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## G = C10.D24order 480 = 25·3·5

### 5th non-split extension by C10 of D24 acting via D24/D12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C10.D24
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×C5⋊2C8 — C10.D24
 Lower central C15 — C30 — C60 — C10.D24
 Upper central C1 — C22 — C2×C4

Generators and relations for C10.D24
G = < a,b,c | a10=b24=1, c2=a5, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 476 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C24, D12 [×2], D12, C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], D4⋊C4, C52C8, C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C4⋊Dic3, C2×C24, C2×D12, Dic15, C60 [×2], S3×C10 [×4], C2×C30, C2×C52C8, C4⋊Dic5, D4×C10, C2.D24, C3×C52C8, C5×D12 [×2], C5×D12, C2×Dic15, C2×C60, S3×C2×C10, D4⋊Dic5, C6×C52C8, C605C4, C10×D12, C10.D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, D4⋊C4, C2×Dic5, C5⋊D4 [×2], C24⋊C2, D24, D6⋊C4, S3×D5, D4⋊D5, D4.D5, C23.D5, C2.D24, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊Dic5, C5⋊D24, D12.D5, D6⋊Dic5, C10.D24

Smallest permutation representation of C10.D24
On 240 points
Generators in S240
```(1 224 96 182 100 46 124 164 68 193)(2 194 69 165 125 47 101 183 73 225)(3 226 74 184 102 48 126 166 70 195)(4 196 71 167 127 25 103 185 75 227)(5 228 76 186 104 26 128 168 72 197)(6 198 49 145 129 27 105 187 77 229)(7 230 78 188 106 28 130 146 50 199)(8 200 51 147 131 29 107 189 79 231)(9 232 80 190 108 30 132 148 52 201)(10 202 53 149 133 31 109 191 81 233)(11 234 82 192 110 32 134 150 54 203)(12 204 55 151 135 33 111 169 83 235)(13 236 84 170 112 34 136 152 56 205)(14 206 57 153 137 35 113 171 85 237)(15 238 86 172 114 36 138 154 58 207)(16 208 59 155 139 37 115 173 87 239)(17 240 88 174 116 38 140 156 60 209)(18 210 61 157 141 39 117 175 89 217)(19 218 90 176 118 40 142 158 62 211)(20 212 63 159 143 41 119 177 91 219)(21 220 92 178 120 42 144 160 64 213)(22 214 65 161 121 43 97 179 93 221)(23 222 94 180 98 44 122 162 66 215)(24 216 67 163 123 45 99 181 95 223)
(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 24 46 45)(2 44 47 23)(3 22 48 43)(4 42 25 21)(5 20 26 41)(6 40 27 19)(7 18 28 39)(8 38 29 17)(9 16 30 37)(10 36 31 15)(11 14 32 35)(12 34 33 13)(49 176 187 62)(50 61 188 175)(51 174 189 60)(52 59 190 173)(53 172 191 58)(54 57 192 171)(55 170 169 56)(63 186 177 72)(64 71 178 185)(65 184 179 70)(66 69 180 183)(67 182 181 68)(73 162 165 94)(74 93 166 161)(75 160 167 92)(76 91 168 159)(77 158 145 90)(78 89 146 157)(79 156 147 88)(80 87 148 155)(81 154 149 86)(82 85 150 153)(83 152 151 84)(95 164 163 96)(97 195 214 102)(98 101 215 194)(99 193 216 100)(103 213 196 120)(104 119 197 212)(105 211 198 118)(106 117 199 210)(107 209 200 116)(108 115 201 208)(109 207 202 114)(110 113 203 206)(111 205 204 112)(121 226 221 126)(122 125 222 225)(123 224 223 124)(127 220 227 144)(128 143 228 219)(129 218 229 142)(130 141 230 217)(131 240 231 140)(132 139 232 239)(133 238 233 138)(134 137 234 237)(135 236 235 136)```

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

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

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

66 conjugacy classes

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

66 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 D6 D8 SD16 Dic5 D10 C4×S3 C3⋊D4 D12 C5⋊D4 C5⋊D4 C24⋊C2 D24 S3×D5 D4⋊D5 D4.D5 S3×Dic5 C15⋊D4 C5⋊D12 C5⋊D24 D12.D5 kernel C10.D24 C6×C5⋊2C8 C60⋊5C4 C10×D12 C5×D12 C2×C5⋊2C8 C60 C2×C30 C2×D12 C2×C20 C30 C30 D12 C2×C12 C20 C20 C2×C10 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 4 2 2 2 2 4 4 4 4 2 2 2 2 2 2 4 4

Matrix representation of C10.D24 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 190 189 0 0 51 0
,
 127 105 0 0 136 232 0 0 0 0 182 181 0 0 50 59
,
 136 232 0 0 127 105 0 0 0 0 59 60 0 0 191 182
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,190,51,0,0,189,0],[127,136,0,0,105,232,0,0,0,0,182,50,0,0,181,59],[136,127,0,0,232,105,0,0,0,0,59,191,0,0,60,182] >;`

C10.D24 in GAP, Magma, Sage, TeX

`C_{10}.D_{24}`
`% in TeX`

`G:=Group("C10.D24");`
`// GroupNames label`

`G:=SmallGroup(480,43);`
`// by ID`

`G=gap.SmallGroup(480,43);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^24=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

׿
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