direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D23, C23⋊C18, C69.C6, C207⋊2C2, C3.(C3×D23), (C3×D23).C3, SmallGroup(414,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C9×D23 |
Generators and relations for C9×D23
G = < a,b,c | a9=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D23
►On 207 points
Generators in S207
(1 187 136 55 172 111 43 157 72)(2 188 137 56 173 112 44 158 73)(3 189 138 57 174 113 45 159 74)(4 190 116 58 175 114 46 160 75)(5 191 117 59 176 115 24 161 76)(6 192 118 60 177 93 25 139 77)(7 193 119 61 178 94 26 140 78)(8 194 120 62 179 95 27 141 79)(9 195 121 63 180 96 28 142 80)(10 196 122 64 181 97 29 143 81)(11 197 123 65 182 98 30 144 82)(12 198 124 66 183 99 31 145 83)(13 199 125 67 184 100 32 146 84)(14 200 126 68 162 101 33 147 85)(15 201 127 69 163 102 34 148 86)(16 202 128 47 164 103 35 149 87)(17 203 129 48 165 104 36 150 88)(18 204 130 49 166 105 37 151 89)(19 205 131 50 167 106 38 152 90)(20 206 132 51 168 107 39 153 91)(21 207 133 52 169 108 40 154 92)(22 185 134 53 170 109 41 155 70)(23 186 135 54 171 110 42 156 71)
(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)(162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(39 46)(40 45)(41 44)(42 43)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(63 69)(64 68)(65 67)(70 73)(71 72)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(106 115)(107 114)(108 113)(109 112)(110 111)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(133 138)(134 137)(135 136)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(152 161)(153 160)(154 159)(155 158)(156 157)(162 181)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(182 184)(185 188)(186 187)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)
G:=sub<Sym(207)| (1,187,136,55,172,111,43,157,72)(2,188,137,56,173,112,44,158,73)(3,189,138,57,174,113,45,159,74)(4,190,116,58,175,114,46,160,75)(5,191,117,59,176,115,24,161,76)(6,192,118,60,177,93,25,139,77)(7,193,119,61,178,94,26,140,78)(8,194,120,62,179,95,27,141,79)(9,195,121,63,180,96,28,142,80)(10,196,122,64,181,97,29,143,81)(11,197,123,65,182,98,30,144,82)(12,198,124,66,183,99,31,145,83)(13,199,125,67,184,100,32,146,84)(14,200,126,68,162,101,33,147,85)(15,201,127,69,163,102,34,148,86)(16,202,128,47,164,103,35,149,87)(17,203,129,48,165,104,36,150,88)(18,204,130,49,166,105,37,151,89)(19,205,131,50,167,106,38,152,90)(20,206,132,51,168,107,39,153,91)(21,207,133,52,169,108,40,154,92)(22,185,134,53,170,109,41,155,70)(23,186,135,54,171,110,42,156,71), (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)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(39,46)(40,45)(41,44)(42,43)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,69)(64,68)(65,67)(70,73)(71,72)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,115)(107,114)(108,113)(109,112)(110,111)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,138)(134,137)(135,136)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(152,161)(153,160)(154,159)(155,158)(156,157)(162,181)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(182,184)(185,188)(186,187)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)>;
G:=Group( (1,187,136,55,172,111,43,157,72)(2,188,137,56,173,112,44,158,73)(3,189,138,57,174,113,45,159,74)(4,190,116,58,175,114,46,160,75)(5,191,117,59,176,115,24,161,76)(6,192,118,60,177,93,25,139,77)(7,193,119,61,178,94,26,140,78)(8,194,120,62,179,95,27,141,79)(9,195,121,63,180,96,28,142,80)(10,196,122,64,181,97,29,143,81)(11,197,123,65,182,98,30,144,82)(12,198,124,66,183,99,31,145,83)(13,199,125,67,184,100,32,146,84)(14,200,126,68,162,101,33,147,85)(15,201,127,69,163,102,34,148,86)(16,202,128,47,164,103,35,149,87)(17,203,129,48,165,104,36,150,88)(18,204,130,49,166,105,37,151,89)(19,205,131,50,167,106,38,152,90)(20,206,132,51,168,107,39,153,91)(21,207,133,52,169,108,40,154,92)(22,185,134,53,170,109,41,155,70)(23,186,135,54,171,110,42,156,71), (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)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(39,46)(40,45)(41,44)(42,43)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(63,69)(64,68)(65,67)(70,73)(71,72)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,115)(107,114)(108,113)(109,112)(110,111)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,138)(134,137)(135,136)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(152,161)(153,160)(154,159)(155,158)(156,157)(162,181)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(182,184)(185,188)(186,187)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199) );
G=PermutationGroup([[(1,187,136,55,172,111,43,157,72),(2,188,137,56,173,112,44,158,73),(3,189,138,57,174,113,45,159,74),(4,190,116,58,175,114,46,160,75),(5,191,117,59,176,115,24,161,76),(6,192,118,60,177,93,25,139,77),(7,193,119,61,178,94,26,140,78),(8,194,120,62,179,95,27,141,79),(9,195,121,63,180,96,28,142,80),(10,196,122,64,181,97,29,143,81),(11,197,123,65,182,98,30,144,82),(12,198,124,66,183,99,31,145,83),(13,199,125,67,184,100,32,146,84),(14,200,126,68,162,101,33,147,85),(15,201,127,69,163,102,34,148,86),(16,202,128,47,164,103,35,149,87),(17,203,129,48,165,104,36,150,88),(18,204,130,49,166,105,37,151,89),(19,205,131,50,167,106,38,152,90),(20,206,132,51,168,107,39,153,91),(21,207,133,52,169,108,40,154,92),(22,185,134,53,170,109,41,155,70),(23,186,135,54,171,110,42,156,71)], [(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),(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(39,46),(40,45),(41,44),(42,43),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(63,69),(64,68),(65,67),(70,73),(71,72),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(106,115),(107,114),(108,113),(109,112),(110,111),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(133,138),(134,137),(135,136),(139,151),(140,150),(141,149),(142,148),(143,147),(144,146),(152,161),(153,160),(154,159),(155,158),(156,157),(162,181),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(182,184),(185,188),(186,187),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199)]])
117 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 23A | ··· | 23K | 69A | ··· | 69V | 207A | ··· | 207BN |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 23 | ··· | 23 | 69 | ··· | 69 | 207 | ··· | 207 |
| size | 1 | 23 | 1 | 1 | 23 | 23 | 1 | ··· | 1 | 23 | ··· | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D23 | C3×D23 | C9×D23 |
| kernel | C9×D23 | C207 | C3×D23 | C69 | D23 | C23 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 11 | 22 | 66 |
Matrix representation of C9×D23 ►in GL3(𝔽829) generated by
| 166 | 0 | 0 |
| 0 | 125 | 0 |
| 0 | 0 | 125 |
| 1 | 0 | 0 |
| 0 | 194 | 1 |
| 0 | 778 | 692 |
| 828 | 0 | 0 |
| 0 | 425 | 283 |
| 0 | 595 | 404 |
G:=sub<GL(3,GF(829))| [166,0,0,0,125,0,0,0,125],[1,0,0,0,194,778,0,1,692],[828,0,0,0,425,595,0,283,404] >;
C9×D23 in GAP, Magma, Sage, TeX
C_9\times D_{23} % in TeX
G:=Group("C9xD23"); // GroupNames label
G:=SmallGroup(414,2);
// by ID
G=gap.SmallGroup(414,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-23,29,6339]);
// Polycyclic
G:=Group<a,b,c|a^9=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export