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G = C1522(C4×D4)  order 480 = 25·3·5

18th semidirect product of C15 and C4×D4 acting via C4×D4/C2×C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D68(C4×D5), C1522(C4×D4), C5⋊D125C4, D3013(C2×C4), Dic54(C4×S3), C55(Dic35D4), D6⋊Dic514C2, (C5×Dic3)⋊12D4, (C2×C20).267D6, C10.134(S3×D4), C30.146(C2×D4), D303C432C2, C6.13(C4○D20), C30.81(C4○D4), (C2×C12).198D10, C10.D421S3, Dic37(C5⋊D4), C30.65(C22×C4), (Dic3×Dic5)⋊22C2, C2.5(D60⋊C2), (C2×C30).136C23, (C2×C60).411C22, (C2×Dic5).117D6, (C22×S3).72D10, C10.18(Q83S3), (C2×Dic3).184D10, (C6×Dic5).83C22, (C22×D15).45C22, (C10×Dic3).187C22, (C2×Dic15).105C22, C32(C4×C5⋊D4), (S3×C2×C4)⋊11D5, C6.33(C2×C4×D5), C2.35(C4×S3×D5), (S3×C2×C20)⋊19C2, C10.66(S3×C2×C4), C2.3(S3×C5⋊D4), (C2×C4).79(S3×D5), C6.35(C2×C5⋊D4), (S3×C10)⋊21(C2×C4), C22.68(C2×S3×D5), (C3×Dic5)⋊4(C2×C4), (C2×C5⋊D12).9C2, (C2×D30.C2)⋊10C2, (S3×C2×C10).87C22, (C3×C10.D4)⋊33C2, (C2×C6).148(C22×D5), (C2×C10).148(C22×S3), SmallGroup(480,522)

Series: Derived Chief Lower central Upper central

C1C30 — C1522(C4×D4)
C1C5C15C30C2×C30C6×Dic5C2×C5⋊D12 — C1522(C4×D4)
C15C30 — C1522(C4×D4)
C1C22C2×C4

Generators and relations for C1522(C4×D4)
 G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=dad=a11, cac-1=a4, bc=cb, bd=db, dcd=c-1 >

Subgroups: 940 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3, C12 [×4], D6 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×4], C2×C10, C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5, C22×C10, C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15, C60, S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, Dic35D4, D30.C2 [×2], C5⋊D12 [×4], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C4×C5⋊D4, Dic3×Dic5, D6⋊Dic5, C3×C10.D4, D303C4, C2×D30.C2, C2×C5⋊D12, S3×C2×C20, C1522(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, Q83S3, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, Dic35D4, C2×S3×D5, C4×C5⋊D4, D60⋊C2, C4×S3×D5, S3×C5⋊D4, C1522(C4×D4)

Smallest permutation representation of C1522(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)
(1 222 31 191)(2 218 32 187)(3 214 33 183)(4 225 34 194)(5 221 35 190)(6 217 36 186)(7 213 37 182)(8 224 38 193)(9 220 39 189)(10 216 40 185)(11 212 41 181)(12 223 42 192)(13 219 43 188)(14 215 44 184)(15 211 45 195)(16 238 54 206)(17 234 55 202)(18 230 56 198)(19 226 57 209)(20 237 58 205)(21 233 59 201)(22 229 60 197)(23 240 46 208)(24 236 47 204)(25 232 48 200)(26 228 49 196)(27 239 50 207)(28 235 51 203)(29 231 52 199)(30 227 53 210)(61 123 99 165)(62 134 100 161)(63 130 101 157)(64 126 102 153)(65 122 103 164)(66 133 104 160)(67 129 105 156)(68 125 91 152)(69 121 92 163)(70 132 93 159)(71 128 94 155)(72 124 95 151)(73 135 96 162)(74 131 97 158)(75 127 98 154)(76 140 106 180)(77 136 107 176)(78 147 108 172)(79 143 109 168)(80 139 110 179)(81 150 111 175)(82 146 112 171)(83 142 113 167)(84 138 114 178)(85 149 115 174)(86 145 116 170)(87 141 117 166)(88 137 118 177)(89 148 119 173)(90 144 120 169)
(1 123 24 138)(2 127 25 142)(3 131 26 146)(4 135 27 150)(5 124 28 139)(6 128 29 143)(7 132 30 147)(8 121 16 136)(9 125 17 140)(10 129 18 144)(11 133 19 148)(12 122 20 137)(13 126 21 141)(14 130 22 145)(15 134 23 149)(31 165 47 178)(32 154 48 167)(33 158 49 171)(34 162 50 175)(35 151 51 179)(36 155 52 168)(37 159 53 172)(38 163 54 176)(39 152 55 180)(40 156 56 169)(41 160 57 173)(42 164 58 177)(43 153 59 166)(44 157 60 170)(45 161 46 174)(61 204 84 191)(62 208 85 195)(63 197 86 184)(64 201 87 188)(65 205 88 192)(66 209 89 181)(67 198 90 185)(68 202 76 189)(69 206 77 193)(70 210 78 182)(71 199 79 186)(72 203 80 190)(73 207 81 194)(74 196 82 183)(75 200 83 187)(91 234 106 220)(92 238 107 224)(93 227 108 213)(94 231 109 217)(95 235 110 221)(96 239 111 225)(97 228 112 214)(98 232 113 218)(99 236 114 222)(100 240 115 211)(101 229 116 215)(102 233 117 219)(103 237 118 223)(104 226 119 212)(105 230 120 216)
(1 165)(2 161)(3 157)(4 153)(5 164)(6 160)(7 156)(8 152)(9 163)(10 159)(11 155)(12 151)(13 162)(14 158)(15 154)(16 180)(17 176)(18 172)(19 168)(20 179)(21 175)(22 171)(23 167)(24 178)(25 174)(26 170)(27 166)(28 177)(29 173)(30 169)(31 123)(32 134)(33 130)(34 126)(35 122)(36 133)(37 129)(38 125)(39 121)(40 132)(41 128)(42 124)(43 135)(44 131)(45 127)(46 142)(47 138)(48 149)(49 145)(50 141)(51 137)(52 148)(53 144)(54 140)(55 136)(56 147)(57 143)(58 139)(59 150)(60 146)(61 222)(62 218)(63 214)(64 225)(65 221)(66 217)(67 213)(68 224)(69 220)(70 216)(71 212)(72 223)(73 219)(74 215)(75 211)(76 238)(77 234)(78 230)(79 226)(80 237)(81 233)(82 229)(83 240)(84 236)(85 232)(86 228)(87 239)(88 235)(89 231)(90 227)(91 193)(92 189)(93 185)(94 181)(95 192)(96 188)(97 184)(98 195)(99 191)(100 187)(101 183)(102 194)(103 190)(104 186)(105 182)(106 206)(107 202)(108 198)(109 209)(110 205)(111 201)(112 197)(113 208)(114 204)(115 200)(116 196)(117 207)(118 203)(119 199)(120 210)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A···20H20I···20P30A···30F60A···60H
order1222222234444444444445566610···1010···10121212121212151520···2020···2030···3060···60
size11116630302223333101010103030222222···26···64420202020442···26···64···44···4

78 irreducible representations

dim11111111122222222222224444444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10D10C4×S3C5⋊D4C4×D5C4○D20S3×D4Q83S3S3×D5C2×S3×D5D60⋊C2C4×S3×D5S3×C5⋊D4
kernelC1522(C4×D4)Dic3×Dic5D6⋊Dic5C3×C10.D4D303C4C2×D30.C2C2×C5⋊D12S3×C2×C20C5⋊D12C10.D4C5×Dic3S3×C2×C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5Dic3D6C6C10C10C2×C4C22C2C2C2
# reps11111111812221222248881122444

Matrix representation of C1522(C4×D4) in GL4(𝔽61) generated by

0100
606000
004317
00430
,
11000
505000
00600
00060
,
1000
0100
005347
0098
,
60000
1100
003045
006031
G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,43,43,0,0,17,0],[11,50,0,0,0,50,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,53,9,0,0,47,8],[60,1,0,0,0,1,0,0,0,0,30,60,0,0,45,31] >;

C1522(C4×D4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_{22}(C_4\times D_4)
% in TeX

G:=Group("C15:22(C4xD4)");
// GroupNames label

G:=SmallGroup(480,522);
// by ID

G=gap.SmallGroup(480,522);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,422,58,1356,18822]);
// 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^4,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽