direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C9×D20, C45⋊6D4, C36⋊3D5, C20⋊1C18, C180⋊4C2, C60.4C6, D10⋊1C18, C18.15D10, C90.20C22, C4⋊(C9×D5), C5⋊1(D4×C9), C3.(C3×D20), (C3×D20).C3, (D5×C18)⋊4C2, (C6×D5).2C6, C12.4(C3×D5), C15.1(C3×D4), C2.4(D5×C18), C6.15(C6×D5), C10.3(C2×C18), C30.15(C2×C6), SmallGroup(360,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D20
G = < a,b,c | a9=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D20
►On 180 points
Generators in S180
(1 118 140 159 94 47 66 21 162)(2 119 121 160 95 48 67 22 163)(3 120 122 141 96 49 68 23 164)(4 101 123 142 97 50 69 24 165)(5 102 124 143 98 51 70 25 166)(6 103 125 144 99 52 71 26 167)(7 104 126 145 100 53 72 27 168)(8 105 127 146 81 54 73 28 169)(9 106 128 147 82 55 74 29 170)(10 107 129 148 83 56 75 30 171)(11 108 130 149 84 57 76 31 172)(12 109 131 150 85 58 77 32 173)(13 110 132 151 86 59 78 33 174)(14 111 133 152 87 60 79 34 175)(15 112 134 153 88 41 80 35 176)(16 113 135 154 89 42 61 36 177)(17 114 136 155 90 43 62 37 178)(18 115 137 156 91 44 63 38 179)(19 116 138 157 92 45 64 39 180)(20 117 139 158 93 46 65 40 161)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 140)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)
G:=sub<Sym(180)| (1,118,140,159,94,47,66,21,162)(2,119,121,160,95,48,67,22,163)(3,120,122,141,96,49,68,23,164)(4,101,123,142,97,50,69,24,165)(5,102,124,143,98,51,70,25,166)(6,103,125,144,99,52,71,26,167)(7,104,126,145,100,53,72,27,168)(8,105,127,146,81,54,73,28,169)(9,106,128,147,82,55,74,29,170)(10,107,129,148,83,56,75,30,171)(11,108,130,149,84,57,76,31,172)(12,109,131,150,85,58,77,32,173)(13,110,132,151,86,59,78,33,174)(14,111,133,152,87,60,79,34,175)(15,112,134,153,88,41,80,35,176)(16,113,135,154,89,42,61,36,177)(17,114,136,155,90,43,62,37,178)(18,115,137,156,91,44,63,38,179)(19,116,138,157,92,45,64,39,180)(20,117,139,158,93,46,65,40,161), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)>;
G:=Group( (1,118,140,159,94,47,66,21,162)(2,119,121,160,95,48,67,22,163)(3,120,122,141,96,49,68,23,164)(4,101,123,142,97,50,69,24,165)(5,102,124,143,98,51,70,25,166)(6,103,125,144,99,52,71,26,167)(7,104,126,145,100,53,72,27,168)(8,105,127,146,81,54,73,28,169)(9,106,128,147,82,55,74,29,170)(10,107,129,148,83,56,75,30,171)(11,108,130,149,84,57,76,31,172)(12,109,131,150,85,58,77,32,173)(13,110,132,151,86,59,78,33,174)(14,111,133,152,87,60,79,34,175)(15,112,134,153,88,41,80,35,176)(16,113,135,154,89,42,61,36,177)(17,114,136,155,90,43,62,37,178)(18,115,137,156,91,44,63,38,179)(19,116,138,157,92,45,64,39,180)(20,117,139,158,93,46,65,40,161), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172) );
G=PermutationGroup([[(1,118,140,159,94,47,66,21,162),(2,119,121,160,95,48,67,22,163),(3,120,122,141,96,49,68,23,164),(4,101,123,142,97,50,69,24,165),(5,102,124,143,98,51,70,25,166),(6,103,125,144,99,52,71,26,167),(7,104,126,145,100,53,72,27,168),(8,105,127,146,81,54,73,28,169),(9,106,128,147,82,55,74,29,170),(10,107,129,148,83,56,75,30,171),(11,108,130,149,84,57,76,31,172),(12,109,131,150,85,58,77,32,173),(13,110,132,151,86,59,78,33,174),(14,111,133,152,87,60,79,34,175),(15,112,134,153,88,41,80,35,176),(16,113,135,154,89,42,61,36,177),(17,114,136,155,90,43,62,37,178),(18,115,137,156,91,44,63,38,179),(19,116,138,157,92,45,64,39,180),(20,117,139,158,93,46,65,40,161)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(139,140),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,160),(158,159),(161,162),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 18G | ··· | 18R | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 36A | ··· | 36F | 45A | ··· | 45L | 60A | ··· | 60H | 90A | ··· | 90L | 180A | ··· | 180X |
| 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 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 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 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | D5 | D10 | C3×D4 | C3×D5 | D20 | C6×D5 | D4×C9 | C9×D5 | C3×D20 | D5×C18 | C9×D20 |
| kernel | C9×D20 | C180 | D5×C18 | C3×D20 | C60 | C6×D5 | D20 | C20 | D10 | C45 | C36 | C18 | C15 | C12 | C9 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C9×D20 ►in GL2(𝔽19) generated by
| 9 | 0 |
| 0 | 9 |
| 9 | 1 |
| 10 | 16 |
| 15 | 6 |
| 7 | 4 |
G:=sub<GL(2,GF(19))| [9,0,0,9],[9,10,1,16],[15,7,6,4] >;
C9×D20 in GAP, Magma, Sage, TeX
C_9\times D_{20} % in TeX
G:=Group("C9xD20"); // GroupNames label
G:=SmallGroup(360,17);
// by ID
G=gap.SmallGroup(360,17);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,122,10373]);
// Polycyclic
G:=Group<a,b,c|a^9=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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