Copied to
clipboard

## G = C24.2D10order 480 = 25·3·5

### 2nd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C24.2D10
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×Dic6 — C24.2D10
 Lower central C15 — C30 — C60 — C24.2D10
 Upper central C1 — C2 — C4 — C8

Generators and relations for C24.2D10
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, cac-1=a19, cbc-1=b-1 >

Subgroups: 732 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5 [×2], C10, Dic3 [×3], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5, C20, C20 [×2], D10, D10, C24, C24, Dic6 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D5, D15, C30, C8.C22, C52C8, C40, Dic10 [×2], C4×D5, C4×D5 [×2], D20 [×2], C5×Q8 [×2], C24⋊C2 [×2], Dic12, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, C8.D6, C3×C52C8, C120, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, D5×C12, C5×Dic6 [×2], Dic30, D60, Q16⋊D5, Dic6⋊D5, C5⋊Dic12, C3×C8⋊D5, C5×Dic12, C24⋊D5, D5×Dic6, C12.28D10, C24.2D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8.C22, C22×D5, C2×D12, S3×D5, D4×D5, C8.D6, C2×S3×D5, Q16⋊D5, D5×D12, C24.2D10

Smallest permutation representation of C24.2D10
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)
(1 107 47)(2 96 48 30 108 76)(3 85 49 19 109 65)(4 114 50 8 110 54)(5 103 51 37 111 43)(6 92 52 26 112 72)(7 81 53 15 113 61)(9 99 55 33 115 79)(10 88 56 22 116 68)(11 117 57)(12 106 58 40 118 46)(13 95 59 29 119 75)(14 84 60 18 120 64)(16 102 62 36 82 42)(17 91 63 25 83 71)(20 98 66 32 86 78)(21 87 67)(23 105 69 39 89 45)(24 94 70 28 90 74)(27 101 73 35 93 41)(31 97 77)(34 104 80 38 100 44)(121 161 217 157 165 213)(122 190 218 146 166 202)(123 179 219 135 167 231)(124 168 220)(125 197 221 153 169 209)(126 186 222 142 170 238)(127 175 223 131 171 227)(128 164 224 160 172 216)(129 193 225 149 173 205)(130 182 226 138 174 234)(132 200 228 156 176 212)(133 189 229 145 177 201)(134 178 230)(136 196 232 152 180 208)(137 185 233 141 181 237)(139 163 235 159 183 215)(140 192 236 148 184 204)(143 199 239 155 187 211)(144 188 240)(147 195 203 151 191 207)(150 162 206 158 194 214)(154 198 210)
(1 124 21 144)(2 143 22 123)(3 122 23 142)(4 141 24 121)(5 160 25 140)(6 139 26 159)(7 158 27 138)(8 137 28 157)(9 156 29 136)(10 135 30 155)(11 154 31 134)(12 133 32 153)(13 152 33 132)(14 131 34 151)(15 150 35 130)(16 129 36 149)(17 148 37 128)(18 127 38 147)(19 146 39 126)(20 125 40 145)(41 174 61 194)(42 193 62 173)(43 172 63 192)(44 191 64 171)(45 170 65 190)(46 189 66 169)(47 168 67 188)(48 187 68 167)(49 166 69 186)(50 185 70 165)(51 164 71 184)(52 183 72 163)(53 162 73 182)(54 181 74 161)(55 200 75 180)(56 179 76 199)(57 198 77 178)(58 177 78 197)(59 196 79 176)(60 175 80 195)(81 206 101 226)(82 225 102 205)(83 204 103 224)(84 223 104 203)(85 202 105 222)(86 221 106 201)(87 240 107 220)(88 219 108 239)(89 238 109 218)(90 217 110 237)(91 236 111 216)(92 215 112 235)(93 234 113 214)(94 213 114 233)(95 232 115 212)(96 211 116 231)(97 230 117 210)(98 209 118 229)(99 228 119 208)(100 207 120 227)

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D12 D12 C8.C22 S3×D5 D4×D5 C8.D6 C2×S3×D5 Q16⋊D5 D5×D12 C24.2D10 kernel C24.2D10 Dic6⋊D5 C5⋊Dic12 C3×C8⋊D5 C5×Dic12 C24⋊D5 D5×Dic6 C12.28D10 C8⋊D5 C3×Dic5 C6×D5 Dic12 C5⋊2C8 C40 C4×D5 C24 Dic6 Dic5 D10 C15 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 4 2 2 1 2 2 2 2 4 4 8

Matrix representation of C24.2D10 in GL8(𝔽241)

 0 12 0 24 0 0 0 0 87 87 174 174 0 0 0 0 0 217 0 229 0 0 0 0 67 67 154 154 0 0 0 0 0 0 0 0 142 98 5 0 0 0 0 0 143 44 0 5 0 0 0 0 125 9 99 143 0 0 0 0 232 116 98 197
,
 52 51 52 51 0 0 0 0 188 189 188 189 0 0 0 0 189 190 0 0 0 0 0 0 53 52 0 0 0 0 0 0 0 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 114 105 1 1 0 0 0 0 136 9 240 0
,
 77 6 192 40 0 0 0 0 93 164 138 49 0 0 0 0 115 34 164 235 0 0 0 0 45 126 148 77 0 0 0 0 0 0 0 0 148 95 0 0 0 0 0 0 188 93 0 0 0 0 0 0 130 20 180 137 0 0 0 0 131 111 198 61

G:=sub<GL(8,GF(241))| [0,87,0,67,0,0,0,0,12,87,217,67,0,0,0,0,0,174,0,154,0,0,0,0,24,174,229,154,0,0,0,0,0,0,0,0,142,143,125,232,0,0,0,0,98,44,9,116,0,0,0,0,5,0,99,98,0,0,0,0,0,5,143,197],[52,188,189,53,0,0,0,0,51,189,190,52,0,0,0,0,52,188,0,0,0,0,0,0,51,189,0,0,0,0,0,0,0,0,0,0,240,1,114,136,0,0,0,0,240,0,105,9,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0],[77,93,115,45,0,0,0,0,6,164,34,126,0,0,0,0,192,138,164,148,0,0,0,0,40,49,235,77,0,0,0,0,0,0,0,0,148,188,130,131,0,0,0,0,95,93,20,111,0,0,0,0,0,0,180,198,0,0,0,0,0,0,137,61] >;

C24.2D10 in GAP, Magma, Sage, TeX

C_{24}._2D_{10}
% in TeX

G:=Group("C24.2D10");
// GroupNames label

G:=SmallGroup(480,337);
// by ID

G=gap.SmallGroup(480,337);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,142,346,80,1356,18822]);
// 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^19,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽