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G = Dic1518D4order 480 = 25·3·5

8th semidirect product of Dic15 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1518D4, (C2×C30)⋊10D4, C6.91(D4×D5), C10.93(S3×D4), C1524(C4⋊D4), D6⋊Dic537C2, C30.251(C2×D4), C23.54(S3×D5), C36(Dic5⋊D4), C56(C23.14D6), C222(C15⋊D4), Dic155C438C2, (C2×Dic5).66D6, (C22×D5).33D6, (C22×C6).47D10, (C22×C10).62D6, C30.159(C4○D4), D10⋊Dic337C2, C6.86(D42D5), (C2×C30).213C23, (C2×Dic3).65D10, (C22×S3).31D10, C10.86(D42S3), C2.43(D10⋊D6), (C22×Dic15)⋊16C2, (C22×C30).75C22, C2.30(C30.C23), (C6×Dic5).123C22, (C10×Dic3).123C22, (C2×Dic15).229C22, (C2×C5⋊D4)⋊8S3, (C2×C3⋊D4)⋊8D5, (C6×C5⋊D4)⋊8C2, (C10×C3⋊D4)⋊8C2, (C2×C6)⋊3(C5⋊D4), C6.96(C2×C5⋊D4), (C2×C15⋊D4)⋊16C2, (C2×C10)⋊7(C3⋊D4), C10.96(C2×C3⋊D4), C2.26(C2×C15⋊D4), (D5×C2×C6).56C22, C22.242(C2×S3×D5), (S3×C2×C10).56C22, (C2×C6).225(C22×D5), (C2×C10).225(C22×S3), SmallGroup(480,647)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic1518D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — Dic1518D4
C15C2×C30 — Dic1518D4
C1C22C23

Generators and relations for Dic1518D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a11, cbc-1=dbd=a15b, dcd=c-1 >

Subgroups: 924 in 188 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], C5, S3, C6 [×3], C6 [×3], C2×C4 [×6], D4 [×6], C23, C23 [×2], D5, C10 [×3], C10 [×3], Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×4], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30 [×3], C30 [×2], C4⋊D4, C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C6×D5 [×3], S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23.14D6, C15⋊D4 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15 [×2], C2×Dic15 [×2], D5×C2×C6, S3×C2×C10, C22×C30, Dic5⋊D4, D10⋊Dic3, D6⋊Dic5, Dic155C4, C2×C15⋊D4, C6×C5⋊D4, C10×C3⋊D4, C22×Dic15, Dic1518D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C23.14D6, C15⋊D4 [×2], C2×S3×D5, Dic5⋊D4, C30.C23, C2×C15⋊D4, D10⋊D6, Dic1518D4

Smallest permutation representation of Dic1518D4
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 33 16 48)(2 32 17 47)(3 31 18 46)(4 60 19 45)(5 59 20 44)(6 58 21 43)(7 57 22 42)(8 56 23 41)(9 55 24 40)(10 54 25 39)(11 53 26 38)(12 52 27 37)(13 51 28 36)(14 50 29 35)(15 49 30 34)(61 198 76 183)(62 197 77 182)(63 196 78 181)(64 195 79 210)(65 194 80 209)(66 193 81 208)(67 192 82 207)(68 191 83 206)(69 190 84 205)(70 189 85 204)(71 188 86 203)(72 187 87 202)(73 186 88 201)(74 185 89 200)(75 184 90 199)(91 126 106 141)(92 125 107 140)(93 124 108 139)(94 123 109 138)(95 122 110 137)(96 121 111 136)(97 150 112 135)(98 149 113 134)(99 148 114 133)(100 147 115 132)(101 146 116 131)(102 145 117 130)(103 144 118 129)(104 143 119 128)(105 142 120 127)(151 233 166 218)(152 232 167 217)(153 231 168 216)(154 230 169 215)(155 229 170 214)(156 228 171 213)(157 227 172 212)(158 226 173 211)(159 225 174 240)(160 224 175 239)(161 223 176 238)(162 222 177 237)(163 221 178 236)(164 220 179 235)(165 219 180 234)
(1 172 205 95)(2 153 206 106)(3 164 207 117)(4 175 208 98)(5 156 209 109)(6 167 210 120)(7 178 181 101)(8 159 182 112)(9 170 183 93)(10 151 184 104)(11 162 185 115)(12 173 186 96)(13 154 187 107)(14 165 188 118)(15 176 189 99)(16 157 190 110)(17 168 191 91)(18 179 192 102)(19 160 193 113)(20 171 194 94)(21 152 195 105)(22 163 196 116)(23 174 197 97)(24 155 198 108)(25 166 199 119)(26 177 200 100)(27 158 201 111)(28 169 202 92)(29 180 203 103)(30 161 204 114)(31 235 67 145)(32 216 68 126)(33 227 69 137)(34 238 70 148)(35 219 71 129)(36 230 72 140)(37 211 73 121)(38 222 74 132)(39 233 75 143)(40 214 76 124)(41 225 77 135)(42 236 78 146)(43 217 79 127)(44 228 80 138)(45 239 81 149)(46 220 82 130)(47 231 83 141)(48 212 84 122)(49 223 85 133)(50 234 86 144)(51 215 87 125)(52 226 88 136)(53 237 89 147)(54 218 90 128)(55 229 61 139)(56 240 62 150)(57 221 63 131)(58 232 64 142)(59 213 65 123)(60 224 66 134)
(1 95)(2 106)(3 117)(4 98)(5 109)(6 120)(7 101)(8 112)(9 93)(10 104)(11 115)(12 96)(13 107)(14 118)(15 99)(16 110)(17 91)(18 102)(19 113)(20 94)(21 105)(22 116)(23 97)(24 108)(25 119)(26 100)(27 111)(28 92)(29 103)(30 114)(31 145)(32 126)(33 137)(34 148)(35 129)(36 140)(37 121)(38 132)(39 143)(40 124)(41 135)(42 146)(43 127)(44 138)(45 149)(46 130)(47 141)(48 122)(49 133)(50 144)(51 125)(52 136)(53 147)(54 128)(55 139)(56 150)(57 131)(58 142)(59 123)(60 134)(61 229)(62 240)(63 221)(64 232)(65 213)(66 224)(67 235)(68 216)(69 227)(70 238)(71 219)(72 230)(73 211)(74 222)(75 233)(76 214)(77 225)(78 236)(79 217)(80 228)(81 239)(82 220)(83 231)(84 212)(85 223)(86 234)(87 215)(88 226)(89 237)(90 218)(151 184)(152 195)(153 206)(154 187)(155 198)(156 209)(157 190)(158 201)(159 182)(160 193)(161 204)(162 185)(163 196)(164 207)(165 188)(166 199)(167 210)(168 191)(169 202)(170 183)(171 194)(172 205)(173 186)(174 197)(175 208)(176 189)(177 200)(178 181)(179 192)(180 203)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A···30N
order12222222344444455666666610···101010101010101010121215152020202030···30
size11112212202122030303030222224420202···2444412121212202044121212124···4

60 irreducible representations

dim111111112222222222222444444444
type+++++++++++++++++++-++--+-+
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10D10C3⋊D4C5⋊D4S3×D4D42S3S3×D5D4×D5D42D5C15⋊D4C2×S3×D5C30.C23D10⋊D6
kernelDic1518D4D10⋊Dic3D6⋊Dic5Dic155C4C2×C15⋊D4C6×C5⋊D4C10×C3⋊D4C22×Dic15C2×C5⋊D4Dic15C2×C30C2×C3⋊D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×S3C22×C6C2×C10C2×C6C10C10C23C6C6C22C22C2C2
# reps111111111222111222248112224244

Matrix representation of Dic1518D4 in GL8(𝔽61)

01000000
6043000000
006000000
000600000
00001000
00000100
000000130
000000247
,
838000000
1653000000
004670000
0020150000
00001000
00000100
0000005637
00000015
,
600000000
060000000
0048410000
0045130000
0000605900
00001100
000000524
0000006056
,
600000000
060000000
0048410000
0045130000
0000605900
00000100
000000524
0000006056

G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,47],[8,16,0,0,0,0,0,0,38,53,0,0,0,0,0,0,0,0,46,20,0,0,0,0,0,0,7,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,56,1,0,0,0,0,0,0,37,5],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,48,45,0,0,0,0,0,0,41,13,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,5,60,0,0,0,0,0,0,24,56],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,48,45,0,0,0,0,0,0,41,13,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,5,60,0,0,0,0,0,0,24,56] >;

Dic1518D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{18}D_4
% in TeX

G:=Group("Dic15:18D4");
// GroupNames label

G:=SmallGroup(480,647);
// by ID

G=gap.SmallGroup(480,647);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽