)(45,140,77,130), (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), (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)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(127,135)(128,134)(129,133)(130,132)(136,146)(137,145)(138,144)(139,143)(140,142)(147,180)(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)>;
G:=Group( (1,141,78,131)(2,142,79,132)(3,143,80,133)(4,144,81,134)(5,145,82,135)(6,146,83,91)(7,147,84,92)(8,148,85,93)(9,149,86,94)(10,150,87,95)(11,151,88,96)(12,152,89,97)(13,153,90,98)(14,154,46,99)(15,155,47,100)(16,156,48,101)(17,157,49,102)(18,158,50,103)(19,159,51,104)(20,160,52,105)(21,161,53,106)(22,162,54,107)(23,163,55,108)(24,164,56,109)(25,165,57,110)(26,166,58,111)(27,167,59,112)(28,168,60,113)(29,169,61,114)(30,170,62,115)(31,171,63,116)(32,172,64,117)(33,173,65,118)(34,174,66,119)(35,175,67,120)(36,176,68,121)(37,177,69,122)(38,178,70,123)(39,179,71,124)(40,180,72,125)(41,136,73,126)(42,137,74,127)(43,138,75,128)(44,139,76,129)(45,140,77,130), (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), (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)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(127,135)(128,134)(129,133)(130,132)(136,146)(137,145)(138,144)(139,143)(140,142)(147,180)(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164) );
G=PermutationGroup([[(1,141,78,131),(2,142,79,132),(3,143,80,133),(4,144,81,134),(5,145,82,135),(6,146,83,91),(7,147,84,92),(8,148,85,93),(9,149,86,94),(10,150,87,95),(11,151,88,96),(12,152,89,97),(13,153,90,98),(14,154,46,99),(15,155,47,100),(16,156,48,101),(17,157,49,102),(18,158,50,103),(19,159,51,104),(20,160,52,105),(21,161,53,106),(22,162,54,107),(23,163,55,108),(24,164,56,109),(25,165,57,110),(26,166,58,111),(27,167,59,112),(28,168,60,113),(29,169,61,114),(30,170,62,115),(31,171,63,116),(32,172,64,117),(33,173,65,118),(34,174,66,119),(35,175,67,120),(36,176,68,121),(37,177,69,122),(38,178,70,123),(39,179,71,124),(40,180,72,125),(41,136,73,126),(42,137,74,127),(43,138,75,128),(44,139,76,129),(45,140,77,130)], [(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)], [(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),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(127,135),(128,134),(129,133),(130,132),(136,146),(137,145),(138,144),(139,143),(140,142),(147,180),(148,179),(149,178),(150,177),(151,176),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 9A | 9B | 9C | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 18A | 18B | 18C | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 36A | ··· | 36F | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 9 | 9 | 9 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 18 | 18 | 18 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 45 | 45 | 2 | 1 | 1 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D9 | D10 | C4×S3 | D15 | D18 | C4×D5 | D30 | C4×D9 | D45 | C4×D15 | D90 | C4×D45 |
| kernel | C4×D45 | Dic45 | C180 | D90 | D45 | C60 | C36 | C30 | C20 | C18 | C15 | C12 | C10 | C9 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 3 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C4×D45 ►in GL2(𝔽181) generated by
| 19 | 0 |
| 0 | 19 |
| 76 | 149 |
| 32 | 108 |
| 119 | 44 |
| 106 | 62 |
G:=sub<GL(2,GF(181))| [19,0,0,19],[76,32,149,108],[119,106,44,62] >;
C4×D45 in GAP, Magma, Sage, TeX
C_4\times D_{45} % in TeX
G:=Group("C4xD45"); // GroupNames label
G:=SmallGroup(360,26);
// by ID
G=gap.SmallGroup(360,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,31,3267,741,2884,8645]);
// Polycyclic
G:=Group<a,b,c|a^4=b^45=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export