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## G = C60.212D4order 480 = 25·3·5

### 12nd non-split extension by C60 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C60.212D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C2×C15⋊3C8 — C60.212D4
 Lower central C15 — C30 — C60.212D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C60.212D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=cac-1=a29, cbc-1=a15b3 >

Subgroups: 276 in 100 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×C8 [×2], C22×C4, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C3⋊C8 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊C8, C52C8 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C3⋊C8 [×2], C22×C12, C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C52C8 [×2], C22×C20, C12.55D4, C153C8 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C20.55D4, C2×C153C8 [×2], C22×C60, C60.212D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C3⋊C8 [×2], C2×Dic3, C3⋊D4 [×2], D15, C22⋊C8, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], C2×C3⋊C8, C4.Dic3, C6.D4, Dic15 [×2], D30, C2×C52C8, C4.Dic5, C23.D5, C12.55D4, C153C8 [×2], C2×Dic15, C157D4 [×2], C20.55D4, C2×C153C8, C60.7C4, C30.38D4, C60.212D4

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

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - - + - + - + - image C1 C2 C2 C4 C4 C8 S3 D4 D5 Dic3 D6 Dic3 M4(2) Dic5 D10 Dic5 C3⋊D4 C3⋊C8 D15 C5⋊D4 C5⋊2C8 C4.Dic3 Dic15 D30 Dic15 C4.Dic5 C15⋊7D4 C15⋊3C8 C60.7C4 kernel C60.212D4 C2×C15⋊3C8 C22×C60 C2×C60 C22×C30 C2×C30 C22×C20 C60 C22×C12 C2×C20 C2×C20 C22×C10 C30 C2×C12 C2×C12 C22×C6 C20 C2×C10 C22×C4 C12 C2×C6 C10 C2×C4 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 2 8 1 2 2 1 1 1 2 2 2 2 4 4 4 8 8 4 4 4 4 8 16 16 16

Matrix representation of C60.212D4 in GL6(𝔽241)

 40 0 0 0 0 0 0 6 0 0 0 0 0 0 201 0 0 0 0 0 21 235 0 0 0 0 0 0 81 0 0 0 0 0 0 122
,
 0 1 0 0 0 0 177 0 0 0 0 0 0 0 4 236 0 0 0 0 16 237 0 0 0 0 0 0 0 74 0 0 0 0 127 0
,
 0 1 0 0 0 0 64 0 0 0 0 0 0 0 4 236 0 0 0 0 16 237 0 0 0 0 0 0 0 74 0 0 0 0 127 0

`G:=sub<GL(6,GF(241))| [40,0,0,0,0,0,0,6,0,0,0,0,0,0,201,21,0,0,0,0,0,235,0,0,0,0,0,0,81,0,0,0,0,0,0,122],[0,177,0,0,0,0,1,0,0,0,0,0,0,0,4,16,0,0,0,0,236,237,0,0,0,0,0,0,0,127,0,0,0,0,74,0],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,4,16,0,0,0,0,236,237,0,0,0,0,0,0,0,127,0,0,0,0,74,0] >;`

C60.212D4 in GAP, Magma, Sage, TeX

`C_{60}._{212}D_4`
`% in TeX`

`G:=Group("C60.212D4");`
`// GroupNames label`

`G:=SmallGroup(480,190);`
`// by ID`

`G=gap.SmallGroup(480,190);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,2693,18822]);`
`// 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^15*b^3>;`
`// generators/relations`

׿
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