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## G = C40.31D6order 480 = 25·3·5

### 31st non-split extension by C40 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C40.31D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D12⋊5D5 — C40.31D6
 Lower central C15 — C30 — C60 — C40.31D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C40.31D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a9, cac-1=a19, cbc-1=a20b-1 >

Subgroups: 764 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, D15, C30, C4○D8, C52C8, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C24⋊C2, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, C4○D24, C3×C52C8, C120, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, D5×C12, C5×Dic6, C5×D12, Dic30, D60, SD163D5, C5⋊D24, C5⋊Dic12, D5×C24, C5×C24⋊C2, C24⋊D5, D125D5, C12.28D10, C40.31D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C4○D8, C22×D5, C2×D12, S3×D5, D4×D5, C4○D24, C2×S3×D5, SD163D5, D5×D12, C40.31D6

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 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 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 D12 D12 C4○D8 C4○D24 S3×D5 D4×D5 C2×S3×D5 SD16⋊3D5 D5×D12 C40.31D6 kernel C40.31D6 C5⋊D24 C5⋊Dic12 D5×C24 C5×C24⋊C2 C24⋊D5 D12⋊5D5 C12.28D10 C8×D5 C3×Dic5 C6×D5 C24⋊C2 C5⋊2C8 C40 C4×D5 C24 Dic6 D12 Dic5 D10 C15 C5 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 4 8 2 2 2 4 4 8

Matrix representation of C40.31D6 in GL6(𝔽241)

 56 62 0 0 0 0 101 185 0 0 0 0 0 0 233 0 0 0 0 0 61 211 0 0 0 0 0 0 190 52 0 0 0 0 190 0
,
 1 49 0 0 0 0 177 239 0 0 0 0 0 0 1 0 0 0 0 0 137 240 0 0 0 0 0 0 51 1 0 0 0 0 51 190
,
 62 135 0 0 0 0 84 179 0 0 0 0 0 0 137 239 0 0 0 0 227 104 0 0 0 0 0 0 190 240 0 0 0 0 190 51

`G:=sub<GL(6,GF(241))| [56,101,0,0,0,0,62,185,0,0,0,0,0,0,233,61,0,0,0,0,0,211,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[1,177,0,0,0,0,49,239,0,0,0,0,0,0,1,137,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,1,190],[62,84,0,0,0,0,135,179,0,0,0,0,0,0,137,227,0,0,0,0,239,104,0,0,0,0,0,0,190,190,0,0,0,0,240,51] >;`

C40.31D6 in GAP, Magma, Sage, TeX

`C_{40}._{31}D_6`
`% in TeX`

`G:=Group("C40.31D6");`
`// GroupNames label`

`G:=SmallGroup(480,345);`
`// by ID`

`G=gap.SmallGroup(480,345);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,58,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^19,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
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