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## G = C60.105D4order 480 = 25·3·5

### 105th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.105D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×C5⋊2C8 — C60.105D4
 Lower central C15 — C30 — C60 — C60.105D4
 Upper central C1 — C4 — C2×C4

Generators and relations for C60.105D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a49, cac-1=a29, cbc-1=a30b3 >

Subgroups: 188 in 60 conjugacy classes, 34 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), C20, C2×C10, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C52C8, C52C8, C40, C2×C20, C4.Dic3, C4.Dic3, C2×C24, C60, C2×C30, C2×C52C8, C4.Dic5, C5×M4(2), C24.C4, C5×C3⋊C8, C3×C52C8, C153C8, C2×C60, C20.53D4, C6×C52C8, C5×C4.Dic3, C60.7C4, C60.105D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, D10, Dic6, D12, C2×Dic3, C8.C4, Dic10, C4×D5, C5⋊D4, C4⋊Dic3, S3×D5, C10.D4, C24.C4, D5×Dic3, C5⋊D12, C15⋊Q8, C20.53D4, C30.Q8, C60.105D4

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + + + - - + - + - image C1 C2 C2 C2 C4 S3 D4 Q8 D5 Dic3 D6 D10 D12 Dic6 C8.C4 C4×D5 C5⋊D4 Dic10 C24.C4 S3×D5 D5×Dic3 C5⋊D12 C15⋊Q8 C20.53D4 C60.105D4 kernel C60.105D4 C6×C5⋊2C8 C5×C4.Dic3 C60.7C4 C3×C5⋊2C8 C2×C5⋊2C8 C60 C2×C30 C4.Dic3 C5⋊2C8 C2×C20 C2×C12 C20 C2×C10 C15 C12 C12 C2×C6 C5 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 1 1 2 2 1 2 2 2 4 4 4 4 8 2 2 2 2 4 8

Matrix representation of C60.105D4 in GL4(𝔽241) generated by

 181 0 0 0 0 237 0 0 0 0 51 51 0 0 190 1
,
 211 0 0 0 0 8 0 0 0 0 63 154 0 0 234 178
,
 0 8 0 0 8 0 0 0 0 0 100 80 0 0 119 141
`G:=sub<GL(4,GF(241))| [181,0,0,0,0,237,0,0,0,0,51,190,0,0,51,1],[211,0,0,0,0,8,0,0,0,0,63,234,0,0,154,178],[0,8,0,0,8,0,0,0,0,0,100,119,0,0,80,141] >;`

C60.105D4 in GAP, Magma, Sage, TeX

`C_{60}._{105}D_4`
`% in TeX`

`G:=Group("C60.105D4");`
`// GroupNames label`

`G:=SmallGroup(480,67);`
`// by ID`

`G=gap.SmallGroup(480,67);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^49,c*a*c^-1=a^29,c*b*c^-1=a^30*b^3>;`
`// generators/relations`

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