metabelian, supersoluble, monomial
Aliases: C72⋊2S3, C24⋊2D9, C6.8D36, C18.8D12, C36.57D6, C12.57D18, C8⋊2(C9⋊S3), (C3×C72)⋊5C2, (C3×C9)⋊9SD16, C9⋊1(C24⋊C2), C3.(C24⋊2S3), C24.4(C3⋊S3), C3⋊1(C72⋊C2), (C3×C24).17S3, (C3×C6).55D12, (C3×C18).30D4, C36⋊S3.2C2, C12.D9⋊2C2, (C3×C12).192D6, C2.4(C36⋊S3), C6.2(C12⋊S3), (C3×C36).59C22, C32.5(C24⋊C2), C4.9(C2×C9⋊S3), C12.60(C2×C3⋊S3), SmallGroup(432,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D9
G = < a,b,c | a24=b9=c2=1, ab=ba, cac=a11, cbc=b-1 >
Subgroups: 876 in 100 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C3⋊S3, C3×C6, C24, C24, Dic6, D12, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, C9⋊S3, C3×C18, C72, Dic18, D36, C3×C24, C32⋊4Q8, C12⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C72⋊C2, C24⋊2S3, C3×C72, C12.D9, C36⋊S3, C24⋊D9
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊S3, D12, D18, C2×C3⋊S3, C24⋊C2, C9⋊S3, D36, C12⋊S3, C2×C9⋊S3, C72⋊C2, C24⋊2S3, C36⋊S3, C24⋊D9
►On 216 points
Generators in S216
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
(1 196 83 40 147 49 102 174 143)(2 197 84 41 148 50 103 175 144)(3 198 85 42 149 51 104 176 121)(4 199 86 43 150 52 105 177 122)(5 200 87 44 151 53 106 178 123)(6 201 88 45 152 54 107 179 124)(7 202 89 46 153 55 108 180 125)(8 203 90 47 154 56 109 181 126)(9 204 91 48 155 57 110 182 127)(10 205 92 25 156 58 111 183 128)(11 206 93 26 157 59 112 184 129)(12 207 94 27 158 60 113 185 130)(13 208 95 28 159 61 114 186 131)(14 209 96 29 160 62 115 187 132)(15 210 73 30 161 63 116 188 133)(16 211 74 31 162 64 117 189 134)(17 212 75 32 163 65 118 190 135)(18 213 76 33 164 66 119 191 136)(19 214 77 34 165 67 120 192 137)(20 215 78 35 166 68 97 169 138)(21 216 79 36 167 69 98 170 139)(22 193 80 37 168 70 99 171 140)(23 194 81 38 145 71 100 172 141)(24 195 82 39 146 72 101 173 142)
(1 143)(2 130)(3 141)(4 128)(5 139)(6 126)(7 137)(8 124)(9 135)(10 122)(11 133)(12 144)(13 131)(14 142)(15 129)(16 140)(17 127)(18 138)(19 125)(20 136)(21 123)(22 134)(23 121)(24 132)(25 52)(26 63)(27 50)(28 61)(29 72)(30 59)(31 70)(32 57)(33 68)(34 55)(35 66)(36 53)(37 64)(38 51)(39 62)(40 49)(41 60)(42 71)(43 58)(44 69)(45 56)(46 67)(47 54)(48 65)(73 112)(74 99)(75 110)(76 97)(77 108)(78 119)(79 106)(80 117)(81 104)(82 115)(83 102)(84 113)(85 100)(86 111)(87 98)(88 109)(89 120)(90 107)(91 118)(92 105)(93 116)(94 103)(95 114)(96 101)(145 149)(146 160)(148 158)(150 156)(151 167)(152 154)(153 165)(155 163)(157 161)(162 168)(164 166)(169 213)(170 200)(171 211)(172 198)(173 209)(174 196)(175 207)(176 194)(177 205)(178 216)(179 203)(180 214)(181 201)(182 212)(183 199)(184 210)(185 197)(186 208)(187 195)(188 206)(189 193)(190 204)(191 215)(192 202)
G:=sub<Sym(216)| (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,196,83,40,147,49,102,174,143)(2,197,84,41,148,50,103,175,144)(3,198,85,42,149,51,104,176,121)(4,199,86,43,150,52,105,177,122)(5,200,87,44,151,53,106,178,123)(6,201,88,45,152,54,107,179,124)(7,202,89,46,153,55,108,180,125)(8,203,90,47,154,56,109,181,126)(9,204,91,48,155,57,110,182,127)(10,205,92,25,156,58,111,183,128)(11,206,93,26,157,59,112,184,129)(12,207,94,27,158,60,113,185,130)(13,208,95,28,159,61,114,186,131)(14,209,96,29,160,62,115,187,132)(15,210,73,30,161,63,116,188,133)(16,211,74,31,162,64,117,189,134)(17,212,75,32,163,65,118,190,135)(18,213,76,33,164,66,119,191,136)(19,214,77,34,165,67,120,192,137)(20,215,78,35,166,68,97,169,138)(21,216,79,36,167,69,98,170,139)(22,193,80,37,168,70,99,171,140)(23,194,81,38,145,71,100,172,141)(24,195,82,39,146,72,101,173,142), (1,143)(2,130)(3,141)(4,128)(5,139)(6,126)(7,137)(8,124)(9,135)(10,122)(11,133)(12,144)(13,131)(14,142)(15,129)(16,140)(17,127)(18,138)(19,125)(20,136)(21,123)(22,134)(23,121)(24,132)(25,52)(26,63)(27,50)(28,61)(29,72)(30,59)(31,70)(32,57)(33,68)(34,55)(35,66)(36,53)(37,64)(38,51)(39,62)(40,49)(41,60)(42,71)(43,58)(44,69)(45,56)(46,67)(47,54)(48,65)(73,112)(74,99)(75,110)(76,97)(77,108)(78,119)(79,106)(80,117)(81,104)(82,115)(83,102)(84,113)(85,100)(86,111)(87,98)(88,109)(89,120)(90,107)(91,118)(92,105)(93,116)(94,103)(95,114)(96,101)(145,149)(146,160)(148,158)(150,156)(151,167)(152,154)(153,165)(155,163)(157,161)(162,168)(164,166)(169,213)(170,200)(171,211)(172,198)(173,209)(174,196)(175,207)(176,194)(177,205)(178,216)(179,203)(180,214)(181,201)(182,212)(183,199)(184,210)(185,197)(186,208)(187,195)(188,206)(189,193)(190,204)(191,215)(192,202)>;
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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,196,83,40,147,49,102,174,143)(2,197,84,41,148,50,103,175,144)(3,198,85,42,149,51,104,176,121)(4,199,86,43,150,52,105,177,122)(5,200,87,44,151,53,106,178,123)(6,201,88,45,152,54,107,179,124)(7,202,89,46,153,55,108,180,125)(8,203,90,47,154,56,109,181,126)(9,204,91,48,155,57,110,182,127)(10,205,92,25,156,58,111,183,128)(11,206,93,26,157,59,112,184,129)(12,207,94,27,158,60,113,185,130)(13,208,95,28,159,61,114,186,131)(14,209,96,29,160,62,115,187,132)(15,210,73,30,161,63,116,188,133)(16,211,74,31,162,64,117,189,134)(17,212,75,32,163,65,118,190,135)(18,213,76,33,164,66,119,191,136)(19,214,77,34,165,67,120,192,137)(20,215,78,35,166,68,97,169,138)(21,216,79,36,167,69,98,170,139)(22,193,80,37,168,70,99,171,140)(23,194,81,38,145,71,100,172,141)(24,195,82,39,146,72,101,173,142), (1,143)(2,130)(3,141)(4,128)(5,139)(6,126)(7,137)(8,124)(9,135)(10,122)(11,133)(12,144)(13,131)(14,142)(15,129)(16,140)(17,127)(18,138)(19,125)(20,136)(21,123)(22,134)(23,121)(24,132)(25,52)(26,63)(27,50)(28,61)(29,72)(30,59)(31,70)(32,57)(33,68)(34,55)(35,66)(36,53)(37,64)(38,51)(39,62)(40,49)(41,60)(42,71)(43,58)(44,69)(45,56)(46,67)(47,54)(48,65)(73,112)(74,99)(75,110)(76,97)(77,108)(78,119)(79,106)(80,117)(81,104)(82,115)(83,102)(84,113)(85,100)(86,111)(87,98)(88,109)(89,120)(90,107)(91,118)(92,105)(93,116)(94,103)(95,114)(96,101)(145,149)(146,160)(148,158)(150,156)(151,167)(152,154)(153,165)(155,163)(157,161)(162,168)(164,166)(169,213)(170,200)(171,211)(172,198)(173,209)(174,196)(175,207)(176,194)(177,205)(178,216)(179,203)(180,214)(181,201)(182,212)(183,199)(184,210)(185,197)(186,208)(187,195)(188,206)(189,193)(190,204)(191,215)(192,202) );
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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,196,83,40,147,49,102,174,143),(2,197,84,41,148,50,103,175,144),(3,198,85,42,149,51,104,176,121),(4,199,86,43,150,52,105,177,122),(5,200,87,44,151,53,106,178,123),(6,201,88,45,152,54,107,179,124),(7,202,89,46,153,55,108,180,125),(8,203,90,47,154,56,109,181,126),(9,204,91,48,155,57,110,182,127),(10,205,92,25,156,58,111,183,128),(11,206,93,26,157,59,112,184,129),(12,207,94,27,158,60,113,185,130),(13,208,95,28,159,61,114,186,131),(14,209,96,29,160,62,115,187,132),(15,210,73,30,161,63,116,188,133),(16,211,74,31,162,64,117,189,134),(17,212,75,32,163,65,118,190,135),(18,213,76,33,164,66,119,191,136),(19,214,77,34,165,67,120,192,137),(20,215,78,35,166,68,97,169,138),(21,216,79,36,167,69,98,170,139),(22,193,80,37,168,70,99,171,140),(23,194,81,38,145,71,100,172,141),(24,195,82,39,146,72,101,173,142)], [(1,143),(2,130),(3,141),(4,128),(5,139),(6,126),(7,137),(8,124),(9,135),(10,122),(11,133),(12,144),(13,131),(14,142),(15,129),(16,140),(17,127),(18,138),(19,125),(20,136),(21,123),(22,134),(23,121),(24,132),(25,52),(26,63),(27,50),(28,61),(29,72),(30,59),(31,70),(32,57),(33,68),(34,55),(35,66),(36,53),(37,64),(38,51),(39,62),(40,49),(41,60),(42,71),(43,58),(44,69),(45,56),(46,67),(47,54),(48,65),(73,112),(74,99),(75,110),(76,97),(77,108),(78,119),(79,106),(80,117),(81,104),(82,115),(83,102),(84,113),(85,100),(86,111),(87,98),(88,109),(89,120),(90,107),(91,118),(92,105),(93,116),(94,103),(95,114),(96,101),(145,149),(146,160),(148,158),(150,156),(151,167),(152,154),(153,165),(155,163),(157,161),(162,168),(164,166),(169,213),(170,200),(171,211),(172,198),(173,209),(174,196),(175,207),(176,194),(177,205),(178,216),(179,203),(180,214),(181,201),(182,212),(183,199),(184,210),(185,197),(186,208),(187,195),(188,206),(189,193),(190,204),(191,215),(192,202)]])
111 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 108 | 2 | 2 | 2 | 2 | 2 | 108 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | SD16 | D9 | D12 | D12 | D18 | C24⋊C2 | C24⋊C2 | D36 | C72⋊C2 |
| kernel | C24⋊D9 | C3×C72 | C12.D9 | C36⋊S3 | C72 | C3×C24 | C3×C18 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 12 | 4 | 18 | 36 |
Matrix representation of C24⋊D9 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 62 | 25 |
| 0 | 0 | 48 | 37 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 42 | 45 |
| 0 | 0 | 28 | 70 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 45 |
| 0 | 0 | 42 | 70 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,62,48,0,0,25,37],[72,1,0,0,72,0,0,0,0,0,42,28,0,0,45,70],[72,0,0,0,72,1,0,0,0,0,3,42,0,0,45,70] >;
C24⋊D9 in GAP, Magma, Sage, TeX
C_{24}\rtimes D_9 % in TeX
G:=Group("C24:D9"); // GroupNames label
G:=SmallGroup(432,171);
// by ID
G=gap.SmallGroup(432,171);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^24=b^9=c^2=1,a*b=b*a,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations