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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, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C30, C30, C22⋊C8, C52C8, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C52C8, C22×C20, C12.55D4, C153C8, C2×C60, C2×C60, C22×C30, C20.55D4, C2×C153C8, C22×C60, C60.212D4
Quotients:

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

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

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

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

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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