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## G = C60.D4order 480 = 25·3·5

### 106th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.D4
 Lower central C15 — C30 — C60 — C60.D4
 Upper central C1 — C4 — C2×C4

Generators and relations for C60.D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a29, cac-1=a49, cbc-1=b3 >

Subgroups: 188 in 60 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×4], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C3⋊C8 [×3], C24, C2×C12, C30, C30, C8.C4, C52C8 [×3], C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60 [×2], C2×C30, C2×C52C8, C4.Dic5, C5×M4(2), C12.53D4, C5×C3⋊C8, C3×C52C8, C153C8 [×2], C2×C60, C20.53D4, C3×C4.Dic5, C5×C4.Dic3, C2×C153C8, C60.D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D10, Dic6, C4×S3, C3⋊D4, C8.C4, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, S3×D5, C10.D4, C12.53D4, D30.C2, C15⋊D4, C15⋊Q8, C20.53D4, Dic155C4, C60.D4

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

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

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

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

60 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 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A ··· 30F 40A ··· 40H 60A ··· 60H 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 20 20 20 20 20 20 24 24 24 24 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 2 1 1 2 2 2 2 4 12 12 20 20 30 30 30 30 2 2 4 4 2 2 4 4 4 2 2 2 2 4 4 20 20 20 20 4 ··· 4 12 ··· 12 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + - + + + - - + + - - image C1 C2 C2 C2 C4 S3 D4 Q8 D5 D6 D10 C4×S3 C3⋊D4 Dic6 C8.C4 C4×D5 C5⋊D4 Dic10 S3×D5 C12.53D4 D30.C2 C15⋊D4 C15⋊Q8 C20.53D4 C60.D4 kernel C60.D4 C3×C4.Dic5 C5×C4.Dic3 C2×C15⋊3C8 C15⋊3C8 C4.Dic5 C60 C2×C30 C4.Dic3 C2×C20 C2×C12 C20 C20 C2×C10 C15 C12 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 4 4 4 2 2 2 2 2 4 8

Matrix representation of C60.D4 in GL6(𝔽241)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 0 51 0 0 0 0 189 52 0 0 0 0 0 0 1 240 0 0 0 0 1 0
,
 30 0 0 0 0 0 0 8 0 0 0 0 0 0 102 56 0 0 0 0 180 139 0 0 0 0 0 0 9 216 0 0 0 0 225 232
,
 0 211 0 0 0 0 30 0 0 0 0 0 0 0 143 62 0 0 0 0 20 98 0 0 0 0 0 0 99 43 0 0 0 0 198 142

`G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,189,0,0,0,0,51,52,0,0,0,0,0,0,1,1,0,0,0,0,240,0],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,102,180,0,0,0,0,56,139,0,0,0,0,0,0,9,225,0,0,0,0,216,232],[0,30,0,0,0,0,211,0,0,0,0,0,0,0,143,20,0,0,0,0,62,98,0,0,0,0,0,0,99,198,0,0,0,0,43,142] >;`

C60.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,68);`
`// by ID`

`G=gap.SmallGroup(480,68);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,100,675,346,80,1356,18822]);`
`// Polycyclic`

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

׿
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