metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.6D6, C30.4D8, C15⋊1SD32, D40.2S3, C60.73D4, C24.6D10, Dic12⋊3D5, C120.33C22, C5⋊2(D8.S3), C8.26(S3×D5), C15⋊3C16⋊5C2, C6.9(D4⋊D5), (C3×D40).3C2, C3⋊2(C5⋊SD32), C10.9(D4⋊S3), (C5×Dic12)⋊5C2, C12.2(C5⋊D4), C20.2(C3⋊D4), C2.5(C15⋊D8), C4.2(C15⋊D4), SmallGroup(480,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.D6
G = < a,b,c | a40=b6=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a25b-1 >
Smallest permutation representation of C40.D6
►On 240 points
Generators in S240
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 86 190)(2 85 191 40 87 189)(3 84 192 39 88 188)(4 83 193 38 89 187)(5 82 194 37 90 186)(6 81 195 36 91 185)(7 120 196 35 92 184)(8 119 197 34 93 183)(9 118 198 33 94 182)(10 117 199 32 95 181)(11 116 200 31 96 180)(12 115 161 30 97 179)(13 114 162 29 98 178)(14 113 163 28 99 177)(15 112 164 27 100 176)(16 111 165 26 101 175)(17 110 166 25 102 174)(18 109 167 24 103 173)(19 108 168 23 104 172)(20 107 169 22 105 171)(21 106 170)(41 140 240 72 149 231)(42 139 201 71 150 230)(43 138 202 70 151 229)(44 137 203 69 152 228)(45 136 204 68 153 227)(46 135 205 67 154 226)(47 134 206 66 155 225)(48 133 207 65 156 224)(49 132 208 64 157 223)(50 131 209 63 158 222)(51 130 210 62 159 221)(52 129 211 61 160 220)(53 128 212 60 121 219)(54 127 213 59 122 218)(55 126 214 58 123 217)(56 125 215 57 124 216)(73 148 232 80 141 239)(74 147 233 79 142 238)(75 146 234 78 143 237)(76 145 235 77 144 236)
(1 157 6 122 11 127 16 132 21 137 26 142 31 147 36 152)(2 126 7 131 12 136 17 141 22 146 27 151 32 156 37 121)(3 135 8 140 13 145 18 150 23 155 28 160 33 125 38 130)(4 144 9 149 14 154 19 159 24 124 29 129 34 134 39 139)(5 153 10 158 15 123 20 128 25 133 30 138 35 143 40 148)(41 99 46 104 51 109 56 114 61 119 66 84 71 89 76 94)(42 108 47 113 52 118 57 83 62 88 67 93 72 98 77 103)(43 117 48 82 53 87 58 92 63 97 68 102 73 107 78 112)(44 86 49 91 54 96 59 101 64 106 69 111 74 116 79 81)(45 95 50 100 55 105 60 110 65 115 70 120 75 85 80 90)(161 227 166 232 171 237 176 202 181 207 186 212 191 217 196 222)(162 236 167 201 172 206 177 211 182 216 187 221 192 226 197 231)(163 205 168 210 173 215 178 220 183 225 188 230 193 235 198 240)(164 214 169 219 174 224 179 229 184 234 189 239 194 204 199 209)(165 223 170 228 175 233 180 238 185 203 190 208 195 213 200 218)
G:=sub<Sym(240)| (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,86,190)(2,85,191,40,87,189)(3,84,192,39,88,188)(4,83,193,38,89,187)(5,82,194,37,90,186)(6,81,195,36,91,185)(7,120,196,35,92,184)(8,119,197,34,93,183)(9,118,198,33,94,182)(10,117,199,32,95,181)(11,116,200,31,96,180)(12,115,161,30,97,179)(13,114,162,29,98,178)(14,113,163,28,99,177)(15,112,164,27,100,176)(16,111,165,26,101,175)(17,110,166,25,102,174)(18,109,167,24,103,173)(19,108,168,23,104,172)(20,107,169,22,105,171)(21,106,170)(41,140,240,72,149,231)(42,139,201,71,150,230)(43,138,202,70,151,229)(44,137,203,69,152,228)(45,136,204,68,153,227)(46,135,205,67,154,226)(47,134,206,66,155,225)(48,133,207,65,156,224)(49,132,208,64,157,223)(50,131,209,63,158,222)(51,130,210,62,159,221)(52,129,211,61,160,220)(53,128,212,60,121,219)(54,127,213,59,122,218)(55,126,214,58,123,217)(56,125,215,57,124,216)(73,148,232,80,141,239)(74,147,233,79,142,238)(75,146,234,78,143,237)(76,145,235,77,144,236), (1,157,6,122,11,127,16,132,21,137,26,142,31,147,36,152)(2,126,7,131,12,136,17,141,22,146,27,151,32,156,37,121)(3,135,8,140,13,145,18,150,23,155,28,160,33,125,38,130)(4,144,9,149,14,154,19,159,24,124,29,129,34,134,39,139)(5,153,10,158,15,123,20,128,25,133,30,138,35,143,40,148)(41,99,46,104,51,109,56,114,61,119,66,84,71,89,76,94)(42,108,47,113,52,118,57,83,62,88,67,93,72,98,77,103)(43,117,48,82,53,87,58,92,63,97,68,102,73,107,78,112)(44,86,49,91,54,96,59,101,64,106,69,111,74,116,79,81)(45,95,50,100,55,105,60,110,65,115,70,120,75,85,80,90)(161,227,166,232,171,237,176,202,181,207,186,212,191,217,196,222)(162,236,167,201,172,206,177,211,182,216,187,221,192,226,197,231)(163,205,168,210,173,215,178,220,183,225,188,230,193,235,198,240)(164,214,169,219,174,224,179,229,184,234,189,239,194,204,199,209)(165,223,170,228,175,233,180,238,185,203,190,208,195,213,200,218)>;
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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,86,190)(2,85,191,40,87,189)(3,84,192,39,88,188)(4,83,193,38,89,187)(5,82,194,37,90,186)(6,81,195,36,91,185)(7,120,196,35,92,184)(8,119,197,34,93,183)(9,118,198,33,94,182)(10,117,199,32,95,181)(11,116,200,31,96,180)(12,115,161,30,97,179)(13,114,162,29,98,178)(14,113,163,28,99,177)(15,112,164,27,100,176)(16,111,165,26,101,175)(17,110,166,25,102,174)(18,109,167,24,103,173)(19,108,168,23,104,172)(20,107,169,22,105,171)(21,106,170)(41,140,240,72,149,231)(42,139,201,71,150,230)(43,138,202,70,151,229)(44,137,203,69,152,228)(45,136,204,68,153,227)(46,135,205,67,154,226)(47,134,206,66,155,225)(48,133,207,65,156,224)(49,132,208,64,157,223)(50,131,209,63,158,222)(51,130,210,62,159,221)(52,129,211,61,160,220)(53,128,212,60,121,219)(54,127,213,59,122,218)(55,126,214,58,123,217)(56,125,215,57,124,216)(73,148,232,80,141,239)(74,147,233,79,142,238)(75,146,234,78,143,237)(76,145,235,77,144,236), (1,157,6,122,11,127,16,132,21,137,26,142,31,147,36,152)(2,126,7,131,12,136,17,141,22,146,27,151,32,156,37,121)(3,135,8,140,13,145,18,150,23,155,28,160,33,125,38,130)(4,144,9,149,14,154,19,159,24,124,29,129,34,134,39,139)(5,153,10,158,15,123,20,128,25,133,30,138,35,143,40,148)(41,99,46,104,51,109,56,114,61,119,66,84,71,89,76,94)(42,108,47,113,52,118,57,83,62,88,67,93,72,98,77,103)(43,117,48,82,53,87,58,92,63,97,68,102,73,107,78,112)(44,86,49,91,54,96,59,101,64,106,69,111,74,116,79,81)(45,95,50,100,55,105,60,110,65,115,70,120,75,85,80,90)(161,227,166,232,171,237,176,202,181,207,186,212,191,217,196,222)(162,236,167,201,172,206,177,211,182,216,187,221,192,226,197,231)(163,205,168,210,173,215,178,220,183,225,188,230,193,235,198,240)(164,214,169,219,174,224,179,229,184,234,189,239,194,204,199,209)(165,223,170,228,175,233,180,238,185,203,190,208,195,213,200,218) );
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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,86,190),(2,85,191,40,87,189),(3,84,192,39,88,188),(4,83,193,38,89,187),(5,82,194,37,90,186),(6,81,195,36,91,185),(7,120,196,35,92,184),(8,119,197,34,93,183),(9,118,198,33,94,182),(10,117,199,32,95,181),(11,116,200,31,96,180),(12,115,161,30,97,179),(13,114,162,29,98,178),(14,113,163,28,99,177),(15,112,164,27,100,176),(16,111,165,26,101,175),(17,110,166,25,102,174),(18,109,167,24,103,173),(19,108,168,23,104,172),(20,107,169,22,105,171),(21,106,170),(41,140,240,72,149,231),(42,139,201,71,150,230),(43,138,202,70,151,229),(44,137,203,69,152,228),(45,136,204,68,153,227),(46,135,205,67,154,226),(47,134,206,66,155,225),(48,133,207,65,156,224),(49,132,208,64,157,223),(50,131,209,63,158,222),(51,130,210,62,159,221),(52,129,211,61,160,220),(53,128,212,60,121,219),(54,127,213,59,122,218),(55,126,214,58,123,217),(56,125,215,57,124,216),(73,148,232,80,141,239),(74,147,233,79,142,238),(75,146,234,78,143,237),(76,145,235,77,144,236)], [(1,157,6,122,11,127,16,132,21,137,26,142,31,147,36,152),(2,126,7,131,12,136,17,141,22,146,27,151,32,156,37,121),(3,135,8,140,13,145,18,150,23,155,28,160,33,125,38,130),(4,144,9,149,14,154,19,159,24,124,29,129,34,134,39,139),(5,153,10,158,15,123,20,128,25,133,30,138,35,143,40,148),(41,99,46,104,51,109,56,114,61,119,66,84,71,89,76,94),(42,108,47,113,52,118,57,83,62,88,67,93,72,98,77,103),(43,117,48,82,53,87,58,92,63,97,68,102,73,107,78,112),(44,86,49,91,54,96,59,101,64,106,69,111,74,116,79,81),(45,95,50,100,55,105,60,110,65,115,70,120,75,85,80,90),(161,227,166,232,171,237,176,202,181,207,186,212,191,217,196,222),(162,236,167,201,172,206,177,211,182,216,187,221,192,226,197,231),(163,205,168,210,173,215,178,220,183,225,188,230,193,235,198,240),(164,214,169,219,174,224,179,229,184,234,189,239,194,204,199,209),(165,223,170,228,175,233,180,238,185,203,190,208,195,213,200,218)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 12 | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 40 | 2 | 2 | 24 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 30 | 30 | 30 | 30 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | SD32 | C5⋊D4 | D4⋊S3 | S3×D5 | D4⋊D5 | D8.S3 | C15⋊D4 | C5⋊SD32 | C15⋊D8 | C40.D6 |
| kernel | C40.D6 | C15⋊3C16 | C3×D40 | C5×Dic12 | D40 | C60 | Dic12 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40.D6 ►in GL6(𝔽241)
| 79 | 129 | 0 | 0 | 0 | 0 |
| 23 | 184 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 189 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 225 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 0 | 116 | 15 |
| 48 | 222 | 0 | 0 | 0 | 0 |
| 62 | 111 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 124 | 83 |
| 0 | 0 | 0 | 0 | 169 | 117 |
G:=sub<GL(6,GF(241))| [79,23,0,0,0,0,129,184,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,225,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,225,116,0,0,0,0,0,15],[48,62,0,0,0,0,222,111,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,124,169,0,0,0,0,83,117] >;
C40.D6 in GAP, Magma, Sage, TeX
C_{40}.D_6 % in TeX
G:=Group("C40.D6"); // GroupNames label
G:=SmallGroup(480,16);
// by ID
G=gap.SmallGroup(480,16);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,120,254,135,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^25*b^-1>;
// generators/relations
Export