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G = C1517(C4×D4)  order 480 = 25·3·5

13rd semidirect product of C15 and C4×D4 acting via C4×D4/C2×C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D66(C4×D5), C1517(C4×D4), D105(C4×S3), C15⋊D44C4, D6⋊Dic513C2, (C5×Dic3)⋊11D4, C10.133(S3×D4), C30.142(C2×D4), (C2×C20).266D6, D10⋊C421S3, Dic1510(C2×C4), C6.57(C4○D20), C30.77(C4○D4), (C2×C12).196D10, C56(Dic34D4), Dic36(C5⋊D4), C30.60(C22×C4), C30.4Q832C2, (C22×D5).55D6, (Dic3×Dic5)⋊20C2, C2.5(D205S3), (C2×C60).410C22, (C2×C30).131C23, (C2×Dic5).115D6, (C22×S3).70D10, C10.31(D42S3), (C2×Dic3).183D10, (C6×Dic5).81C22, (C2×Dic15).103C22, (C10×Dic3).184C22, (S3×C2×C4)⋊9D5, C33(C4×C5⋊D4), C6.28(C2×C4×D5), C2.30(C4×S3×D5), (S3×C2×C20)⋊18C2, (C6×D5)⋊5(C2×C4), C10.61(S3×C2×C4), C2.2(S3×C5⋊D4), (C2×D5×Dic3)⋊10C2, (C2×C4).78(S3×D5), C6.34(C2×C5⋊D4), (S3×C10)⋊19(C2×C4), C22.63(C2×S3×D5), (C2×C15⋊D4).3C2, (D5×C2×C6).25C22, (S3×C2×C10).85C22, (C3×D10⋊C4)⋊32C2, (C2×C6).143(C22×D5), (C2×C10).143(C22×S3), SmallGroup(480,517)

Series: Derived Chief Lower central Upper central

C1C30 — C1517(C4×D4)
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — C1517(C4×D4)
C15C30 — C1517(C4×D4)
C1C22C2×C4

Generators and relations for C1517(C4×D4)
 G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=dad=a11, cac-1=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 844 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, Dic34D4, D5×Dic3 [×2], C15⋊D4 [×4], C6×Dic5, S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, C4×C5⋊D4, Dic3×Dic5, D6⋊Dic5, C3×D10⋊C4, C30.4Q8, C2×D5×Dic3, C2×C15⋊D4, S3×C2×C20, C1517(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, S3×D4, D42S3, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, Dic34D4, C2×S3×D5, C4×C5⋊D4, D205S3, C4×S3×D5, S3×C5⋊D4, C1517(C4×D4)

Smallest permutation representation of C1517(C4×D4)
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)
(1 224 39 181)(2 220 40 192)(3 216 41 188)(4 212 42 184)(5 223 43 195)(6 219 44 191)(7 215 45 187)(8 211 31 183)(9 222 32 194)(10 218 33 190)(11 214 34 186)(12 225 35 182)(13 221 36 193)(14 217 37 189)(15 213 38 185)(16 234 50 204)(17 230 51 200)(18 226 52 196)(19 237 53 207)(20 233 54 203)(21 229 55 199)(22 240 56 210)(23 236 57 206)(24 232 58 202)(25 228 59 198)(26 239 60 209)(27 235 46 205)(28 231 47 201)(29 227 48 197)(30 238 49 208)(61 125 95 162)(62 121 96 158)(63 132 97 154)(64 128 98 165)(65 124 99 161)(66 135 100 157)(67 131 101 153)(68 127 102 164)(69 123 103 160)(70 134 104 156)(71 130 105 152)(72 126 91 163)(73 122 92 159)(74 133 93 155)(75 129 94 151)(76 138 109 180)(77 149 110 176)(78 145 111 172)(79 141 112 168)(80 137 113 179)(81 148 114 175)(82 144 115 171)(83 140 116 167)(84 136 117 178)(85 147 118 174)(86 143 119 170)(87 139 120 166)(88 150 106 177)(89 146 107 173)(90 142 108 169)
(1 177 24 160)(2 176 25 159)(3 175 26 158)(4 174 27 157)(5 173 28 156)(6 172 29 155)(7 171 30 154)(8 170 16 153)(9 169 17 152)(10 168 18 151)(11 167 19 165)(12 166 20 164)(13 180 21 163)(14 179 22 162)(15 178 23 161)(31 143 50 131)(32 142 51 130)(33 141 52 129)(34 140 53 128)(35 139 54 127)(36 138 55 126)(37 137 56 125)(38 136 57 124)(39 150 58 123)(40 149 59 122)(41 148 60 121)(42 147 46 135)(43 146 47 134)(44 145 48 133)(45 144 49 132)(61 217 80 240)(62 216 81 239)(63 215 82 238)(64 214 83 237)(65 213 84 236)(66 212 85 235)(67 211 86 234)(68 225 87 233)(69 224 88 232)(70 223 89 231)(71 222 90 230)(72 221 76 229)(73 220 77 228)(74 219 78 227)(75 218 79 226)(91 193 109 199)(92 192 110 198)(93 191 111 197)(94 190 112 196)(95 189 113 210)(96 188 114 209)(97 187 115 208)(98 186 116 207)(99 185 117 206)(100 184 118 205)(101 183 119 204)(102 182 120 203)(103 181 106 202)(104 195 107 201)(105 194 108 200)
(1 123)(2 134)(3 130)(4 126)(5 122)(6 133)(7 129)(8 125)(9 121)(10 132)(11 128)(12 124)(13 135)(14 131)(15 127)(16 137)(17 148)(18 144)(19 140)(20 136)(21 147)(22 143)(23 139)(24 150)(25 146)(26 142)(27 138)(28 149)(29 145)(30 141)(31 162)(32 158)(33 154)(34 165)(35 161)(36 157)(37 153)(38 164)(39 160)(40 156)(41 152)(42 163)(43 159)(44 155)(45 151)(46 180)(47 176)(48 172)(49 168)(50 179)(51 175)(52 171)(53 167)(54 178)(55 174)(56 170)(57 166)(58 177)(59 173)(60 169)(61 183)(62 194)(63 190)(64 186)(65 182)(66 193)(67 189)(68 185)(69 181)(70 192)(71 188)(72 184)(73 195)(74 191)(75 187)(76 205)(77 201)(78 197)(79 208)(80 204)(81 200)(82 196)(83 207)(84 203)(85 199)(86 210)(87 206)(88 202)(89 198)(90 209)(91 212)(92 223)(93 219)(94 215)(95 211)(96 222)(97 218)(98 214)(99 225)(100 221)(101 217)(102 213)(103 224)(104 220)(105 216)(106 232)(107 228)(108 239)(109 235)(110 231)(111 227)(112 238)(113 234)(114 230)(115 226)(116 237)(117 233)(118 229)(119 240)(120 236)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E10A···10F10G···10N12A12B12C12D15A15B20A···20H20I···20P30A···30F60A···60H
order122222223444444444444556666610···1010···1012121212151520···2020···2030···3060···60
size111166101022233331010303030302222220202···26···6442020442···26···64···44···4

78 irreducible representations

dim111111111222222222222224444444
type++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6D6C4○D4D10D10D10C4×S3C5⋊D4C4×D5C4○D20S3×D4D42S3S3×D5C2×S3×D5D205S3C4×S3×D5S3×C5⋊D4
kernelC1517(C4×D4)Dic3×Dic5D6⋊Dic5C3×D10⋊C4C30.4Q8C2×D5×Dic3C2×C15⋊D4S3×C2×C20C15⋊D4D10⋊C4C5×Dic3S3×C2×C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3D10Dic3D6C6C10C10C2×C4C22C2C2C2
# reps111111118122111222248881122444

Matrix representation of C1517(C4×D4) in GL4(𝔽61) generated by

604300
181800
00601
00600
,
60000
06000
002359
002138
,
304400
533100
00939
004852
,
304400
173100
005222
00139
G:=sub<GL(4,GF(61))| [60,18,0,0,43,18,0,0,0,0,60,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,23,21,0,0,59,38],[30,53,0,0,44,31,0,0,0,0,9,48,0,0,39,52],[30,17,0,0,44,31,0,0,0,0,52,13,0,0,22,9] >;

C1517(C4×D4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_{17}(C_4\times D_4)
% in TeX

G:=Group("C15:17(C4xD4)");
// GroupNames label

G:=SmallGroup(480,517);
// by ID

G=gap.SmallGroup(480,517);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,58,1356,18822]);
// Polycyclic

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

׿
×
𝔽