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## G = C60.31D4order 480 = 25·3·5

### 31st non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.31D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.31D4
 Lower central C15 — C30 — C2×C30 — C60.31D4
 Upper central C1 — C2 — C2×C4

Generators and relations for C60.31D4
G = < a,b,c | a60=1, b4=c2=a30, bab-1=a19, cac-1=a-1, cbc-1=a45b3 >

Subgroups: 380 in 76 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C5, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C10, C10, Dic3 [×2], C12 [×2], C2×C6, C15, M4(2) [×2], C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C30, C30, C4.10D4, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20, C4.Dic3, C3×M4(2), C2×Dic6, Dic15 [×2], C60 [×2], C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.47D4, C5×C3⋊C8, C3×C52C8, Dic30 [×2], C2×Dic15 [×2], C2×C60, C4.12D20, C3×C4.Dic5, C5×C4.Dic3, C2×Dic30, C60.31D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4.10D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C12.47D4, D30.C2, C3⋊D20, C5⋊D12, C4.12D20, D304C4, C60.31D4

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 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 4 5 5 6 6 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 2 2 60 60 2 2 2 4 12 12 20 20 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

57 irreducible representations

 dim 1 1 1 1 1 2 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 D10 D12 C3⋊D4 C4×S3 D20 C5⋊D4 C4×D5 C4.10D4 S3×D5 C12.47D4 C3⋊D20 C5⋊D12 D30.C2 C4.12D20 C60.31D4 kernel C60.31D4 C3×C4.Dic5 C5×C4.Dic3 C2×Dic30 C2×Dic15 C4.Dic5 C60 C4.Dic3 C2×C20 C2×C12 C20 C20 C2×C10 C12 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 2 2 1 2 2 2 2 4 4 4 1 2 2 2 2 2 4 8

Matrix representation of C60.31D4 in GL8(𝔽241)

 0 0 0 189 0 0 0 0 0 0 51 190 0 0 0 0 0 52 0 52 0 0 0 0 190 51 190 51 0 0 0 0 0 0 0 0 125 210 0 0 0 0 0 0 115 172 0 0 0 0 0 0 214 195 142 99 0 0 0 0 100 53 142 43
,
 149 99 34 78 0 0 0 0 132 92 184 207 0 0 0 0 207 163 115 21 0 0 0 0 57 34 189 126 0 0 0 0 0 0 0 0 47 0 146 86 0 0 0 0 195 0 151 186 0 0 0 0 128 174 170 11 0 0 0 0 49 36 108 24
,
 213 53 229 126 0 0 0 0 196 28 84 12 0 0 0 0 16 73 28 188 0 0 0 0 129 225 45 213 0 0 0 0 0 0 0 0 84 150 0 0 0 0 0 0 157 157 0 0 0 0 0 0 178 36 36 67 0 0 0 0 122 210 31 205

`G:=sub<GL(8,GF(241))| [0,0,0,190,0,0,0,0,0,0,52,51,0,0,0,0,0,51,0,190,0,0,0,0,189,190,52,51,0,0,0,0,0,0,0,0,125,115,214,100,0,0,0,0,210,172,195,53,0,0,0,0,0,0,142,142,0,0,0,0,0,0,99,43],[149,132,207,57,0,0,0,0,99,92,163,34,0,0,0,0,34,184,115,189,0,0,0,0,78,207,21,126,0,0,0,0,0,0,0,0,47,195,128,49,0,0,0,0,0,0,174,36,0,0,0,0,146,151,170,108,0,0,0,0,86,186,11,24],[213,196,16,129,0,0,0,0,53,28,73,225,0,0,0,0,229,84,28,45,0,0,0,0,126,12,188,213,0,0,0,0,0,0,0,0,84,157,178,122,0,0,0,0,150,157,36,210,0,0,0,0,0,0,36,31,0,0,0,0,0,0,67,205] >;`

C60.31D4 in GAP, Magma, Sage, TeX

`C_{60}._{31}D_4`
`% in TeX`

`G:=Group("C60.31D4");`
`// GroupNames label`

`G:=SmallGroup(480,39);`
`// by ID`

`G=gap.SmallGroup(480,39);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,204,219,100,675,346,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=1,b^4=c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^45*b^3>;`
`// generators/relations`

׿
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