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G = C1528(C4×D4)  order 480 = 25·3·5

4th semidirect product of C15 and C4×D4 acting via C4×D4/C23=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C1528(C4×D4), C157D45C4, D3017(C2×C4), (C5×Dic3)⋊15D4, C30.236(C2×D4), C10.160(S3×D4), C23.D515S3, D304C430C2, C23.31(S3×D5), Dic1512(C2×C4), C6.85(C4○D20), C57(Dic34D4), Dic38(C5⋊D4), Dic155C436C2, (C22×Dic3)⋊6D5, (C22×C6).35D10, (Dic3×Dic5)⋊36C2, C30.150(C4○D4), C30.148(C22×C4), (C2×C30).198C23, C222(D30.C2), (C2×Dic5).130D6, (C22×C10).109D6, C10.57(D42S3), (C2×Dic3).188D10, C2.7(Dic5.D6), (C22×C30).60C22, (C6×Dic5).114C22, (C22×D15).64C22, (C10×Dic3).203C22, (C2×Dic15).137C22, C34(C4×C5⋊D4), (C2×C6)⋊2(C4×D5), C6.55(C2×C4×D5), C10.87(S3×C2×C4), C2.7(S3×C5⋊D4), (C2×C30)⋊19(C2×C4), (C2×C10)⋊17(C4×S3), (Dic3×C2×C10)⋊6C2, C6.63(C2×C5⋊D4), C22.88(C2×S3×D5), (C3×C23.D5)⋊9C2, (C2×D30.C2)⋊15C2, (C2×C157D4).12C2, C2.19(C2×D30.C2), (C2×C6).210(C22×D5), (C2×C10).210(C22×S3), SmallGroup(480,632)

Series: Derived Chief Lower central Upper central

C1C30 — C1528(C4×D4)
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — C1528(C4×D4)
C15C30 — C1528(C4×D4)
C1C22C23

Generators and relations for C1528(C4×D4)
 G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=a11, cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 908 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 [×2], Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], C20 [×3], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, D15 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20 [×4], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, Dic34D4, D30.C2 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×4], C22×D15, C22×C30, C4×C5⋊D4, Dic3×Dic5, D304C4, Dic155C4, C3×C23.D5, C2×D30.C2, Dic3×C2×C10, C2×C157D4, C1528(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, D30.C2 [×2], C2×S3×D5, C4×C5⋊D4, Dic5.D6, C2×D30.C2, S3×C5⋊D4, C1528(C4×D4)

Smallest permutation representation of C1528(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)
(1 164 43 135)(2 160 44 131)(3 156 45 127)(4 152 31 123)(5 163 32 134)(6 159 33 130)(7 155 34 126)(8 151 35 122)(9 162 36 133)(10 158 37 129)(11 154 38 125)(12 165 39 121)(13 161 40 132)(14 157 41 128)(15 153 42 124)(16 171 46 149)(17 167 47 145)(18 178 48 141)(19 174 49 137)(20 170 50 148)(21 166 51 144)(22 177 52 140)(23 173 53 136)(24 169 54 147)(25 180 55 143)(26 176 56 139)(27 172 57 150)(28 168 58 146)(29 179 59 142)(30 175 60 138)(61 223 91 189)(62 219 92 185)(63 215 93 181)(64 211 94 192)(65 222 95 188)(66 218 96 184)(67 214 97 195)(68 225 98 191)(69 221 99 187)(70 217 100 183)(71 213 101 194)(72 224 102 190)(73 220 103 186)(74 216 104 182)(75 212 105 193)(76 238 106 206)(77 234 107 202)(78 230 108 198)(79 226 109 209)(80 237 110 205)(81 233 111 201)(82 229 112 197)(83 240 113 208)(84 236 114 204)(85 232 115 200)(86 228 116 196)(87 239 117 207)(88 235 118 203)(89 231 119 199)(90 227 120 210)
(1 80 16 65)(2 79 17 64)(3 78 18 63)(4 77 19 62)(5 76 20 61)(6 90 21 75)(7 89 22 74)(8 88 23 73)(9 87 24 72)(10 86 25 71)(11 85 26 70)(12 84 27 69)(13 83 28 68)(14 82 29 67)(15 81 30 66)(31 107 49 92)(32 106 50 91)(33 120 51 105)(34 119 52 104)(35 118 53 103)(36 117 54 102)(37 116 55 101)(38 115 56 100)(39 114 57 99)(40 113 58 98)(41 112 59 97)(42 111 60 96)(43 110 46 95)(44 109 47 94)(45 108 48 93)(121 204 150 187)(122 203 136 186)(123 202 137 185)(124 201 138 184)(125 200 139 183)(126 199 140 182)(127 198 141 181)(128 197 142 195)(129 196 143 194)(130 210 144 193)(131 209 145 192)(132 208 146 191)(133 207 147 190)(134 206 148 189)(135 205 149 188)(151 235 173 220)(152 234 174 219)(153 233 175 218)(154 232 176 217)(155 231 177 216)(156 230 178 215)(157 229 179 214)(158 228 180 213)(159 227 166 212)(160 226 167 211)(161 240 168 225)(162 239 169 224)(163 238 170 223)(164 237 171 222)(165 236 172 221)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 105)(12 91)(13 92)(14 93)(15 94)(16 110)(17 111)(18 112)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 106)(28 107)(29 108)(30 109)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 76)(58 77)(59 78)(60 79)(121 223)(122 224)(123 225)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 239)(137 240)(138 226)(139 227)(140 228)(141 229)(142 230)(143 231)(144 232)(145 233)(146 234)(147 235)(148 236)(149 237)(150 238)(151 190)(152 191)(153 192)(154 193)(155 194)(156 195)(157 181)(158 182)(159 183)(160 184)(161 185)(162 186)(163 187)(164 188)(165 189)(166 200)(167 201)(168 202)(169 203)(170 204)(171 205)(172 206)(173 207)(174 208)(175 209)(176 210)(177 196)(178 197)(179 198)(180 199)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E10A···10N12A12B12C12D15A15B20A···20P30A···30N
order122222223444444444444556666610···1012121212151520···2030···30
size1111223030233336610101010303022222442···220202020446···64···4

78 irreducible representations

dim1111111112222222222224444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10C4×S3C5⋊D4C4×D5C4○D20S3×D4D42S3S3×D5D30.C2C2×S3×D5Dic5.D6S3×C5⋊D4
kernelC1528(C4×D4)Dic3×Dic5D304C4Dic155C4C3×C23.D5C2×D30.C2Dic3×C2×C10C2×C157D4C157D4C23.D5C5×Dic3C22×Dic3C2×Dic5C22×C10C30C2×Dic3C22×C6C2×C10Dic3C2×C6C6C10C10C23C22C22C2C2
# reps1111111181222124248881124244

Matrix representation of C1528(C4×D4) in GL6(𝔽61)

60170000
44440000
0004300
00171700
0000601
0000600
,
5000000
0500000
0011000
0001100
0000060
0000600
,
14450000
39470000
00171800
00454400
000001
000010
,
47160000
45140000
0060000
0006000
000010
000001

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,0,17,0,0,0,0,43,17,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[14,39,0,0,0,0,45,47,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[47,45,0,0,0,0,16,14,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C1528(C4×D4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_{28}(C_4\times D_4)
% in TeX

G:=Group("C15:28(C4xD4)");
// GroupNames label

G:=SmallGroup(480,632);
// by ID

G=gap.SmallGroup(480,632);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,1356,18822]);
// Polycyclic

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

׿
×
𝔽