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## G = C40.35D6order 480 = 25·3·5

### 35th non-split extension by C40 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C40.35D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D6.D10 — C40.35D6
 Lower central C15 — C30 — C40.35D6
 Upper central C1 — C4 — C8

Generators and relations for C40.35D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, 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 [×2], C8⋊S3, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5 [×2], C8⋊D5, C8⋊D5, C2×C52C8, C5×M4(2), C4○D20, D12.C4, C5×C3⋊C8, C3×C52C8, C153C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.2C4, D5×C3⋊C8, S3×C52C8, D30.5C4, C3×C8⋊D5, C5×C8⋊S3, C8×D15, D6.D10, C40.35D6
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, D12.C4, C2×S3×D5, D20.2C4, C4×S3×D5, C40.35D6

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

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

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

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

66 conjugacy classes

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

66 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 4 4 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 S3×D5 D12.C4 C2×S3×D5 D20.2C4 C4×S3×D5 C40.35D6 kernel C40.35D6 D5×C3⋊C8 S3×C5⋊2C8 D30.5C4 C3×C8⋊D5 C5×C8⋊S3 C8×D15 D6.D10 C15⋊D4 C3⋊D20 C5⋊D12 C15⋊Q8 C8⋊D5 C8⋊S3 C5⋊2C8 C40 C4×D5 C3⋊C8 C24 C4×S3 Dic5 D10 C15 Dic3 D6 C8 C5 C4 C3 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 2 2 2 4 4 8

Matrix representation of C40.35D6 in GL6(𝔽241)

 64 54 0 0 0 0 86 177 0 0 0 0 0 0 0 51 0 0 0 0 189 189 0 0 0 0 0 0 177 0 0 0 0 0 0 177
,
 1 0 0 0 0 0 194 240 0 0 0 0 0 0 189 190 0 0 0 0 53 52 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 30 191 0 0 0 0 206 211 0 0 0 0 0 0 189 190 0 0 0 0 53 52 0 0 0 0 0 0 89 81 0 0 0 0 170 152

`G:=sub<GL(6,GF(241))| [64,86,0,0,0,0,54,177,0,0,0,0,0,0,0,189,0,0,0,0,51,189,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,194,0,0,0,0,0,240,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[30,206,0,0,0,0,191,211,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,89,170,0,0,0,0,81,152] >;`

C40.35D6 in GAP, Magma, Sage, TeX

`C_{40}._{35}D_6`
`% in TeX`

`G:=Group("C40.35D6");`
`// GroupNames label`

`G:=SmallGroup(480,344);`
`// by ID`

`G=gap.SmallGroup(480,344);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,219,58,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
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