direct product, abelian, monomial, 3-elementary
Aliases: C3×C72, SmallGroup(216,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C72 |
| C1 — C3×C72 |
| C1 — C3×C72 |
Generators and relations for C3×C72
G = < a,b | a3=b72=1, ab=ba >
Smallest permutation representation of C3×C72
►Regular action on 216 points
Generators in S216
(1 117 200)(2 118 201)(3 119 202)(4 120 203)(5 121 204)(6 122 205)(7 123 206)(8 124 207)(9 125 208)(10 126 209)(11 127 210)(12 128 211)(13 129 212)(14 130 213)(15 131 214)(16 132 215)(17 133 216)(18 134 145)(19 135 146)(20 136 147)(21 137 148)(22 138 149)(23 139 150)(24 140 151)(25 141 152)(26 142 153)(27 143 154)(28 144 155)(29 73 156)(30 74 157)(31 75 158)(32 76 159)(33 77 160)(34 78 161)(35 79 162)(36 80 163)(37 81 164)(38 82 165)(39 83 166)(40 84 167)(41 85 168)(42 86 169)(43 87 170)(44 88 171)(45 89 172)(46 90 173)(47 91 174)(48 92 175)(49 93 176)(50 94 177)(51 95 178)(52 96 179)(53 97 180)(54 98 181)(55 99 182)(56 100 183)(57 101 184)(58 102 185)(59 103 186)(60 104 187)(61 105 188)(62 106 189)(63 107 190)(64 108 191)(65 109 192)(66 110 193)(67 111 194)(68 112 195)(69 113 196)(70 114 197)(71 115 198)(72 116 199)
(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)| (1,117,200)(2,118,201)(3,119,202)(4,120,203)(5,121,204)(6,122,205)(7,123,206)(8,124,207)(9,125,208)(10,126,209)(11,127,210)(12,128,211)(13,129,212)(14,130,213)(15,131,214)(16,132,215)(17,133,216)(18,134,145)(19,135,146)(20,136,147)(21,137,148)(22,138,149)(23,139,150)(24,140,151)(25,141,152)(26,142,153)(27,143,154)(28,144,155)(29,73,156)(30,74,157)(31,75,158)(32,76,159)(33,77,160)(34,78,161)(35,79,162)(36,80,163)(37,81,164)(38,82,165)(39,83,166)(40,84,167)(41,85,168)(42,86,169)(43,87,170)(44,88,171)(45,89,172)(46,90,173)(47,91,174)(48,92,175)(49,93,176)(50,94,177)(51,95,178)(52,96,179)(53,97,180)(54,98,181)(55,99,182)(56,100,183)(57,101,184)(58,102,185)(59,103,186)(60,104,187)(61,105,188)(62,106,189)(63,107,190)(64,108,191)(65,109,192)(66,110,193)(67,111,194)(68,112,195)(69,113,196)(70,114,197)(71,115,198)(72,116,199), (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( (1,117,200)(2,118,201)(3,119,202)(4,120,203)(5,121,204)(6,122,205)(7,123,206)(8,124,207)(9,125,208)(10,126,209)(11,127,210)(12,128,211)(13,129,212)(14,130,213)(15,131,214)(16,132,215)(17,133,216)(18,134,145)(19,135,146)(20,136,147)(21,137,148)(22,138,149)(23,139,150)(24,140,151)(25,141,152)(26,142,153)(27,143,154)(28,144,155)(29,73,156)(30,74,157)(31,75,158)(32,76,159)(33,77,160)(34,78,161)(35,79,162)(36,80,163)(37,81,164)(38,82,165)(39,83,166)(40,84,167)(41,85,168)(42,86,169)(43,87,170)(44,88,171)(45,89,172)(46,90,173)(47,91,174)(48,92,175)(49,93,176)(50,94,177)(51,95,178)(52,96,179)(53,97,180)(54,98,181)(55,99,182)(56,100,183)(57,101,184)(58,102,185)(59,103,186)(60,104,187)(61,105,188)(62,106,189)(63,107,190)(64,108,191)(65,109,192)(66,110,193)(67,111,194)(68,112,195)(69,113,196)(70,114,197)(71,115,198)(72,116,199), (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([[(1,117,200),(2,118,201),(3,119,202),(4,120,203),(5,121,204),(6,122,205),(7,123,206),(8,124,207),(9,125,208),(10,126,209),(11,127,210),(12,128,211),(13,129,212),(14,130,213),(15,131,214),(16,132,215),(17,133,216),(18,134,145),(19,135,146),(20,136,147),(21,137,148),(22,138,149),(23,139,150),(24,140,151),(25,141,152),(26,142,153),(27,143,154),(28,144,155),(29,73,156),(30,74,157),(31,75,158),(32,76,159),(33,77,160),(34,78,161),(35,79,162),(36,80,163),(37,81,164),(38,82,165),(39,83,166),(40,84,167),(41,85,168),(42,86,169),(43,87,170),(44,88,171),(45,89,172),(46,90,173),(47,91,174),(48,92,175),(49,93,176),(50,94,177),(51,95,178),(52,96,179),(53,97,180),(54,98,181),(55,99,182),(56,100,183),(57,101,184),(58,102,185),(59,103,186),(60,104,187),(61,105,188),(62,106,189),(63,107,190),(64,108,191),(65,109,192),(66,110,193),(67,111,194),(68,112,195),(69,113,196),(70,114,197),(71,115,198),(72,116,199)], [(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)]])
C3×C72 is a maximal subgroup of
C72.S3 C24.D9 C72⋊S3 C24⋊D9 C72⋊1S3
216 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 9A | ··· | 9R | 12A | ··· | 12P | 18A | ··· | 18R | 24A | ··· | 24AF | 36A | ··· | 36AJ | 72A | ··· | 72BT |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C8 | C9 | C12 | C12 | C18 | C24 | C24 | C36 | C72 |
| kernel | C3×C72 | C3×C36 | C72 | C3×C24 | C3×C18 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 4 | 18 | 12 | 4 | 18 | 24 | 8 | 36 | 72 |
Matrix representation of C3×C72 ►in GL2(𝔽73) generated by
| 8 | 0 |
| 0 | 8 |
| 24 | 0 |
| 0 | 20 |
G:=sub<GL(2,GF(73))| [8,0,0,8],[24,0,0,20] >;
C3×C72 in GAP, Magma, Sage, TeX
C_3\times C_{72} % in TeX
G:=Group("C3xC72"); // GroupNames label
G:=SmallGroup(216,18);
// by ID
G=gap.SmallGroup(216,18);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-2,108,223,165]);
// Polycyclic
G:=Group<a,b|a^3=b^72=1,a*b=b*a>;
// generators/relations
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