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## G = C12.59D20order 480 = 25·3·5

### 13rd non-split extension by C12 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C12.59D20
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C12.59D20
 Lower central C15 — C30 — C60 — C12.59D20
 Upper central C1 — C4 — C2×C4

Generators and relations for C12.59D20
G = < a,b,c | a12=1, b20=a6, c2=a3, bab-1=cac-1=a5, cbc-1=b19 >

Subgroups: 188 in 60 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C3⋊C8, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C52C8, C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60, C2×C30, C4.Dic5, C4.Dic5, C2×C40, C12.53D4, C5×C3⋊C8, C3×C52C8, C153C8, C2×C60, C40.6C4, C3×C4.Dic5, C10×C3⋊C8, C60.7C4, C12.59D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, Dic5, D10, Dic6, C4×S3, C3⋊D4, C8.C4, Dic10, D20, C2×Dic5, Dic3⋊C4, S3×D5, C4⋊Dic5, C12.53D4, S3×Dic5, C3⋊D20, C15⋊Q8, C40.6C4, C6.Dic10, C12.59D20

Smallest permutation representation of C12.59D20
On 240 points
Generators in S240
```(1 198 44 11 168 54 21 178 64 31 188 74)(2 55 189 12 65 199 22 75 169 32 45 179)(3 200 46 13 170 56 23 180 66 33 190 76)(4 57 191 14 67 161 24 77 171 34 47 181)(5 162 48 15 172 58 25 182 68 35 192 78)(6 59 193 16 69 163 26 79 173 36 49 183)(7 164 50 17 174 60 27 184 70 37 194 80)(8 61 195 18 71 165 28 41 175 38 51 185)(9 166 52 19 176 62 29 186 72 39 196 42)(10 63 197 20 73 167 30 43 177 40 53 187)(81 210 143 111 240 133 101 230 123 91 220 153)(82 134 221 112 124 211 102 154 201 92 144 231)(83 212 145 113 202 135 103 232 125 93 222 155)(84 136 223 114 126 213 104 156 203 94 146 233)(85 214 147 115 204 137 105 234 127 95 224 157)(86 138 225 116 128 215 106 158 205 96 148 235)(87 216 149 117 206 139 107 236 129 97 226 159)(88 140 227 118 130 217 108 160 207 98 150 237)(89 218 151 119 208 141 109 238 131 99 228 121)(90 142 229 120 132 219 110 122 209 100 152 239)
(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)
(1 116 11 106 21 96 31 86)(2 95 12 85 22 115 32 105)(3 114 13 104 23 94 33 84)(4 93 14 83 24 113 34 103)(5 112 15 102 25 92 35 82)(6 91 16 81 26 111 36 101)(7 110 17 100 27 90 37 120)(8 89 18 119 28 109 38 99)(9 108 19 98 29 88 39 118)(10 87 20 117 30 107 40 97)(41 121 51 151 61 141 71 131)(42 140 52 130 62 160 72 150)(43 159 53 149 63 139 73 129)(44 138 54 128 64 158 74 148)(45 157 55 147 65 137 75 127)(46 136 56 126 66 156 76 146)(47 155 57 145 67 135 77 125)(48 134 58 124 68 154 78 144)(49 153 59 143 69 133 79 123)(50 132 60 122 70 152 80 142)(161 222 171 212 181 202 191 232)(162 201 172 231 182 221 192 211)(163 220 173 210 183 240 193 230)(164 239 174 229 184 219 194 209)(165 218 175 208 185 238 195 228)(166 237 176 227 186 217 196 207)(167 216 177 206 187 236 197 226)(168 235 178 225 188 215 198 205)(169 214 179 204 189 234 199 224)(170 233 180 223 190 213 200 203)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 12A 12B 12C 15A 15B 20A ··· 20H 24A 24B 24C 24D 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 3 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 15 15 20 ··· 20 24 24 24 24 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 2 1 1 2 2 2 2 4 6 6 6 6 20 20 60 60 2 ··· 2 2 2 4 4 4 2 ··· 2 20 20 20 20 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + + - + - + - + - + - image C1 C2 C2 C2 C4 S3 D4 Q8 D5 D6 Dic5 D10 C4×S3 C3⋊D4 Dic6 C8.C4 D20 Dic10 C40.6C4 S3×D5 C12.53D4 S3×Dic5 C3⋊D20 C15⋊Q8 C12.59D20 kernel C12.59D20 C3×C4.Dic5 C10×C3⋊C8 C60.7C4 C5×C3⋊C8 C4.Dic5 C60 C2×C30 C2×C3⋊C8 C2×C20 C3⋊C8 C2×C12 C20 C20 C2×C10 C15 C12 C2×C6 C3 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 4 1 1 1 2 1 4 2 2 2 2 4 4 4 16 2 2 2 2 2 8

Matrix representation of C12.59D20 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 85 41 0 0 0 0 200 122 0 0 0 0 0 0 211 0 0 0 0 0 229 233 0 0 0 0 0 0 202 37 0 0 0 0 239 39
,
 44 225 0 0 0 0 106 197 0 0 0 0 0 0 148 50 0 0 0 0 33 93 0 0 0 0 0 0 39 204 0 0 0 0 2 202

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[85,200,0,0,0,0,41,122,0,0,0,0,0,0,211,229,0,0,0,0,0,233,0,0,0,0,0,0,202,239,0,0,0,0,37,39],[44,106,0,0,0,0,225,197,0,0,0,0,0,0,148,33,0,0,0,0,50,93,0,0,0,0,0,0,39,2,0,0,0,0,204,202] >;`

C12.59D20 in GAP, Magma, Sage, TeX

`C_{12}._{59}D_{20}`
`% in TeX`

`G:=Group("C12.59D20");`
`// GroupNames label`

`G:=SmallGroup(480,69);`
`// by ID`

`G=gap.SmallGroup(480,69);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^3,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^19>;`
`// generators/relations`

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