metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.51D6, C15⋊5M5(2), C24.58D10, C120.54C22, C3⋊C16⋊4D5, (C6×D5).2C8, (C8×D5).2S3, C6.12(C8×D5), C8.37(S3×D5), C3⋊3(C80⋊C2), C30.22(C2×C8), D10.1(C3⋊C8), (D5×C12).3C4, (D5×C24).4C2, C15⋊3C16⋊11C2, C12.74(C4×D5), C5⋊3(C12.C8), C60.138(C2×C4), Dic5.1(C3⋊C8), (C4×D5).2Dic3, (C3×Dic5).2C8, C4.17(D5×Dic3), C5⋊2C8.2Dic3, C20.43(C2×Dic3), (C5×C3⋊C16)⋊6C2, C2.3(D5×C3⋊C8), C10.11(C2×C3⋊C8), (C3×C5⋊2C8).3C4, SmallGroup(480,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.51D6
G = < a,b,c | a40=b6=1, c2=a5, bab-1=cac-1=a9, cbc-1=a20b-1 >
Smallest permutation representation of C40.51D6
►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 59 179)(2 68 180 10 60 188)(3 77 181 19 61 197)(4 46 182 28 62 166)(5 55 183 37 63 175)(6 64 184)(7 73 185 15 65 193)(8 42 186 24 66 162)(9 51 187 33 67 171)(11 69 189)(12 78 190 20 70 198)(13 47 191 29 71 167)(14 56 192 38 72 176)(16 74 194)(17 43 195 25 75 163)(18 52 196 34 76 172)(21 79 199)(22 48 200 30 80 168)(23 57 161 39 41 177)(26 44 164)(27 53 165 35 45 173)(31 49 169)(32 58 170 40 50 178)(36 54 174)(81 123 225 93 151 237)(82 132 226 102 152 206)(83 141 227 111 153 215)(84 150 228 120 154 224)(85 159 229 89 155 233)(86 128 230 98 156 202)(87 137 231 107 157 211)(88 146 232 116 158 220)(90 124 234 94 160 238)(91 133 235 103 121 207)(92 142 236 112 122 216)(95 129 239 99 125 203)(96 138 240 108 126 212)(97 147 201 117 127 221)(100 134 204 104 130 208)(101 143 205 113 131 217)(105 139 209 109 135 213)(106 148 210 118 136 222)(110 144 214 114 140 218)(115 149 219 119 145 223)
(1 216 6 221 11 226 16 231 21 236 26 201 31 206 36 211)(2 225 7 230 12 235 17 240 22 205 27 210 32 215 37 220)(3 234 8 239 13 204 18 209 23 214 28 219 33 224 38 229)(4 203 9 208 14 213 19 218 24 223 29 228 34 233 39 238)(5 212 10 217 15 222 20 227 25 232 30 237 35 202 40 207)(41 140 46 145 51 150 56 155 61 160 66 125 71 130 76 135)(42 149 47 154 52 159 57 124 62 129 67 134 72 139 77 144)(43 158 48 123 53 128 58 133 63 138 68 143 73 148 78 153)(44 127 49 132 54 137 59 142 64 147 69 152 74 157 79 122)(45 136 50 141 55 146 60 151 65 156 70 121 75 126 80 131)(81 185 86 190 91 195 96 200 101 165 106 170 111 175 116 180)(82 194 87 199 92 164 97 169 102 174 107 179 112 184 117 189)(83 163 88 168 93 173 98 178 103 183 108 188 113 193 118 198)(84 172 89 177 94 182 99 187 104 192 109 197 114 162 119 167)(85 181 90 186 95 191 100 196 105 161 110 166 115 171 120 176)
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), (1,59,179)(2,68,180,10,60,188)(3,77,181,19,61,197)(4,46,182,28,62,166)(5,55,183,37,63,175)(6,64,184)(7,73,185,15,65,193)(8,42,186,24,66,162)(9,51,187,33,67,171)(11,69,189)(12,78,190,20,70,198)(13,47,191,29,71,167)(14,56,192,38,72,176)(16,74,194)(17,43,195,25,75,163)(18,52,196,34,76,172)(21,79,199)(22,48,200,30,80,168)(23,57,161,39,41,177)(26,44,164)(27,53,165,35,45,173)(31,49,169)(32,58,170,40,50,178)(36,54,174)(81,123,225,93,151,237)(82,132,226,102,152,206)(83,141,227,111,153,215)(84,150,228,120,154,224)(85,159,229,89,155,233)(86,128,230,98,156,202)(87,137,231,107,157,211)(88,146,232,116,158,220)(90,124,234,94,160,238)(91,133,235,103,121,207)(92,142,236,112,122,216)(95,129,239,99,125,203)(96,138,240,108,126,212)(97,147,201,117,127,221)(100,134,204,104,130,208)(101,143,205,113,131,217)(105,139,209,109,135,213)(106,148,210,118,136,222)(110,144,214,114,140,218)(115,149,219,119,145,223), (1,216,6,221,11,226,16,231,21,236,26,201,31,206,36,211)(2,225,7,230,12,235,17,240,22,205,27,210,32,215,37,220)(3,234,8,239,13,204,18,209,23,214,28,219,33,224,38,229)(4,203,9,208,14,213,19,218,24,223,29,228,34,233,39,238)(5,212,10,217,15,222,20,227,25,232,30,237,35,202,40,207)(41,140,46,145,51,150,56,155,61,160,66,125,71,130,76,135)(42,149,47,154,52,159,57,124,62,129,67,134,72,139,77,144)(43,158,48,123,53,128,58,133,63,138,68,143,73,148,78,153)(44,127,49,132,54,137,59,142,64,147,69,152,74,157,79,122)(45,136,50,141,55,146,60,151,65,156,70,121,75,126,80,131)(81,185,86,190,91,195,96,200,101,165,106,170,111,175,116,180)(82,194,87,199,92,164,97,169,102,174,107,179,112,184,117,189)(83,163,88,168,93,173,98,178,103,183,108,188,113,193,118,198)(84,172,89,177,94,182,99,187,104,192,109,197,114,162,119,167)(85,181,90,186,95,191,100,196,105,161,110,166,115,171,120,176)>;
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), (1,59,179)(2,68,180,10,60,188)(3,77,181,19,61,197)(4,46,182,28,62,166)(5,55,183,37,63,175)(6,64,184)(7,73,185,15,65,193)(8,42,186,24,66,162)(9,51,187,33,67,171)(11,69,189)(12,78,190,20,70,198)(13,47,191,29,71,167)(14,56,192,38,72,176)(16,74,194)(17,43,195,25,75,163)(18,52,196,34,76,172)(21,79,199)(22,48,200,30,80,168)(23,57,161,39,41,177)(26,44,164)(27,53,165,35,45,173)(31,49,169)(32,58,170,40,50,178)(36,54,174)(81,123,225,93,151,237)(82,132,226,102,152,206)(83,141,227,111,153,215)(84,150,228,120,154,224)(85,159,229,89,155,233)(86,128,230,98,156,202)(87,137,231,107,157,211)(88,146,232,116,158,220)(90,124,234,94,160,238)(91,133,235,103,121,207)(92,142,236,112,122,216)(95,129,239,99,125,203)(96,138,240,108,126,212)(97,147,201,117,127,221)(100,134,204,104,130,208)(101,143,205,113,131,217)(105,139,209,109,135,213)(106,148,210,118,136,222)(110,144,214,114,140,218)(115,149,219,119,145,223), (1,216,6,221,11,226,16,231,21,236,26,201,31,206,36,211)(2,225,7,230,12,235,17,240,22,205,27,210,32,215,37,220)(3,234,8,239,13,204,18,209,23,214,28,219,33,224,38,229)(4,203,9,208,14,213,19,218,24,223,29,228,34,233,39,238)(5,212,10,217,15,222,20,227,25,232,30,237,35,202,40,207)(41,140,46,145,51,150,56,155,61,160,66,125,71,130,76,135)(42,149,47,154,52,159,57,124,62,129,67,134,72,139,77,144)(43,158,48,123,53,128,58,133,63,138,68,143,73,148,78,153)(44,127,49,132,54,137,59,142,64,147,69,152,74,157,79,122)(45,136,50,141,55,146,60,151,65,156,70,121,75,126,80,131)(81,185,86,190,91,195,96,200,101,165,106,170,111,175,116,180)(82,194,87,199,92,164,97,169,102,174,107,179,112,184,117,189)(83,163,88,168,93,173,98,178,103,183,108,188,113,193,118,198)(84,172,89,177,94,182,99,187,104,192,109,197,114,162,119,167)(85,181,90,186,95,191,100,196,105,161,110,166,115,171,120,176) );
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)], [(1,59,179),(2,68,180,10,60,188),(3,77,181,19,61,197),(4,46,182,28,62,166),(5,55,183,37,63,175),(6,64,184),(7,73,185,15,65,193),(8,42,186,24,66,162),(9,51,187,33,67,171),(11,69,189),(12,78,190,20,70,198),(13,47,191,29,71,167),(14,56,192,38,72,176),(16,74,194),(17,43,195,25,75,163),(18,52,196,34,76,172),(21,79,199),(22,48,200,30,80,168),(23,57,161,39,41,177),(26,44,164),(27,53,165,35,45,173),(31,49,169),(32,58,170,40,50,178),(36,54,174),(81,123,225,93,151,237),(82,132,226,102,152,206),(83,141,227,111,153,215),(84,150,228,120,154,224),(85,159,229,89,155,233),(86,128,230,98,156,202),(87,137,231,107,157,211),(88,146,232,116,158,220),(90,124,234,94,160,238),(91,133,235,103,121,207),(92,142,236,112,122,216),(95,129,239,99,125,203),(96,138,240,108,126,212),(97,147,201,117,127,221),(100,134,204,104,130,208),(101,143,205,113,131,217),(105,139,209,109,135,213),(106,148,210,118,136,222),(110,144,214,114,140,218),(115,149,219,119,145,223)], [(1,216,6,221,11,226,16,231,21,236,26,201,31,206,36,211),(2,225,7,230,12,235,17,240,22,205,27,210,32,215,37,220),(3,234,8,239,13,204,18,209,23,214,28,219,33,224,38,229),(4,203,9,208,14,213,19,218,24,223,29,228,34,233,39,238),(5,212,10,217,15,222,20,227,25,232,30,237,35,202,40,207),(41,140,46,145,51,150,56,155,61,160,66,125,71,130,76,135),(42,149,47,154,52,159,57,124,62,129,67,134,72,139,77,144),(43,158,48,123,53,128,58,133,63,138,68,143,73,148,78,153),(44,127,49,132,54,137,59,142,64,147,69,152,74,157,79,122),(45,136,50,141,55,146,60,151,65,156,70,121,75,126,80,131),(81,185,86,190,91,195,96,200,101,165,106,170,111,175,116,180),(82,194,87,199,92,164,97,169,102,174,107,179,112,184,117,189),(83,163,88,168,93,173,98,178,103,183,108,188,113,193,118,198),(84,172,89,177,94,182,99,187,104,192,109,197,114,162,119,167),(85,181,90,186,95,191,100,196,105,161,110,166,115,171,120,176)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 2 | 1 | 1 | 10 | 2 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 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 | Dic3 | D6 | Dic3 | D10 | C3⋊C8 | C3⋊C8 | M5(2) | C4×D5 | C8×D5 | C12.C8 | C80⋊C2 | S3×D5 | D5×Dic3 | D5×C3⋊C8 | C40.51D6 |
| kernel | C40.51D6 | C5×C3⋊C16 | C15⋊3C16 | D5×C24 | C3×C5⋊2C8 | D5×C12 | C3×Dic5 | C6×D5 | C8×D5 | C3⋊C16 | C5⋊2C8 | C40 | C4×D5 | C24 | Dic5 | D10 | C15 | C12 | C6 | C5 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C40.51D6 ►in GL4(𝔽241) generated by
| 110 | 131 | 0 | 0 |
| 110 | 177 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 51 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 62 | 16 |
| 8 | 167 | 0 | 0 |
| 0 | 233 | 0 | 0 |
| 0 | 0 | 148 | 119 |
| 0 | 0 | 2 | 93 |
G:=sub<GL(4,GF(241))| [110,110,0,0,131,177,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,51,240,0,0,0,0,15,62,0,0,0,16],[8,0,0,0,167,233,0,0,0,0,148,2,0,0,119,93] >;
C40.51D6 in GAP, Magma, Sage, TeX
C_{40}._{51}D_6 % in TeX
G:=Group("C40.51D6"); // GroupNames label
G:=SmallGroup(480,10);
// by ID
G=gap.SmallGroup(480,10);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,36,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;
// generators/relations
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