metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.5D4, C8.6D30, Q16⋊1D15, C30.41D8, C40.14D6, C15⋊13SD32, D120.3C2, C24.14D10, C120.11C22, (C5×Q16)⋊1S3, (C3×Q16)⋊1D5, C15⋊3C16⋊3C2, (C15×Q16)⋊1C2, C5⋊3(C8.6D6), C3⋊3(C5⋊SD32), C2.6(D4⋊D15), C6.19(D4⋊D5), C4.3(C15⋊7D4), C10.19(D4⋊S3), C12.19(C5⋊D4), C20.17(C3⋊D4), SmallGroup(480,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.6D30
G = < a,b,c | a8=1, b30=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b29 >
120C2
4C4
60C22
40S3
24D5
2Q8
30D4
4C12
20D6
4C20
12D10
8D15
15D8
15C16
10D12
6D20
4D30
4C60
15SD32
5D24
3D40
2D60
Smallest permutation representation of C8.6D30
►On 240 points
Generators in S240
(1 218 83 175 31 188 113 145)(2 146 114 189 32 176 84 219)(3 220 85 177 33 190 115 147)(4 148 116 191 34 178 86 221)(5 222 87 179 35 192 117 149)(6 150 118 193 36 180 88 223)(7 224 89 121 37 194 119 151)(8 152 120 195 38 122 90 225)(9 226 91 123 39 196 61 153)(10 154 62 197 40 124 92 227)(11 228 93 125 41 198 63 155)(12 156 64 199 42 126 94 229)(13 230 95 127 43 200 65 157)(14 158 66 201 44 128 96 231)(15 232 97 129 45 202 67 159)(16 160 68 203 46 130 98 233)(17 234 99 131 47 204 69 161)(18 162 70 205 48 132 100 235)(19 236 101 133 49 206 71 163)(20 164 72 207 50 134 102 237)(21 238 103 135 51 208 73 165)(22 166 74 209 52 136 104 239)(23 240 105 137 53 210 75 167)(24 168 76 211 54 138 106 181)(25 182 107 139 55 212 77 169)(26 170 78 213 56 140 108 183)(27 184 109 141 57 214 79 171)(28 172 80 215 58 142 110 185)(29 186 111 143 59 216 81 173)(30 174 82 217 60 144 112 187)
(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 30 175 217 113 112 218 174 31 60 145 187 83 82 188 144)(2 143 189 81 84 186 146 59 32 173 219 111 114 216 176 29)(3 28 177 215 115 110 220 172 33 58 147 185 85 80 190 142)(4 141 191 79 86 184 148 57 34 171 221 109 116 214 178 27)(5 26 179 213 117 108 222 170 35 56 149 183 87 78 192 140)(6 139 193 77 88 182 150 55 36 169 223 107 118 212 180 25)(7 24 121 211 119 106 224 168 37 54 151 181 89 76 194 138)(8 137 195 75 90 240 152 53 38 167 225 105 120 210 122 23)(9 22 123 209 61 104 226 166 39 52 153 239 91 74 196 136)(10 135 197 73 92 238 154 51 40 165 227 103 62 208 124 21)(11 20 125 207 63 102 228 164 41 50 155 237 93 72 198 134)(12 133 199 71 94 236 156 49 42 163 229 101 64 206 126 19)(13 18 127 205 65 100 230 162 43 48 157 235 95 70 200 132)(14 131 201 69 96 234 158 47 44 161 231 99 66 204 128 17)(15 16 129 203 67 98 232 160 45 46 159 233 97 68 202 130)
G:=sub<Sym(240)| (1,218,83,175,31,188,113,145)(2,146,114,189,32,176,84,219)(3,220,85,177,33,190,115,147)(4,148,116,191,34,178,86,221)(5,222,87,179,35,192,117,149)(6,150,118,193,36,180,88,223)(7,224,89,121,37,194,119,151)(8,152,120,195,38,122,90,225)(9,226,91,123,39,196,61,153)(10,154,62,197,40,124,92,227)(11,228,93,125,41,198,63,155)(12,156,64,199,42,126,94,229)(13,230,95,127,43,200,65,157)(14,158,66,201,44,128,96,231)(15,232,97,129,45,202,67,159)(16,160,68,203,46,130,98,233)(17,234,99,131,47,204,69,161)(18,162,70,205,48,132,100,235)(19,236,101,133,49,206,71,163)(20,164,72,207,50,134,102,237)(21,238,103,135,51,208,73,165)(22,166,74,209,52,136,104,239)(23,240,105,137,53,210,75,167)(24,168,76,211,54,138,106,181)(25,182,107,139,55,212,77,169)(26,170,78,213,56,140,108,183)(27,184,109,141,57,214,79,171)(28,172,80,215,58,142,110,185)(29,186,111,143,59,216,81,173)(30,174,82,217,60,144,112,187), (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), (1,30,175,217,113,112,218,174,31,60,145,187,83,82,188,144)(2,143,189,81,84,186,146,59,32,173,219,111,114,216,176,29)(3,28,177,215,115,110,220,172,33,58,147,185,85,80,190,142)(4,141,191,79,86,184,148,57,34,171,221,109,116,214,178,27)(5,26,179,213,117,108,222,170,35,56,149,183,87,78,192,140)(6,139,193,77,88,182,150,55,36,169,223,107,118,212,180,25)(7,24,121,211,119,106,224,168,37,54,151,181,89,76,194,138)(8,137,195,75,90,240,152,53,38,167,225,105,120,210,122,23)(9,22,123,209,61,104,226,166,39,52,153,239,91,74,196,136)(10,135,197,73,92,238,154,51,40,165,227,103,62,208,124,21)(11,20,125,207,63,102,228,164,41,50,155,237,93,72,198,134)(12,133,199,71,94,236,156,49,42,163,229,101,64,206,126,19)(13,18,127,205,65,100,230,162,43,48,157,235,95,70,200,132)(14,131,201,69,96,234,158,47,44,161,231,99,66,204,128,17)(15,16,129,203,67,98,232,160,45,46,159,233,97,68,202,130)>;
G:=Group( (1,218,83,175,31,188,113,145)(2,146,114,189,32,176,84,219)(3,220,85,177,33,190,115,147)(4,148,116,191,34,178,86,221)(5,222,87,179,35,192,117,149)(6,150,118,193,36,180,88,223)(7,224,89,121,37,194,119,151)(8,152,120,195,38,122,90,225)(9,226,91,123,39,196,61,153)(10,154,62,197,40,124,92,227)(11,228,93,125,41,198,63,155)(12,156,64,199,42,126,94,229)(13,230,95,127,43,200,65,157)(14,158,66,201,44,128,96,231)(15,232,97,129,45,202,67,159)(16,160,68,203,46,130,98,233)(17,234,99,131,47,204,69,161)(18,162,70,205,48,132,100,235)(19,236,101,133,49,206,71,163)(20,164,72,207,50,134,102,237)(21,238,103,135,51,208,73,165)(22,166,74,209,52,136,104,239)(23,240,105,137,53,210,75,167)(24,168,76,211,54,138,106,181)(25,182,107,139,55,212,77,169)(26,170,78,213,56,140,108,183)(27,184,109,141,57,214,79,171)(28,172,80,215,58,142,110,185)(29,186,111,143,59,216,81,173)(30,174,82,217,60,144,112,187), (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), (1,30,175,217,113,112,218,174,31,60,145,187,83,82,188,144)(2,143,189,81,84,186,146,59,32,173,219,111,114,216,176,29)(3,28,177,215,115,110,220,172,33,58,147,185,85,80,190,142)(4,141,191,79,86,184,148,57,34,171,221,109,116,214,178,27)(5,26,179,213,117,108,222,170,35,56,149,183,87,78,192,140)(6,139,193,77,88,182,150,55,36,169,223,107,118,212,180,25)(7,24,121,211,119,106,224,168,37,54,151,181,89,76,194,138)(8,137,195,75,90,240,152,53,38,167,225,105,120,210,122,23)(9,22,123,209,61,104,226,166,39,52,153,239,91,74,196,136)(10,135,197,73,92,238,154,51,40,165,227,103,62,208,124,21)(11,20,125,207,63,102,228,164,41,50,155,237,93,72,198,134)(12,133,199,71,94,236,156,49,42,163,229,101,64,206,126,19)(13,18,127,205,65,100,230,162,43,48,157,235,95,70,200,132)(14,131,201,69,96,234,158,47,44,161,231,99,66,204,128,17)(15,16,129,203,67,98,232,160,45,46,159,233,97,68,202,130) );
G=PermutationGroup([[(1,218,83,175,31,188,113,145),(2,146,114,189,32,176,84,219),(3,220,85,177,33,190,115,147),(4,148,116,191,34,178,86,221),(5,222,87,179,35,192,117,149),(6,150,118,193,36,180,88,223),(7,224,89,121,37,194,119,151),(8,152,120,195,38,122,90,225),(9,226,91,123,39,196,61,153),(10,154,62,197,40,124,92,227),(11,228,93,125,41,198,63,155),(12,156,64,199,42,126,94,229),(13,230,95,127,43,200,65,157),(14,158,66,201,44,128,96,231),(15,232,97,129,45,202,67,159),(16,160,68,203,46,130,98,233),(17,234,99,131,47,204,69,161),(18,162,70,205,48,132,100,235),(19,236,101,133,49,206,71,163),(20,164,72,207,50,134,102,237),(21,238,103,135,51,208,73,165),(22,166,74,209,52,136,104,239),(23,240,105,137,53,210,75,167),(24,168,76,211,54,138,106,181),(25,182,107,139,55,212,77,169),(26,170,78,213,56,140,108,183),(27,184,109,141,57,214,79,171),(28,172,80,215,58,142,110,185),(29,186,111,143,59,216,81,173),(30,174,82,217,60,144,112,187)], [(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)], [(1,30,175,217,113,112,218,174,31,60,145,187,83,82,188,144),(2,143,189,81,84,186,146,59,32,173,219,111,114,216,176,29),(3,28,177,215,115,110,220,172,33,58,147,185,85,80,190,142),(4,141,191,79,86,184,148,57,34,171,221,109,116,214,178,27),(5,26,179,213,117,108,222,170,35,56,149,183,87,78,192,140),(6,139,193,77,88,182,150,55,36,169,223,107,118,212,180,25),(7,24,121,211,119,106,224,168,37,54,151,181,89,76,194,138),(8,137,195,75,90,240,152,53,38,167,225,105,120,210,122,23),(9,22,123,209,61,104,226,166,39,52,153,239,91,74,196,136),(10,135,197,73,92,238,154,51,40,165,227,103,62,208,124,21),(11,20,125,207,63,102,228,164,41,50,155,237,93,72,198,134),(12,133,199,71,94,236,156,49,42,163,229,101,64,206,126,19),(13,18,127,205,65,100,230,162,43,48,157,235,95,70,200,132),(14,131,201,69,96,234,158,47,44,161,231,99,66,204,128,17),(15,16,129,203,67,98,232,160,45,46,159,233,97,68,202,130)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 30C | 30D | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 120 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | D15 | SD32 | C5⋊D4 | D30 | C15⋊7D4 | D4⋊S3 | D4⋊D5 | C8.6D6 | C5⋊SD32 | D4⋊D15 | C8.6D30 |
| kernel | C8.6D30 | C15⋊3C16 | D120 | C15×Q16 | C5×Q16 | C60 | C3×Q16 | C40 | C30 | C24 | C20 | Q16 | C15 | C12 | C8 | C4 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C8.6D30 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 230 | 11 |
| 0 | 0 | 230 | 230 |
| 110 | 98 | 0 | 0 |
| 108 | 2 | 0 | 0 |
| 0 | 0 | 103 | 200 |
| 0 | 0 | 200 | 138 |
| 69 | 33 | 0 | 0 |
| 104 | 172 | 0 | 0 |
| 0 | 0 | 103 | 41 |
| 0 | 0 | 200 | 103 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[110,108,0,0,98,2,0,0,0,0,103,200,0,0,200,138],[69,104,0,0,33,172,0,0,0,0,103,200,0,0,41,103] >;
C8.6D30 in GAP, Magma, Sage, TeX
C_8._6D_{30} % in TeX
G:=Group("C8.6D30"); // GroupNames label
G:=SmallGroup(480,188);
// by ID
G=gap.SmallGroup(480,188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,120,254,135,142,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^30=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^29>;
// generators/relations
Export