direct product, abelian, monomial, 3-elementary
Aliases: C32×C24, SmallGroup(216,85)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C24 |
| C1 — C32×C24 |
| C1 — C32×C24 |
Generators and relations for C32×C24
G = < a,b,c | a3=b3=c24=1, ab=ba, ac=ca, bc=cb >
Subgroups: 112, all normal (8 characteristic)
C1, C2, C3, C4, C6, C8, C32, C12, C3×C6, C24, C33, C3×C12, C32×C6, C3×C24, C32×C12, C32×C24
Quotients: C1, C2, C3, C4, C6, C8, C32, C12, C3×C6, C24, C33, C3×C12, C32×C6, C3×C24, C32×C12, C32×C24
►Regular action on 216 points
Generators in S216
(1 55 105)(2 56 106)(3 57 107)(4 58 108)(5 59 109)(6 60 110)(7 61 111)(8 62 112)(9 63 113)(10 64 114)(11 65 115)(12 66 116)(13 67 117)(14 68 118)(15 69 119)(16 70 120)(17 71 97)(18 72 98)(19 49 99)(20 50 100)(21 51 101)(22 52 102)(23 53 103)(24 54 104)(25 187 149)(26 188 150)(27 189 151)(28 190 152)(29 191 153)(30 192 154)(31 169 155)(32 170 156)(33 171 157)(34 172 158)(35 173 159)(36 174 160)(37 175 161)(38 176 162)(39 177 163)(40 178 164)(41 179 165)(42 180 166)(43 181 167)(44 182 168)(45 183 145)(46 184 146)(47 185 147)(48 186 148)(73 194 137)(74 195 138)(75 196 139)(76 197 140)(77 198 141)(78 199 142)(79 200 143)(80 201 144)(81 202 121)(82 203 122)(83 204 123)(84 205 124)(85 206 125)(86 207 126)(87 208 127)(88 209 128)(89 210 129)(90 211 130)(91 212 131)(92 213 132)(93 214 133)(94 215 134)(95 216 135)(96 193 136)
(1 34 206)(2 35 207)(3 36 208)(4 37 209)(5 38 210)(6 39 211)(7 40 212)(8 41 213)(9 42 214)(10 43 215)(11 44 216)(12 45 193)(13 46 194)(14 47 195)(15 48 196)(16 25 197)(17 26 198)(18 27 199)(19 28 200)(20 29 201)(21 30 202)(22 31 203)(23 32 204)(24 33 205)(49 190 143)(50 191 144)(51 192 121)(52 169 122)(53 170 123)(54 171 124)(55 172 125)(56 173 126)(57 174 127)(58 175 128)(59 176 129)(60 177 130)(61 178 131)(62 179 132)(63 180 133)(64 181 134)(65 182 135)(66 183 136)(67 184 137)(68 185 138)(69 186 139)(70 187 140)(71 188 141)(72 189 142)(73 117 146)(74 118 147)(75 119 148)(76 120 149)(77 97 150)(78 98 151)(79 99 152)(80 100 153)(81 101 154)(82 102 155)(83 103 156)(84 104 157)(85 105 158)(86 106 159)(87 107 160)(88 108 161)(89 109 162)(90 110 163)(91 111 164)(92 112 165)(93 113 166)(94 114 167)(95 115 168)(96 116 145)
(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)
G:=sub<Sym(216)| (1,55,105)(2,56,106)(3,57,107)(4,58,108)(5,59,109)(6,60,110)(7,61,111)(8,62,112)(9,63,113)(10,64,114)(11,65,115)(12,66,116)(13,67,117)(14,68,118)(15,69,119)(16,70,120)(17,71,97)(18,72,98)(19,49,99)(20,50,100)(21,51,101)(22,52,102)(23,53,103)(24,54,104)(25,187,149)(26,188,150)(27,189,151)(28,190,152)(29,191,153)(30,192,154)(31,169,155)(32,170,156)(33,171,157)(34,172,158)(35,173,159)(36,174,160)(37,175,161)(38,176,162)(39,177,163)(40,178,164)(41,179,165)(42,180,166)(43,181,167)(44,182,168)(45,183,145)(46,184,146)(47,185,147)(48,186,148)(73,194,137)(74,195,138)(75,196,139)(76,197,140)(77,198,141)(78,199,142)(79,200,143)(80,201,144)(81,202,121)(82,203,122)(83,204,123)(84,205,124)(85,206,125)(86,207,126)(87,208,127)(88,209,128)(89,210,129)(90,211,130)(91,212,131)(92,213,132)(93,214,133)(94,215,134)(95,216,135)(96,193,136), (1,34,206)(2,35,207)(3,36,208)(4,37,209)(5,38,210)(6,39,211)(7,40,212)(8,41,213)(9,42,214)(10,43,215)(11,44,216)(12,45,193)(13,46,194)(14,47,195)(15,48,196)(16,25,197)(17,26,198)(18,27,199)(19,28,200)(20,29,201)(21,30,202)(22,31,203)(23,32,204)(24,33,205)(49,190,143)(50,191,144)(51,192,121)(52,169,122)(53,170,123)(54,171,124)(55,172,125)(56,173,126)(57,174,127)(58,175,128)(59,176,129)(60,177,130)(61,178,131)(62,179,132)(63,180,133)(64,181,134)(65,182,135)(66,183,136)(67,184,137)(68,185,138)(69,186,139)(70,187,140)(71,188,141)(72,189,142)(73,117,146)(74,118,147)(75,119,148)(76,120,149)(77,97,150)(78,98,151)(79,99,152)(80,100,153)(81,101,154)(82,102,155)(83,103,156)(84,104,157)(85,105,158)(86,106,159)(87,107,160)(88,108,161)(89,109,162)(90,110,163)(91,111,164)(92,112,165)(93,113,166)(94,114,167)(95,115,168)(96,116,145), (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)>;
G:=Group( (1,55,105)(2,56,106)(3,57,107)(4,58,108)(5,59,109)(6,60,110)(7,61,111)(8,62,112)(9,63,113)(10,64,114)(11,65,115)(12,66,116)(13,67,117)(14,68,118)(15,69,119)(16,70,120)(17,71,97)(18,72,98)(19,49,99)(20,50,100)(21,51,101)(22,52,102)(23,53,103)(24,54,104)(25,187,149)(26,188,150)(27,189,151)(28,190,152)(29,191,153)(30,192,154)(31,169,155)(32,170,156)(33,171,157)(34,172,158)(35,173,159)(36,174,160)(37,175,161)(38,176,162)(39,177,163)(40,178,164)(41,179,165)(42,180,166)(43,181,167)(44,182,168)(45,183,145)(46,184,146)(47,185,147)(48,186,148)(73,194,137)(74,195,138)(75,196,139)(76,197,140)(77,198,141)(78,199,142)(79,200,143)(80,201,144)(81,202,121)(82,203,122)(83,204,123)(84,205,124)(85,206,125)(86,207,126)(87,208,127)(88,209,128)(89,210,129)(90,211,130)(91,212,131)(92,213,132)(93,214,133)(94,215,134)(95,216,135)(96,193,136), (1,34,206)(2,35,207)(3,36,208)(4,37,209)(5,38,210)(6,39,211)(7,40,212)(8,41,213)(9,42,214)(10,43,215)(11,44,216)(12,45,193)(13,46,194)(14,47,195)(15,48,196)(16,25,197)(17,26,198)(18,27,199)(19,28,200)(20,29,201)(21,30,202)(22,31,203)(23,32,204)(24,33,205)(49,190,143)(50,191,144)(51,192,121)(52,169,122)(53,170,123)(54,171,124)(55,172,125)(56,173,126)(57,174,127)(58,175,128)(59,176,129)(60,177,130)(61,178,131)(62,179,132)(63,180,133)(64,181,134)(65,182,135)(66,183,136)(67,184,137)(68,185,138)(69,186,139)(70,187,140)(71,188,141)(72,189,142)(73,117,146)(74,118,147)(75,119,148)(76,120,149)(77,97,150)(78,98,151)(79,99,152)(80,100,153)(81,101,154)(82,102,155)(83,103,156)(84,104,157)(85,105,158)(86,106,159)(87,107,160)(88,108,161)(89,109,162)(90,110,163)(91,111,164)(92,112,165)(93,113,166)(94,114,167)(95,115,168)(96,116,145), (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) );
G=PermutationGroup([[(1,55,105),(2,56,106),(3,57,107),(4,58,108),(5,59,109),(6,60,110),(7,61,111),(8,62,112),(9,63,113),(10,64,114),(11,65,115),(12,66,116),(13,67,117),(14,68,118),(15,69,119),(16,70,120),(17,71,97),(18,72,98),(19,49,99),(20,50,100),(21,51,101),(22,52,102),(23,53,103),(24,54,104),(25,187,149),(26,188,150),(27,189,151),(28,190,152),(29,191,153),(30,192,154),(31,169,155),(32,170,156),(33,171,157),(34,172,158),(35,173,159),(36,174,160),(37,175,161),(38,176,162),(39,177,163),(40,178,164),(41,179,165),(42,180,166),(43,181,167),(44,182,168),(45,183,145),(46,184,146),(47,185,147),(48,186,148),(73,194,137),(74,195,138),(75,196,139),(76,197,140),(77,198,141),(78,199,142),(79,200,143),(80,201,144),(81,202,121),(82,203,122),(83,204,123),(84,205,124),(85,206,125),(86,207,126),(87,208,127),(88,209,128),(89,210,129),(90,211,130),(91,212,131),(92,213,132),(93,214,133),(94,215,134),(95,216,135),(96,193,136)], [(1,34,206),(2,35,207),(3,36,208),(4,37,209),(5,38,210),(6,39,211),(7,40,212),(8,41,213),(9,42,214),(10,43,215),(11,44,216),(12,45,193),(13,46,194),(14,47,195),(15,48,196),(16,25,197),(17,26,198),(18,27,199),(19,28,200),(20,29,201),(21,30,202),(22,31,203),(23,32,204),(24,33,205),(49,190,143),(50,191,144),(51,192,121),(52,169,122),(53,170,123),(54,171,124),(55,172,125),(56,173,126),(57,174,127),(58,175,128),(59,176,129),(60,177,130),(61,178,131),(62,179,132),(63,180,133),(64,181,134),(65,182,135),(66,183,136),(67,184,137),(68,185,138),(69,186,139),(70,187,140),(71,188,141),(72,189,142),(73,117,146),(74,118,147),(75,119,148),(76,120,149),(77,97,150),(78,98,151),(79,99,152),(80,100,153),(81,101,154),(82,102,155),(83,103,156),(84,104,157),(85,105,158),(86,106,159),(87,107,160),(88,108,161),(89,109,162),(90,110,163),(91,111,164),(92,112,165),(93,113,166),(94,114,167),(95,115,168),(96,116,145)], [(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)]])
C32×C24 is a maximal subgroup of
C33⋊7C16 C33⋊15M4(2) C33⋊21SD16 C33⋊12D8 C33⋊12Q16
216 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 4A | 4B | 6A | ··· | 6Z | 8A | 8B | 8C | 8D | 12A | ··· | 12AZ | 24A | ··· | 24CZ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 |
| kernel | C32×C24 | C32×C12 | C3×C24 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 26 | 2 | 26 | 4 | 52 | 104 |
Matrix representation of C32×C24 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 1 |
| 24 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 7 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,64],[64,0,0,0,8,0,0,0,1],[24,0,0,0,24,0,0,0,7] >;
C32×C24 in GAP, Magma, Sage, TeX
C_3^2\times C_{24} % in TeX
G:=Group("C3^2xC24"); // GroupNames label
G:=SmallGroup(216,85);
// by ID
G=gap.SmallGroup(216,85);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,-2,324,88]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations