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## G = C40.54D6order 480 = 25·3·5

### 15th non-split extension by C40 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C40.54D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D6.D10 — C40.54D6
 Lower central C15 — C30 — C40.54D6
 Upper central C1 — C8

Generators and relations for C40.54D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=cac-1=a9, cbc-1=a20b-1 >

Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4 [×2], C2×C20, S3×C8, S3×C8, C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C8×D5, C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, C8○D12, C5×C3⋊C8, C3×C52C8, C153C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.3C4, C20.32D6, D6.Dic5, D30.5C4, D5×C24, S3×C40, C8×D15, D6.D10, C40.54D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C8○D12, C2×S3×D5, D20.3C4, C4×S3×D5, C40.54D6

Smallest permutation representation of C40.54D6
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)
(1 222 70)(2 231 71 10 223 79)(3 240 72 19 224 48)(4 209 73 28 225 57)(5 218 74 37 226 66)(6 227 75)(7 236 76 15 228 44)(8 205 77 24 229 53)(9 214 78 33 230 62)(11 232 80)(12 201 41 20 233 49)(13 210 42 29 234 58)(14 219 43 38 235 67)(16 237 45)(17 206 46 25 238 54)(18 215 47 34 239 63)(21 202 50)(22 211 51 30 203 59)(23 220 52 39 204 68)(26 207 55)(27 216 56 35 208 64)(31 212 60)(32 221 61 40 213 69)(36 217 65)(81 136 190 117 140 186)(82 145 191 86 141 195)(83 154 192 95 142 164)(84 123 193 104 143 173)(85 132 194 113 144 182)(87 150 196 91 146 200)(88 159 197 100 147 169)(89 128 198 109 148 178)(90 137 199 118 149 187)(92 155 161 96 151 165)(93 124 162 105 152 174)(94 133 163 114 153 183)(97 160 166 101 156 170)(98 129 167 110 157 179)(99 138 168 119 158 188)(102 125 171 106 121 175)(103 134 172 115 122 184)(107 130 176 111 126 180)(108 139 177 120 127 189)(112 135 181 116 131 185)
(1 173 21 193)(2 182 22 162)(3 191 23 171)(4 200 24 180)(5 169 25 189)(6 178 26 198)(7 187 27 167)(8 196 28 176)(9 165 29 185)(10 174 30 194)(11 183 31 163)(12 192 32 172)(13 161 33 181)(14 170 34 190)(15 179 35 199)(16 188 36 168)(17 197 37 177)(18 166 38 186)(19 175 39 195)(20 184 40 164)(41 83 61 103)(42 92 62 112)(43 101 63 81)(44 110 64 90)(45 119 65 99)(46 88 66 108)(47 97 67 117)(48 106 68 86)(49 115 69 95)(50 84 70 104)(51 93 71 113)(52 102 72 82)(53 111 73 91)(54 120 74 100)(55 89 75 109)(56 98 76 118)(57 107 77 87)(58 116 78 96)(59 85 79 105)(60 94 80 114)(121 224 141 204)(122 233 142 213)(123 202 143 222)(124 211 144 231)(125 220 145 240)(126 229 146 209)(127 238 147 218)(128 207 148 227)(129 216 149 236)(130 225 150 205)(131 234 151 214)(132 203 152 223)(133 212 153 232)(134 221 154 201)(135 230 155 210)(136 239 156 219)(137 208 157 228)(138 217 158 237)(139 226 159 206)(140 235 160 215)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 12 12 12 12 15 15 20 20 20 20 20 20 20 20 24 24 24 24 24 24 24 24 30 30 40 ··· 40 40 ··· 40 60 60 60 60 120 ··· 120 size 1 1 6 10 30 2 1 1 6 10 30 2 2 2 10 10 1 1 1 1 6 6 10 10 30 30 2 2 6 6 6 6 2 2 10 10 4 4 2 2 2 2 6 6 6 6 2 2 2 2 10 10 10 10 4 4 2 ··· 2 6 ··· 6 4 4 4 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×S3 C8○D4 C4×D5 C4×D5 C8○D12 D20.3C4 S3×D5 C2×S3×D5 C4×S3×D5 C40.54D6 kernel C40.54D6 C20.32D6 D6.Dic5 D30.5C4 D5×C24 S3×C40 C8×D15 D6.D10 C15⋊D4 C3⋊D20 C5⋊D12 C15⋊Q8 C8×D5 S3×C8 C5⋊2C8 C40 C4×D5 C3⋊C8 C24 C4×S3 Dic5 D10 C15 Dic3 D6 C5 C3 C8 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 2 2 2 2 2 4 4 4 8 16 2 2 4 8

Matrix representation of C40.54D6 in GL4(𝔽241) generated by

 30 0 0 0 0 30 0 0 0 0 0 84 0 0 127 157
,
 225 233 0 0 0 226 0 0 0 0 51 51 0 0 1 190
,
 15 233 0 0 209 226 0 0 0 0 69 20 0 0 27 172
`G:=sub<GL(4,GF(241))| [30,0,0,0,0,30,0,0,0,0,0,127,0,0,84,157],[225,0,0,0,233,226,0,0,0,0,51,1,0,0,51,190],[15,209,0,0,233,226,0,0,0,0,69,27,0,0,20,172] >;`

C40.54D6 in GAP, Magma, Sage, TeX

`C_{40}._{54}D_6`
`% in TeX`

`G:=Group("C40.54D6");`
`// GroupNames label`

`G:=SmallGroup(480,341);`
`// by ID`

`G=gap.SmallGroup(480,341);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
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