Copied to
clipboard

## G = C60.210D4order 480 = 25·3·5

### 10th non-split extension by C60 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.210D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C2×C15⋊3C8 — C60.210D4
 Lower central C15 — C30 — C60 — C60.210D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C60.210D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=cac-1=a29, cbc-1=a30b3 >

Subgroups: 212 in 60 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), M4(2), C20, C2×C10, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C52C8, C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60, C2×C30, C2×C52C8, C4.Dic5, C5×M4(2), C12.53D4, C153C8, C153C8, C120, C2×C60, C20.53D4, C2×C153C8, C60.7C4, C15×M4(2), C60.210D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D10, Dic6, C4×S3, C3⋊D4, D15, C8.C4, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D30, C10.D4, C12.53D4, Dic30, C4×D15, C157D4, C20.53D4, C30.4Q8, C60.210D4

Smallest permutation representation of C60.210D4
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)
(1 153 16 168 31 123 46 138)(2 122 17 137 32 152 47 167)(3 151 18 166 33 121 48 136)(4 180 19 135 34 150 49 165)(5 149 20 164 35 179 50 134)(6 178 21 133 36 148 51 163)(7 147 22 162 37 177 52 132)(8 176 23 131 38 146 53 161)(9 145 24 160 39 175 54 130)(10 174 25 129 40 144 55 159)(11 143 26 158 41 173 56 128)(12 172 27 127 42 142 57 157)(13 141 28 156 43 171 58 126)(14 170 29 125 44 140 59 155)(15 139 30 154 45 169 60 124)(61 213 106 198 91 183 76 228)(62 182 107 227 92 212 77 197)(63 211 108 196 93 181 78 226)(64 240 109 225 94 210 79 195)(65 209 110 194 95 239 80 224)(66 238 111 223 96 208 81 193)(67 207 112 192 97 237 82 222)(68 236 113 221 98 206 83 191)(69 205 114 190 99 235 84 220)(70 234 115 219 100 204 85 189)(71 203 116 188 101 233 86 218)(72 232 117 217 102 202 87 187)(73 201 118 186 103 231 88 216)(74 230 119 215 104 200 89 185)(75 199 120 184 105 229 90 214)
(1 189 46 234 31 219 16 204)(2 218 47 203 32 188 17 233)(3 187 48 232 33 217 18 202)(4 216 49 201 34 186 19 231)(5 185 50 230 35 215 20 200)(6 214 51 199 36 184 21 229)(7 183 52 228 37 213 22 198)(8 212 53 197 38 182 23 227)(9 181 54 226 39 211 24 196)(10 210 55 195 40 240 25 225)(11 239 56 224 41 209 26 194)(12 208 57 193 42 238 27 223)(13 237 58 222 43 207 28 192)(14 206 59 191 44 236 29 221)(15 235 60 220 45 205 30 190)(61 162 106 147 91 132 76 177)(62 131 107 176 92 161 77 146)(63 160 108 145 93 130 78 175)(64 129 109 174 94 159 79 144)(65 158 110 143 95 128 80 173)(66 127 111 172 96 157 81 142)(67 156 112 141 97 126 82 171)(68 125 113 170 98 155 83 140)(69 154 114 139 99 124 84 169)(70 123 115 168 100 153 85 138)(71 152 116 137 101 122 86 167)(72 121 117 166 102 151 87 136)(73 150 118 135 103 180 88 165)(74 179 119 164 104 149 89 134)(75 148 120 133 105 178 90 163)```

`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), (1,153,16,168,31,123,46,138)(2,122,17,137,32,152,47,167)(3,151,18,166,33,121,48,136)(4,180,19,135,34,150,49,165)(5,149,20,164,35,179,50,134)(6,178,21,133,36,148,51,163)(7,147,22,162,37,177,52,132)(8,176,23,131,38,146,53,161)(9,145,24,160,39,175,54,130)(10,174,25,129,40,144,55,159)(11,143,26,158,41,173,56,128)(12,172,27,127,42,142,57,157)(13,141,28,156,43,171,58,126)(14,170,29,125,44,140,59,155)(15,139,30,154,45,169,60,124)(61,213,106,198,91,183,76,228)(62,182,107,227,92,212,77,197)(63,211,108,196,93,181,78,226)(64,240,109,225,94,210,79,195)(65,209,110,194,95,239,80,224)(66,238,111,223,96,208,81,193)(67,207,112,192,97,237,82,222)(68,236,113,221,98,206,83,191)(69,205,114,190,99,235,84,220)(70,234,115,219,100,204,85,189)(71,203,116,188,101,233,86,218)(72,232,117,217,102,202,87,187)(73,201,118,186,103,231,88,216)(74,230,119,215,104,200,89,185)(75,199,120,184,105,229,90,214), (1,189,46,234,31,219,16,204)(2,218,47,203,32,188,17,233)(3,187,48,232,33,217,18,202)(4,216,49,201,34,186,19,231)(5,185,50,230,35,215,20,200)(6,214,51,199,36,184,21,229)(7,183,52,228,37,213,22,198)(8,212,53,197,38,182,23,227)(9,181,54,226,39,211,24,196)(10,210,55,195,40,240,25,225)(11,239,56,224,41,209,26,194)(12,208,57,193,42,238,27,223)(13,237,58,222,43,207,28,192)(14,206,59,191,44,236,29,221)(15,235,60,220,45,205,30,190)(61,162,106,147,91,132,76,177)(62,131,107,176,92,161,77,146)(63,160,108,145,93,130,78,175)(64,129,109,174,94,159,79,144)(65,158,110,143,95,128,80,173)(66,127,111,172,96,157,81,142)(67,156,112,141,97,126,82,171)(68,125,113,170,98,155,83,140)(69,154,114,139,99,124,84,169)(70,123,115,168,100,153,85,138)(71,152,116,137,101,122,86,167)(72,121,117,166,102,151,87,136)(73,150,118,135,103,180,88,165)(74,179,119,164,104,149,89,134)(75,148,120,133,105,178,90,163)>;`

`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), (1,153,16,168,31,123,46,138)(2,122,17,137,32,152,47,167)(3,151,18,166,33,121,48,136)(4,180,19,135,34,150,49,165)(5,149,20,164,35,179,50,134)(6,178,21,133,36,148,51,163)(7,147,22,162,37,177,52,132)(8,176,23,131,38,146,53,161)(9,145,24,160,39,175,54,130)(10,174,25,129,40,144,55,159)(11,143,26,158,41,173,56,128)(12,172,27,127,42,142,57,157)(13,141,28,156,43,171,58,126)(14,170,29,125,44,140,59,155)(15,139,30,154,45,169,60,124)(61,213,106,198,91,183,76,228)(62,182,107,227,92,212,77,197)(63,211,108,196,93,181,78,226)(64,240,109,225,94,210,79,195)(65,209,110,194,95,239,80,224)(66,238,111,223,96,208,81,193)(67,207,112,192,97,237,82,222)(68,236,113,221,98,206,83,191)(69,205,114,190,99,235,84,220)(70,234,115,219,100,204,85,189)(71,203,116,188,101,233,86,218)(72,232,117,217,102,202,87,187)(73,201,118,186,103,231,88,216)(74,230,119,215,104,200,89,185)(75,199,120,184,105,229,90,214), (1,189,46,234,31,219,16,204)(2,218,47,203,32,188,17,233)(3,187,48,232,33,217,18,202)(4,216,49,201,34,186,19,231)(5,185,50,230,35,215,20,200)(6,214,51,199,36,184,21,229)(7,183,52,228,37,213,22,198)(8,212,53,197,38,182,23,227)(9,181,54,226,39,211,24,196)(10,210,55,195,40,240,25,225)(11,239,56,224,41,209,26,194)(12,208,57,193,42,238,27,223)(13,237,58,222,43,207,28,192)(14,206,59,191,44,236,29,221)(15,235,60,220,45,205,30,190)(61,162,106,147,91,132,76,177)(62,131,107,176,92,161,77,146)(63,160,108,145,93,130,78,175)(64,129,109,174,94,159,79,144)(65,158,110,143,95,128,80,173)(66,127,111,172,96,157,81,142)(67,156,112,141,97,126,82,171)(68,125,113,170,98,155,83,140)(69,154,114,139,99,124,84,169)(70,123,115,168,100,153,85,138)(71,152,116,137,101,122,86,167)(72,121,117,166,102,151,87,136)(73,150,118,135,103,180,88,165)(74,179,119,164,104,149,89,134)(75,148,120,133,105,178,90,163) );`

`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)], [(1,153,16,168,31,123,46,138),(2,122,17,137,32,152,47,167),(3,151,18,166,33,121,48,136),(4,180,19,135,34,150,49,165),(5,149,20,164,35,179,50,134),(6,178,21,133,36,148,51,163),(7,147,22,162,37,177,52,132),(8,176,23,131,38,146,53,161),(9,145,24,160,39,175,54,130),(10,174,25,129,40,144,55,159),(11,143,26,158,41,173,56,128),(12,172,27,127,42,142,57,157),(13,141,28,156,43,171,58,126),(14,170,29,125,44,140,59,155),(15,139,30,154,45,169,60,124),(61,213,106,198,91,183,76,228),(62,182,107,227,92,212,77,197),(63,211,108,196,93,181,78,226),(64,240,109,225,94,210,79,195),(65,209,110,194,95,239,80,224),(66,238,111,223,96,208,81,193),(67,207,112,192,97,237,82,222),(68,236,113,221,98,206,83,191),(69,205,114,190,99,235,84,220),(70,234,115,219,100,204,85,189),(71,203,116,188,101,233,86,218),(72,232,117,217,102,202,87,187),(73,201,118,186,103,231,88,216),(74,230,119,215,104,200,89,185),(75,199,120,184,105,229,90,214)], [(1,189,46,234,31,219,16,204),(2,218,47,203,32,188,17,233),(3,187,48,232,33,217,18,202),(4,216,49,201,34,186,19,231),(5,185,50,230,35,215,20,200),(6,214,51,199,36,184,21,229),(7,183,52,228,37,213,22,198),(8,212,53,197,38,182,23,227),(9,181,54,226,39,211,24,196),(10,210,55,195,40,240,25,225),(11,239,56,224,41,209,26,194),(12,208,57,193,42,238,27,223),(13,237,58,222,43,207,28,192),(14,206,59,191,44,236,29,221),(15,235,60,220,45,205,30,190),(61,162,106,147,91,132,76,177),(62,131,107,176,92,161,77,146),(63,160,108,145,93,130,78,175),(64,129,109,174,94,159,79,144),(65,158,110,143,95,128,80,173),(66,127,111,172,96,157,81,142),(67,156,112,141,97,126,82,171),(68,125,113,170,98,155,83,140),(69,154,114,139,99,124,84,169),(70,123,115,168,100,153,85,138),(71,152,116,137,101,122,86,167),(72,121,117,166,102,151,87,136),(73,150,118,135,103,180,88,165),(74,179,119,164,104,149,89,134),(75,148,120,133,105,178,90,163)]])`

84 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 12A 12B 12C 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 3 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 15 15 15 15 20 20 20 20 20 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 2 1 1 2 2 2 2 4 4 4 30 30 30 30 60 60 2 2 4 4 2 2 4 2 2 2 2 2 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - + - + - image C1 C2 C2 C2 C4 S3 D4 Q8 D5 D6 D10 C4×S3 C3⋊D4 Dic6 D15 C8.C4 C4×D5 C5⋊D4 Dic10 D30 C4×D15 C15⋊7D4 Dic30 C12.53D4 C20.53D4 C60.210D4 kernel C60.210D4 C2×C15⋊3C8 C60.7C4 C15×M4(2) C15⋊3C8 C5×M4(2) C60 C2×C30 C3×M4(2) C2×C20 C2×C12 C20 C20 C2×C10 M4(2) C15 C12 C12 C2×C6 C2×C4 C4 C4 C22 C5 C3 C1 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 4 4 4 4 4 8 8 8 2 4 8

Matrix representation of C60.210D4 in GL4(𝔽241) generated by

 80 16 0 0 77 157 0 0 0 0 64 0 0 0 0 64
,
 143 138 0 0 9 98 0 0 0 0 30 0 0 0 189 233
,
 143 138 0 0 9 98 0 0 0 0 5 50 0 0 102 236
`G:=sub<GL(4,GF(241))| [80,77,0,0,16,157,0,0,0,0,64,0,0,0,0,64],[143,9,0,0,138,98,0,0,0,0,30,189,0,0,0,233],[143,9,0,0,138,98,0,0,0,0,5,102,0,0,50,236] >;`

C60.210D4 in GAP, Magma, Sage, TeX

`C_{60}._{210}D_4`
`% in TeX`

`G:=Group("C60.210D4");`
`// GroupNames label`

`G:=SmallGroup(480,182);`
`// by ID`

`G=gap.SmallGroup(480,182);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,346,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=c*a*c^-1=a^29,c*b*c^-1=a^30*b^3>;`
`// generators/relations`

׿
×
𝔽