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G = C6.D40order 480 = 25·3·5

5th non-split extension by C6 of D40 acting via D40/D20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C6.D40
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×D20 — C6.D40
 Lower central C15 — C30 — C60 — C6.D40
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.D40
G = < a,b,c | a6=b40=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 572 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×3], C23, D5 [×2], C10 [×3], Dic3, C12 [×2], C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], C30 [×3], D4⋊C4, C40, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, Dic15, C60 [×2], C6×D5 [×4], C2×C30, C4⋊Dic5, C2×C40, C2×D20, D4⋊Dic3, C5×C3⋊C8, C3×D20 [×2], C3×D20, C2×Dic15, C2×C60, D5×C2×C6, D205C4, C10×C3⋊C8, C605C4, C6×D20, C6.D40
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D8, SD16, D10, C2×Dic3, C3⋊D4 [×2], D4⋊C4, C4×D5, D20, C5⋊D4, D4⋊S3, D4.S3, C6.D4, S3×D5, C40⋊C2, D40, D10⋊C4, D4⋊Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D205C4, C3⋊D40, C6.D20, D10⋊Dic3, C6.D40

Smallest permutation representation of C6.D40
On 240 points
Generators in S240
```(1 232 63 117 148 165)(2 166 149 118 64 233)(3 234 65 119 150 167)(4 168 151 120 66 235)(5 236 67 81 152 169)(6 170 153 82 68 237)(7 238 69 83 154 171)(8 172 155 84 70 239)(9 240 71 85 156 173)(10 174 157 86 72 201)(11 202 73 87 158 175)(12 176 159 88 74 203)(13 204 75 89 160 177)(14 178 121 90 76 205)(15 206 77 91 122 179)(16 180 123 92 78 207)(17 208 79 93 124 181)(18 182 125 94 80 209)(19 210 41 95 126 183)(20 184 127 96 42 211)(21 212 43 97 128 185)(22 186 129 98 44 213)(23 214 45 99 130 187)(24 188 131 100 46 215)(25 216 47 101 132 189)(26 190 133 102 48 217)(27 218 49 103 134 191)(28 192 135 104 50 219)(29 220 51 105 136 193)(30 194 137 106 52 221)(31 222 53 107 138 195)(32 196 139 108 54 223)(33 224 55 109 140 197)(34 198 141 110 56 225)(35 226 57 111 142 199)(36 200 143 112 58 227)(37 228 59 113 144 161)(38 162 145 114 60 229)(39 230 61 115 146 163)(40 164 147 116 62 231)
(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)
(1 116 117 40)(2 39 118 115)(3 114 119 38)(4 37 120 113)(5 112 81 36)(6 35 82 111)(7 110 83 34)(8 33 84 109)(9 108 85 32)(10 31 86 107)(11 106 87 30)(12 29 88 105)(13 104 89 28)(14 27 90 103)(15 102 91 26)(16 25 92 101)(17 100 93 24)(18 23 94 99)(19 98 95 22)(20 21 96 97)(41 186 183 44)(42 43 184 185)(45 182 187 80)(46 79 188 181)(47 180 189 78)(48 77 190 179)(49 178 191 76)(50 75 192 177)(51 176 193 74)(52 73 194 175)(53 174 195 72)(54 71 196 173)(55 172 197 70)(56 69 198 171)(57 170 199 68)(58 67 200 169)(59 168 161 66)(60 65 162 167)(61 166 163 64)(62 63 164 165)(121 134 205 218)(122 217 206 133)(123 132 207 216)(124 215 208 131)(125 130 209 214)(126 213 210 129)(127 128 211 212)(135 160 219 204)(136 203 220 159)(137 158 221 202)(138 201 222 157)(139 156 223 240)(140 239 224 155)(141 154 225 238)(142 237 226 153)(143 152 227 236)(144 235 228 151)(145 150 229 234)(146 233 230 149)(147 148 231 232)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10F 12A 12B 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 20 20 2 2 2 60 60 2 2 2 2 2 20 20 20 20 6 6 6 6 2 ··· 2 4 4 4 4 2 ··· 2 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 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - + - - + + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 Dic3 D6 D8 SD16 D10 C3⋊D4 C3⋊D4 C4×D5 C5⋊D4 D20 C40⋊C2 D40 D4⋊S3 D4.S3 S3×D5 D5×Dic3 C15⋊D4 C3⋊D20 C3⋊D40 C6.D20 kernel C6.D40 C10×C3⋊C8 C60⋊5C4 C6×D20 C3×D20 C2×D20 C60 C2×C30 C2×C3⋊C8 D20 C2×C20 C30 C30 C2×C12 C20 C2×C10 C12 C12 C2×C6 C6 C6 C10 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 2 1 2 2 2 2 2 4 4 4 8 8 1 1 2 2 2 2 4 4

Matrix representation of C6.D40 in GL4(𝔽241) generated by

 0 1 0 0 240 1 0 0 0 0 1 0 0 0 0 1
,
 84 202 0 0 45 157 0 0 0 0 194 227 0 0 14 199
,
 84 202 0 0 45 157 0 0 0 0 194 227 0 0 20 47
`G:=sub<GL(4,GF(241))| [0,240,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[84,45,0,0,202,157,0,0,0,0,194,14,0,0,227,199],[84,45,0,0,202,157,0,0,0,0,194,20,0,0,227,47] >;`

C6.D40 in GAP, Magma, Sage, TeX

`C_6.D_{40}`
`% in TeX`

`G:=Group("C6.D40");`
`// GroupNames label`

`G:=SmallGroup(480,41);`
`// by ID`

`G=gap.SmallGroup(480,41);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,422,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^40=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;`
`// generators/relations`

׿
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