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## G = C15⋊2- 1+4order 480 = 25·3·5

### The semidirect product of C15 and 2- 1+4 acting via 2- 1+4/D4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C15⋊2- 1+4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D5×Dic6 — C15⋊2- 1+4
 Lower central C15 — C30 — C15⋊2- 1+4
 Upper central C1 — C2 — D4

Generators and relations for C15⋊2- 1+4
G = < a,b,c,d,e | a15=b4=c2=1, d2=e2=b2, bab-1=dad-1=a4, ac=ca, eae-1=a11, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1308 in 292 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×9], C22 [×2], C22 [×3], C5, S3 [×2], C6, C6 [×3], C2×C4 [×15], D4, D4 [×9], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3, Dic3 [×2], Dic3 [×3], C12, C12 [×3], D6, D6, C2×C6 [×2], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5, Dic5 [×2], Dic5 [×3], C20, C20 [×3], D10, D10, C2×C10 [×2], C2×C10, Dic6, Dic6 [×8], C4×S3, C4×S3 [×5], D12, C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C5×S3, C3×D5, D15, C30, C30 [×2], 2- 1+4, Dic10, Dic10 [×8], C4×D5, C4×D5 [×5], D20, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C2×Dic6 [×3], C4○D12 [×3], D42S3, D42S3 [×5], S3×Q8 [×2], C3×C4○D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, C3×Dic5 [×2], Dic15, Dic15 [×2], C60, C6×D5, S3×C10, D30, C2×C30 [×2], C2×Dic10 [×3], C4○D20 [×3], D42D5, D42D5 [×5], Q8×D5 [×2], C5×C4○D4, Q8○D12, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C15⋊Q8 [×6], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], Dic30, C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15, D4.10D10, D5×Dic6, S3×Dic10, D15⋊Q8, D6.D10, Dic5.D6 [×2], C30.C23 [×2], Dic3.D10 [×2], C2×C15⋊Q8 [×2], C3×D42D5, C5×D42S3, D42D15, C15⋊2- 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, C15⋊2- 1+4

Smallest permutation representation of C15⋊2- 1+4
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)
(1 132 16 136)(2 121 17 140)(3 125 18 144)(4 129 19 148)(5 133 20 137)(6 122 21 141)(7 126 22 145)(8 130 23 149)(9 134 24 138)(10 123 25 142)(11 127 26 146)(12 131 27 150)(13 135 28 139)(14 124 29 143)(15 128 30 147)(31 176 54 161)(32 180 55 165)(33 169 56 154)(34 173 57 158)(35 177 58 162)(36 166 59 151)(37 170 60 155)(38 174 46 159)(39 178 47 163)(40 167 48 152)(41 171 49 156)(42 175 50 160)(43 179 51 164)(44 168 52 153)(45 172 53 157)(61 185 82 210)(62 189 83 199)(63 193 84 203)(64 182 85 207)(65 186 86 196)(66 190 87 200)(67 194 88 204)(68 183 89 208)(69 187 90 197)(70 191 76 201)(71 195 77 205)(72 184 78 209)(73 188 79 198)(74 192 80 202)(75 181 81 206)(91 232 117 221)(92 236 118 225)(93 240 119 214)(94 229 120 218)(95 233 106 222)(96 237 107 211)(97 226 108 215)(98 230 109 219)(99 234 110 223)(100 238 111 212)(101 227 112 216)(102 231 113 220)(103 235 114 224)(104 239 115 213)(105 228 116 217)
(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)(91 117)(92 118)(93 119)(94 120)(95 106)(96 107)(97 108)(98 109)(99 110)(100 111)(101 112)(102 113)(103 114)(104 115)(105 116)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 136)(133 137)(134 138)(135 139)(181 206)(182 207)(183 208)(184 209)(185 210)(186 196)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)(193 203)(194 204)(195 205)
(1 177 16 162)(2 166 17 151)(3 170 18 155)(4 174 19 159)(5 178 20 163)(6 167 21 152)(7 171 22 156)(8 175 23 160)(9 179 24 164)(10 168 25 153)(11 172 26 157)(12 176 27 161)(13 180 28 165)(14 169 29 154)(15 173 30 158)(31 131 54 150)(32 135 55 139)(33 124 56 143)(34 128 57 147)(35 132 58 136)(36 121 59 140)(37 125 60 144)(38 129 46 148)(39 133 47 137)(40 122 48 141)(41 126 49 145)(42 130 50 149)(43 134 51 138)(44 123 52 142)(45 127 53 146)(61 212 82 238)(62 216 83 227)(63 220 84 231)(64 224 85 235)(65 213 86 239)(66 217 87 228)(67 221 88 232)(68 225 89 236)(69 214 90 240)(70 218 76 229)(71 222 77 233)(72 211 78 237)(73 215 79 226)(74 219 80 230)(75 223 81 234)(91 204 117 194)(92 208 118 183)(93 197 119 187)(94 201 120 191)(95 205 106 195)(96 209 107 184)(97 198 108 188)(98 202 109 192)(99 206 110 181)(100 210 111 185)(101 199 112 189)(102 203 113 193)(103 207 114 182)(104 196 115 186)(105 200 116 190)
(1 76 16 70)(2 87 17 66)(3 83 18 62)(4 79 19 73)(5 90 20 69)(6 86 21 65)(7 82 22 61)(8 78 23 72)(9 89 24 68)(10 85 25 64)(11 81 26 75)(12 77 27 71)(13 88 28 67)(14 84 29 63)(15 80 30 74)(31 106 54 95)(32 117 55 91)(33 113 56 102)(34 109 57 98)(35 120 58 94)(36 116 59 105)(37 112 60 101)(38 108 46 97)(39 119 47 93)(40 115 48 104)(41 111 49 100)(42 107 50 96)(43 118 51 92)(44 114 52 103)(45 110 53 99)(121 200 140 190)(122 196 141 186)(123 207 142 182)(124 203 143 193)(125 199 144 189)(126 210 145 185)(127 206 146 181)(128 202 147 192)(129 198 148 188)(130 209 149 184)(131 205 150 195)(132 201 136 191)(133 197 137 187)(134 208 138 183)(135 204 139 194)(151 228 166 217)(152 239 167 213)(153 235 168 224)(154 231 169 220)(155 227 170 216)(156 238 171 212)(157 234 172 223)(158 230 173 219)(159 226 174 215)(160 237 175 211)(161 233 176 222)(162 229 177 218)(163 240 178 214)(164 236 179 225)(165 232 180 221)```

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

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

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

57 conjugacy classes

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

57 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 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + - + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 D10 D10 D10 D10 D10 2- 1+4 S3×D5 Q8○D12 C2×S3×D5 C2×S3×D5 D4.10D10 C15⋊2- 1+4 kernel C15⋊2- 1+4 D5×Dic6 S3×Dic10 D15⋊Q8 D6.D10 Dic5.D6 C30.C23 Dic3.D10 C2×C15⋊Q8 C3×D4⋊2D5 C5×D4⋊2S3 D4⋊2D15 D4⋊2D5 D4⋊2S3 Dic10 C4×D5 C2×Dic5 C5⋊D4 C5×D4 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 C15 D4 C5 C4 C22 C3 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 2 1 1 2 2 1 2 2 4 4 2 1 2 2 2 4 4 2

Matrix representation of C15⋊2- 1+4 in GL6(𝔽61)

 59 5 0 0 0 0 36 1 0 0 0 0 0 0 0 43 0 0 0 0 17 17 0 0 0 0 0 0 60 43 0 0 0 0 18 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 53 58 17 53 0 0 42 8 36 30 0 0 0 0 56 29 0 0 0 0 58 5
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 47 47 60 0 0 0 14 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 3 0 0 0 0 19 53 0 0 0 0 0 0 56 29 0 0 0 0 58 5
,
 58 2 0 0 0 0 57 3 0 0 0 0 0 0 32 57 0 0 0 0 58 29 0 0 0 0 21 16 36 4 0 0 21 5 57 25

`G:=sub<GL(6,GF(61))| [59,36,0,0,0,0,5,1,0,0,0,0,0,0,0,17,0,0,0,0,43,17,0,0,0,0,0,0,60,18,0,0,0,0,43,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,42,0,0,0,0,58,8,0,0,0,0,17,36,56,58,0,0,53,30,29,5],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,47,14,0,0,0,1,47,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,19,0,0,0,0,3,53,0,0,0,0,0,0,56,58,0,0,0,0,29,5],[58,57,0,0,0,0,2,3,0,0,0,0,0,0,32,58,21,21,0,0,57,29,16,5,0,0,0,0,36,57,0,0,0,0,4,25] >;`

C15⋊2- 1+4 in GAP, Magma, Sage, TeX

`C_{15}\rtimes 2_-^{1+4}`
`% in TeX`

`G:=Group("C15:ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(480,1096);`
`// by ID`

`G=gap.SmallGroup(480,1096);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,185,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^15=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=d*a*d^-1=a^4,a*c=c*a,e*a*e^-1=a^11,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

׿
×
𝔽