metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.52D6, C15⋊6M5(2), C24.59D10, C120.55C22, C5⋊2C16⋊4S3, C5⋊5(D6.C8), D6.(C5⋊2C8), (S3×C8).2D5, C8.38(S3×D5), C3⋊C8.2Dic5, (S3×C40).3C2, (S3×C10).4C8, (S3×C20).9C4, C10.21(S3×C8), C30.23(C2×C8), C15⋊3C16⋊12C2, C3⋊1(C20.4C8), C20.106(C4×S3), C60.139(C2×C4), Dic3.(C5⋊2C8), (C4×S3).2Dic5, (C5×Dic3).4C8, C4.17(S3×Dic5), C12.22(C2×Dic5), (C5×C3⋊C8).7C4, C6.2(C2×C5⋊2C8), C2.3(S3×C5⋊2C8), (C3×C5⋊2C16)⋊9C2, SmallGroup(480,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.52D6
G = < a,b,c | a40=1, b6=a35, c2=a5, bab-1=cac-1=a9, cbc-1=a10b5 >
Smallest permutation representation of C40.52D6
►On 240 points
Generators in S240
(1 185 219 74 120 43 179 213 68 114 37 173 207 62 108 31 167 201 56 102 25 161 195 50 144 19 155 237 92 138 13 149 231 86 132 7 191 225 80 126)(2 115 57 238 192 44 109 51 232 186 38 103 93 226 180 32 97 87 220 174 26 139 81 214 168 20 133 75 208 162 14 127 69 202 156 8 121 63 196 150)(3 187 221 76 122 45 181 215 70 116 39 175 209 64 110 33 169 203 58 104 27 163 197 52 98 21 157 239 94 140 15 151 233 88 134 9 145 227 82 128)(4 117 59 240 146 46 111 53 234 188 40 105 95 228 182 34 99 89 222 176 28 141 83 216 170 22 135 77 210 164 16 129 71 204 158 10 123 65 198 152)(5 189 223 78 124 47 183 217 72 118 41 177 211 66 112 35 171 205 60 106 29 165 199 54 100 23 159 193 96 142 17 153 235 90 136 11 147 229 84 130)(6 119 61 194 148 48 113 55 236 190 42 107 49 230 184 36 101 91 224 178 30 143 85 218 172 24 137 79 212 166 18 131 73 206 160 12 125 67 200 154)
(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 18 43 12 37 6 31 48 25 42 19 36 13 30 7 24)(2 11 44 5 38 47 32 41 26 35 20 29 14 23 8 17)(3 4 45 46 39 40 33 34 27 28 21 22 15 16 9 10)(49 74 91 68 85 62 79 56 73 50 67 92 61 86 55 80)(51 60 93 54 87 96 81 90 75 84 69 78 63 72 57 66)(52 53 94 95 88 89 82 83 76 77 70 71 64 65 58 59)(97 106 139 100 133 142 127 136 121 130 115 124 109 118 103 112)(98 99 140 141 134 135 128 129 122 123 116 117 110 111 104 105)(101 126 143 120 137 114 131 108 125 102 119 144 113 138 107 132)(145 146 187 188 181 182 175 176 169 170 163 164 157 158 151 152)(147 180 189 174 183 168 177 162 171 156 165 150 159 192 153 186)(148 173 190 167 184 161 178 155 172 149 166 191 160 185 154 179)(193 226 235 220 229 214 223 208 217 202 211 196 205 238 199 232)(194 219 236 213 230 207 224 201 218 195 212 237 206 231 200 225)(197 198 239 240 233 234 227 228 221 222 215 216 209 210 203 204)
G:=sub<Sym(240)| (1,185,219,74,120,43,179,213,68,114,37,173,207,62,108,31,167,201,56,102,25,161,195,50,144,19,155,237,92,138,13,149,231,86,132,7,191,225,80,126)(2,115,57,238,192,44,109,51,232,186,38,103,93,226,180,32,97,87,220,174,26,139,81,214,168,20,133,75,208,162,14,127,69,202,156,8,121,63,196,150)(3,187,221,76,122,45,181,215,70,116,39,175,209,64,110,33,169,203,58,104,27,163,197,52,98,21,157,239,94,140,15,151,233,88,134,9,145,227,82,128)(4,117,59,240,146,46,111,53,234,188,40,105,95,228,182,34,99,89,222,176,28,141,83,216,170,22,135,77,210,164,16,129,71,204,158,10,123,65,198,152)(5,189,223,78,124,47,183,217,72,118,41,177,211,66,112,35,171,205,60,106,29,165,199,54,100,23,159,193,96,142,17,153,235,90,136,11,147,229,84,130)(6,119,61,194,148,48,113,55,236,190,42,107,49,230,184,36,101,91,224,178,30,143,85,218,172,24,137,79,212,166,18,131,73,206,160,12,125,67,200,154), (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), (1,18,43,12,37,6,31,48,25,42,19,36,13,30,7,24)(2,11,44,5,38,47,32,41,26,35,20,29,14,23,8,17)(3,4,45,46,39,40,33,34,27,28,21,22,15,16,9,10)(49,74,91,68,85,62,79,56,73,50,67,92,61,86,55,80)(51,60,93,54,87,96,81,90,75,84,69,78,63,72,57,66)(52,53,94,95,88,89,82,83,76,77,70,71,64,65,58,59)(97,106,139,100,133,142,127,136,121,130,115,124,109,118,103,112)(98,99,140,141,134,135,128,129,122,123,116,117,110,111,104,105)(101,126,143,120,137,114,131,108,125,102,119,144,113,138,107,132)(145,146,187,188,181,182,175,176,169,170,163,164,157,158,151,152)(147,180,189,174,183,168,177,162,171,156,165,150,159,192,153,186)(148,173,190,167,184,161,178,155,172,149,166,191,160,185,154,179)(193,226,235,220,229,214,223,208,217,202,211,196,205,238,199,232)(194,219,236,213,230,207,224,201,218,195,212,237,206,231,200,225)(197,198,239,240,233,234,227,228,221,222,215,216,209,210,203,204)>;
G:=Group( (1,185,219,74,120,43,179,213,68,114,37,173,207,62,108,31,167,201,56,102,25,161,195,50,144,19,155,237,92,138,13,149,231,86,132,7,191,225,80,126)(2,115,57,238,192,44,109,51,232,186,38,103,93,226,180,32,97,87,220,174,26,139,81,214,168,20,133,75,208,162,14,127,69,202,156,8,121,63,196,150)(3,187,221,76,122,45,181,215,70,116,39,175,209,64,110,33,169,203,58,104,27,163,197,52,98,21,157,239,94,140,15,151,233,88,134,9,145,227,82,128)(4,117,59,240,146,46,111,53,234,188,40,105,95,228,182,34,99,89,222,176,28,141,83,216,170,22,135,77,210,164,16,129,71,204,158,10,123,65,198,152)(5,189,223,78,124,47,183,217,72,118,41,177,211,66,112,35,171,205,60,106,29,165,199,54,100,23,159,193,96,142,17,153,235,90,136,11,147,229,84,130)(6,119,61,194,148,48,113,55,236,190,42,107,49,230,184,36,101,91,224,178,30,143,85,218,172,24,137,79,212,166,18,131,73,206,160,12,125,67,200,154), (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), (1,18,43,12,37,6,31,48,25,42,19,36,13,30,7,24)(2,11,44,5,38,47,32,41,26,35,20,29,14,23,8,17)(3,4,45,46,39,40,33,34,27,28,21,22,15,16,9,10)(49,74,91,68,85,62,79,56,73,50,67,92,61,86,55,80)(51,60,93,54,87,96,81,90,75,84,69,78,63,72,57,66)(52,53,94,95,88,89,82,83,76,77,70,71,64,65,58,59)(97,106,139,100,133,142,127,136,121,130,115,124,109,118,103,112)(98,99,140,141,134,135,128,129,122,123,116,117,110,111,104,105)(101,126,143,120,137,114,131,108,125,102,119,144,113,138,107,132)(145,146,187,188,181,182,175,176,169,170,163,164,157,158,151,152)(147,180,189,174,183,168,177,162,171,156,165,150,159,192,153,186)(148,173,190,167,184,161,178,155,172,149,166,191,160,185,154,179)(193,226,235,220,229,214,223,208,217,202,211,196,205,238,199,232)(194,219,236,213,230,207,224,201,218,195,212,237,206,231,200,225)(197,198,239,240,233,234,227,228,221,222,215,216,209,210,203,204) );
G=PermutationGroup([[(1,185,219,74,120,43,179,213,68,114,37,173,207,62,108,31,167,201,56,102,25,161,195,50,144,19,155,237,92,138,13,149,231,86,132,7,191,225,80,126),(2,115,57,238,192,44,109,51,232,186,38,103,93,226,180,32,97,87,220,174,26,139,81,214,168,20,133,75,208,162,14,127,69,202,156,8,121,63,196,150),(3,187,221,76,122,45,181,215,70,116,39,175,209,64,110,33,169,203,58,104,27,163,197,52,98,21,157,239,94,140,15,151,233,88,134,9,145,227,82,128),(4,117,59,240,146,46,111,53,234,188,40,105,95,228,182,34,99,89,222,176,28,141,83,216,170,22,135,77,210,164,16,129,71,204,158,10,123,65,198,152),(5,189,223,78,124,47,183,217,72,118,41,177,211,66,112,35,171,205,60,106,29,165,199,54,100,23,159,193,96,142,17,153,235,90,136,11,147,229,84,130),(6,119,61,194,148,48,113,55,236,190,42,107,49,230,184,36,101,91,224,178,30,143,85,218,172,24,137,79,212,166,18,131,73,206,160,12,125,67,200,154)], [(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)], [(1,18,43,12,37,6,31,48,25,42,19,36,13,30,7,24),(2,11,44,5,38,47,32,41,26,35,20,29,14,23,8,17),(3,4,45,46,39,40,33,34,27,28,21,22,15,16,9,10),(49,74,91,68,85,62,79,56,73,50,67,92,61,86,55,80),(51,60,93,54,87,96,81,90,75,84,69,78,63,72,57,66),(52,53,94,95,88,89,82,83,76,77,70,71,64,65,58,59),(97,106,139,100,133,142,127,136,121,130,115,124,109,118,103,112),(98,99,140,141,134,135,128,129,122,123,116,117,110,111,104,105),(101,126,143,120,137,114,131,108,125,102,119,144,113,138,107,132),(145,146,187,188,181,182,175,176,169,170,163,164,157,158,151,152),(147,180,189,174,183,168,177,162,171,156,165,150,159,192,153,186),(148,173,190,167,184,161,178,155,172,149,166,191,160,185,154,179),(193,226,235,220,229,214,223,208,217,202,211,196,205,238,199,232),(194,219,236,213,230,207,224,201,218,195,212,237,206,231,200,225),(197,198,239,240,233,234,227,228,221,222,215,216,209,210,203,204)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D5 | D6 | Dic5 | D10 | Dic5 | C4×S3 | M5(2) | C5⋊2C8 | C5⋊2C8 | S3×C8 | D6.C8 | C20.4C8 | S3×D5 | S3×Dic5 | S3×C5⋊2C8 | C40.52D6 |
| kernel | C40.52D6 | C3×C5⋊2C16 | C15⋊3C16 | S3×C40 | C5×C3⋊C8 | S3×C20 | C5×Dic3 | S3×C10 | C5⋊2C16 | S3×C8 | C40 | C3⋊C8 | C24 | C4×S3 | C20 | C15 | Dic3 | D6 | C10 | C5 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C40.52D6 ►in GL4(𝔽241) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 180 | 0 |
| 0 | 0 | 13 | 236 |
| 0 | 33 | 0 | 0 |
| 139 | 82 | 0 | 0 |
| 0 | 0 | 240 | 144 |
| 0 | 0 | 15 | 1 |
| 0 | 208 | 0 | 0 |
| 139 | 0 | 0 | 0 |
| 0 | 0 | 1 | 97 |
| 0 | 0 | 92 | 240 |
G:=sub<GL(4,GF(241))| [8,0,0,0,0,8,0,0,0,0,180,13,0,0,0,236],[0,139,0,0,33,82,0,0,0,0,240,15,0,0,144,1],[0,139,0,0,208,0,0,0,0,0,1,92,0,0,97,240] >;
C40.52D6 in GAP, Magma, Sage, TeX
C_{40}._{52}D_6 % in TeX
G:=Group("C40.52D6"); // GroupNames label
G:=SmallGroup(480,11);
// by ID
G=gap.SmallGroup(480,11);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^6=a^35,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^5>;
// generators/relations
Export