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G = C1526(C4×D4)  order 480 = 25·3·5

2nd semidirect product of C15 and C4×D4 acting via C4×D4/C23=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C1526(C4×D4), C157D44C4, D3016(C2×C4), C6.160(D4×D5), (C3×Dic5)⋊15D4, C30.232(C2×D4), D304C429C2, C23.42(S3×D5), Dic1511(C2×C4), C6.D415D5, Dic58(C3⋊D4), C36(Dic54D4), Dic155C434C2, (C22×Dic5)⋊9S3, (C22×C6).92D10, (C22×C10).49D6, (Dic3×Dic5)⋊34C2, C10.82(C4○D12), C30.147(C4○D4), C6.55(D42D5), (C2×C30).194C23, C30.145(C22×C4), C223(D30.C2), (C2×Dic5).194D6, (C2×Dic3).122D10, C2.7(Dic3.D10), (C22×C30).56C22, (C6×Dic5).223C22, (C22×D15).63C22, (C2×Dic15).134C22, (C10×Dic3).113C22, C55(C4×C3⋊D4), (C2×C6)⋊8(C4×D5), C6.54(C2×C4×D5), C10.86(S3×C2×C4), (C2×C6×Dic5)⋊6C2, C2.7(D5×C3⋊D4), (C2×C30)⋊16(C2×C4), (C2×C10)⋊11(C4×S3), C22.85(C2×S3×D5), C10.64(C2×C3⋊D4), (C2×D30.C2)⋊14C2, (C2×C157D4).11C2, C2.18(C2×D30.C2), (C5×C6.D4)⋊11C2, (C2×C6).206(C22×D5), (C2×C10).206(C22×S3), SmallGroup(480,628)

Series: Derived Chief Lower central Upper central

C1C30 — C1526(C4×D4)
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — C1526(C4×D4)
C15C30 — C1526(C4×D4)
C1C22C23

Generators and relations for C1526(C4×D4)
 G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=a4, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 924 in 188 conjugacy classes, 64 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×9], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×4], C12 [×3], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12 [×4], C22×S3, C22×C6, D15 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C3⋊D4, D30.C2 [×2], C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×4], C22×D15, C22×C30, Dic54D4, Dic3×Dic5, D304C4, Dic155C4, C5×C6.D4, C2×D30.C2, C2×C6×Dic5, C2×C157D4, C1526(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], C3⋊D4 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, D42D5, C4×C3⋊D4, D30.C2 [×2], C2×S3×D5, Dic54D4, Dic3.D10, C2×D30.C2, D5×C3⋊D4, C1526(C4×D4)

Smallest permutation representation of C1526(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)
(1 160 35 125)(2 164 36 129)(3 153 37 133)(4 157 38 122)(5 161 39 126)(6 165 40 130)(7 154 41 134)(8 158 42 123)(9 162 43 127)(10 151 44 131)(11 155 45 135)(12 159 31 124)(13 163 32 128)(14 152 33 132)(15 156 34 121)(16 179 49 136)(17 168 50 140)(18 172 51 144)(19 176 52 148)(20 180 53 137)(21 169 54 141)(22 173 55 145)(23 177 56 149)(24 166 57 138)(25 170 58 142)(26 174 59 146)(27 178 60 150)(28 167 46 139)(29 171 47 143)(30 175 48 147)(61 236 95 199)(62 240 96 203)(63 229 97 207)(64 233 98 196)(65 237 99 200)(66 226 100 204)(67 230 101 208)(68 234 102 197)(69 238 103 201)(70 227 104 205)(71 231 105 209)(72 235 91 198)(73 239 92 202)(74 228 93 206)(75 232 94 210)(76 213 109 191)(77 217 110 195)(78 221 111 184)(79 225 112 188)(80 214 113 192)(81 218 114 181)(82 222 115 185)(83 211 116 189)(84 215 117 193)(85 219 118 182)(86 223 119 186)(87 212 120 190)(88 216 106 194)(89 220 107 183)(90 224 108 187)
(1 99 17 110)(2 98 18 109)(3 97 19 108)(4 96 20 107)(5 95 21 106)(6 94 22 120)(7 93 23 119)(8 92 24 118)(9 91 25 117)(10 105 26 116)(11 104 27 115)(12 103 28 114)(13 102 29 113)(14 101 30 112)(15 100 16 111)(31 69 46 81)(32 68 47 80)(33 67 48 79)(34 66 49 78)(35 65 50 77)(36 64 51 76)(37 63 52 90)(38 62 53 89)(39 61 54 88)(40 75 55 87)(41 74 56 86)(42 73 57 85)(43 72 58 84)(44 71 59 83)(45 70 60 82)(121 226 136 221)(122 240 137 220)(123 239 138 219)(124 238 139 218)(125 237 140 217)(126 236 141 216)(127 235 142 215)(128 234 143 214)(129 233 144 213)(130 232 145 212)(131 231 146 211)(132 230 147 225)(133 229 148 224)(134 228 149 223)(135 227 150 222)(151 209 174 189)(152 208 175 188)(153 207 176 187)(154 206 177 186)(155 205 178 185)(156 204 179 184)(157 203 180 183)(158 202 166 182)(159 201 167 181)(160 200 168 195)(161 199 169 194)(162 198 170 193)(163 197 171 192)(164 196 172 191)(165 210 173 190)
(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(16 18)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 120)(105 119)(121 129)(122 128)(123 127)(124 126)(130 135)(131 134)(132 133)(136 144)(137 143)(138 142)(139 141)(145 150)(146 149)(147 148)(151 154)(152 153)(155 165)(156 164)(157 163)(158 162)(159 161)(166 170)(167 169)(171 180)(172 179)(173 178)(174 177)(175 176)(181 199)(182 198)(183 197)(184 196)(185 210)(186 209)(187 208)(188 207)(189 206)(190 205)(191 204)(192 203)(193 202)(194 201)(195 200)(211 228)(212 227)(213 226)(214 240)(215 239)(216 238)(217 237)(218 236)(219 235)(220 234)(221 233)(222 232)(223 231)(224 230)(225 229)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A···6G10A···10F10G10H10I10J12A···12H15A15B20A···20H30A···30N
order122222223444444444444556···610···101010101012···12151520···2030···30
size111122303025555666610103030222···22···2444410···104412···124···4

72 irreducible representations

dim1111111112222222222224444444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10C3⋊D4C4×S3C4×D5C4○D12S3×D5D4×D5D42D5D30.C2C2×S3×D5Dic3.D10D5×C3⋊D4
kernelC1526(C4×D4)Dic3×Dic5D304C4Dic155C4C5×C6.D4C2×D30.C2C2×C6×Dic5C2×C157D4C157D4C22×Dic5C3×Dic5C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6Dic5C2×C10C2×C6C10C23C6C6C22C22C2C2
# reps1111111181222124244842224244

Matrix representation of C1526(C4×D4) in GL6(𝔽61)

60170000
44440000
001000
000100
0000601
0000600
,
5000000
57110000
001000
000100
0000500
0000050
,
100000
17600000
0011500
0086000
0000060
0000600
,
100000
17600000
001000
0086000
000001
000010

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[50,57,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,8,0,0,0,0,15,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,8,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C1526(C4×D4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_{26}(C_4\times D_4)
% in TeX

G:=Group("C15:26(C4xD4)");
// GroupNames label

G:=SmallGroup(480,628);
// by ID

G=gap.SmallGroup(480,628);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,219,1356,18822]);
// Polycyclic

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

׿
×
𝔽