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## G = C60.54D4order 480 = 25·3·5

### 54th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.54D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.54D4
 Lower central C15 — C30 — C2×C30 — C60.54D4
 Upper central C1 — C2 — C2×C4

Generators and relations for C60.54D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a49, cbc-1=a45b3 >

Subgroups: 252 in 76 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, M4(2), C2×Q8, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C30, C30, C4.10D4, C52C8, C2×C20, C2×C20, C5×Q8, C4.Dic3, C3×M4(2), C2×Dic6, C5×Dic3, C60, C2×C30, C4.Dic5, C4.Dic5, Q8×C10, C12.47D4, C3×C52C8, C153C8, C5×Dic6, C10×Dic3, C2×C60, C20.10D4, C3×C4.Dic5, C60.7C4, C10×Dic6, C60.54D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, Dic5, D10, C4×S3, D12, C3⋊D4, C4.10D4, C2×Dic5, C5⋊D4, D6⋊C4, S3×D5, C23.D5, C12.47D4, S3×Dic5, C15⋊D4, C5⋊D12, C20.10D4, D6⋊Dic5, C60.54D4

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

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + + - + - - + - image C1 C2 C2 C2 C4 S3 D4 D5 D6 Dic5 D10 D12 C3⋊D4 C4×S3 C5⋊D4 C4.10D4 S3×D5 C12.47D4 C15⋊D4 C5⋊D12 S3×Dic5 C20.10D4 C60.54D4 kernel C60.54D4 C3×C4.Dic5 C60.7C4 C10×Dic6 C10×Dic3 C4.Dic5 C60 C2×Dic6 C2×C20 C2×Dic3 C2×C12 C20 C20 C2×C10 C12 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 2 2 1 4 2 2 2 2 8 1 2 2 2 2 2 4 8

Matrix representation of C60.54D4 in GL6(𝔽241)

 226 0 0 0 0 0 210 16 0 0 0 0 0 0 87 26 0 0 0 0 11 154 0 0 0 0 95 24 0 205 0 0 10 24 36 0
,
 168 73 0 0 0 0 135 73 0 0 0 0 0 0 33 0 179 207 0 0 176 0 1 210 0 0 92 15 54 154 0 0 162 16 54 154
,
 177 0 0 0 0 0 113 64 0 0 0 0 0 0 141 0 28 0 0 0 0 0 240 1 0 0 13 1 100 0 0 0 99 0 100 0

`G:=sub<GL(6,GF(241))| [226,210,0,0,0,0,0,16,0,0,0,0,0,0,87,11,95,10,0,0,26,154,24,24,0,0,0,0,0,36,0,0,0,0,205,0],[168,135,0,0,0,0,73,73,0,0,0,0,0,0,33,176,92,162,0,0,0,0,15,16,0,0,179,1,54,54,0,0,207,210,154,154],[177,113,0,0,0,0,0,64,0,0,0,0,0,0,141,0,13,99,0,0,0,0,1,0,0,0,28,240,100,100,0,0,0,1,0,0] >;`

C60.54D4 in GAP, Magma, Sage, TeX

`C_{60}._{54}D_4`
`% in TeX`

`G:=Group("C60.54D4");`
`// GroupNames label`

`G:=SmallGroup(480,38);`
`// by ID`

`G=gap.SmallGroup(480,38);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,1356,18822]);`
`// Polycyclic`

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

׿
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