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G = Dic15.19D4order 480 = 25·3·5

19th non-split extension by Dic15 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.19D4, C6.83(D4×D5), C10.84(S3×D4), C23.D54S3, C30.212(C2×D4), C6.D43D5, D304C425C2, C23.12(S3×D5), C6.74(C4○D20), C1514(C4.4D4), (C2×Dic5).52D6, (C22×C10).33D6, (C22×C6).18D10, (Dic3×Dic5)⋊26C2, C30.129(C4○D4), C10.75(C4○D12), C6.49(D42D5), (C2×C30).168C23, C54(C23.11D6), C34(Dic5.5D4), C2.36(D10⋊D6), C10.48(D42S3), (C2×Dic3).116D10, (C22×C30).30C22, (C6×Dic5).98C22, C2.21(Dic3.D10), C2.20(Dic5.D6), (C10×Dic3).98C22, (C22×D15).58C22, (C2×Dic15).119C22, (C2×C15⋊Q8)⋊14C2, (C2×C157D4).8C2, (C3×C23.D5)⋊4C2, C22.216(C2×S3×D5), (C5×C6.D4)⋊3C2, (C2×C6).180(C22×D5), (C2×C10).180(C22×S3), SmallGroup(480,602)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.19D4
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic15.19D4
C15C2×C30 — Dic15.19D4
C1C22C23

Generators and relations for Dic15.19D4
 G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, cac-1=dad-1=a19, bc=cb, dbd-1=a15b, dcd-1=a15c-1 >

Subgroups: 876 in 152 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, S3, C6 [×3], C6, C2×C4 [×5], D4 [×2], Q8 [×2], C23, C23, D5, C10 [×3], C10, Dic3 [×4], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], Dic6 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, D15, C30 [×3], C30, C4.4D4, Dic10 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], D30 [×3], C2×C30, C2×C30 [×3], C4×Dic5, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.11D6, C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], C22×D15, C22×C30, Dic5.5D4, Dic3×Dic5, D304C4 [×2], C3×C23.D5, C5×C6.D4, C2×C15⋊Q8, C2×C157D4, Dic15.19D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C4.4D4, C22×D5, C4○D12, S3×D4, D42S3, S3×D5, C4○D20, D4×D5, D42D5, C23.11D6, C2×S3×D5, Dic5.5D4, Dic5.D6, Dic3.D10, D10⋊D6, Dic15.19D4

Smallest permutation representation of Dic15.19D4
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)
(1 225 16 240)(2 224 17 239)(3 223 18 238)(4 222 19 237)(5 221 20 236)(6 220 21 235)(7 219 22 234)(8 218 23 233)(9 217 24 232)(10 216 25 231)(11 215 26 230)(12 214 27 229)(13 213 28 228)(14 212 29 227)(15 211 30 226)(31 82 46 67)(32 81 47 66)(33 80 48 65)(34 79 49 64)(35 78 50 63)(36 77 51 62)(37 76 52 61)(38 75 53 90)(39 74 54 89)(40 73 55 88)(41 72 56 87)(42 71 57 86)(43 70 58 85)(44 69 59 84)(45 68 60 83)(91 208 106 193)(92 207 107 192)(93 206 108 191)(94 205 109 190)(95 204 110 189)(96 203 111 188)(97 202 112 187)(98 201 113 186)(99 200 114 185)(100 199 115 184)(101 198 116 183)(102 197 117 182)(103 196 118 181)(104 195 119 210)(105 194 120 209)(121 159 136 174)(122 158 137 173)(123 157 138 172)(124 156 139 171)(125 155 140 170)(126 154 141 169)(127 153 142 168)(128 152 143 167)(129 151 144 166)(130 180 145 165)(131 179 146 164)(132 178 147 163)(133 177 148 162)(134 176 149 161)(135 175 150 160)
(1 191 38 172)(2 210 39 161)(3 199 40 180)(4 188 41 169)(5 207 42 158)(6 196 43 177)(7 185 44 166)(8 204 45 155)(9 193 46 174)(10 182 47 163)(11 201 48 152)(12 190 49 171)(13 209 50 160)(14 198 51 179)(15 187 52 168)(16 206 53 157)(17 195 54 176)(18 184 55 165)(19 203 56 154)(20 192 57 173)(21 181 58 162)(22 200 59 151)(23 189 60 170)(24 208 31 159)(25 197 32 178)(26 186 33 167)(27 205 34 156)(28 194 35 175)(29 183 36 164)(30 202 37 153)(61 127 211 97)(62 146 212 116)(63 135 213 105)(64 124 214 94)(65 143 215 113)(66 132 216 102)(67 121 217 91)(68 140 218 110)(69 129 219 99)(70 148 220 118)(71 137 221 107)(72 126 222 96)(73 145 223 115)(74 134 224 104)(75 123 225 93)(76 142 226 112)(77 131 227 101)(78 150 228 120)(79 139 229 109)(80 128 230 98)(81 147 231 117)(82 136 232 106)(83 125 233 95)(84 144 234 114)(85 133 235 103)(86 122 236 92)(87 141 237 111)(88 130 238 100)(89 149 239 119)(90 138 240 108)
(1 108 16 93)(2 97 17 112)(3 116 18 101)(4 105 19 120)(5 94 20 109)(6 113 21 98)(7 102 22 117)(8 91 23 106)(9 110 24 95)(10 99 25 114)(11 118 26 103)(12 107 27 92)(13 96 28 111)(14 115 29 100)(15 104 30 119)(31 125 46 140)(32 144 47 129)(33 133 48 148)(34 122 49 137)(35 141 50 126)(36 130 51 145)(37 149 52 134)(38 138 53 123)(39 127 54 142)(40 146 55 131)(41 135 56 150)(42 124 57 139)(43 143 58 128)(44 132 59 147)(45 121 60 136)(61 161 76 176)(62 180 77 165)(63 169 78 154)(64 158 79 173)(65 177 80 162)(66 166 81 151)(67 155 82 170)(68 174 83 159)(69 163 84 178)(70 152 85 167)(71 171 86 156)(72 160 87 175)(73 179 88 164)(74 168 89 153)(75 157 90 172)(181 215 196 230)(182 234 197 219)(183 223 198 238)(184 212 199 227)(185 231 200 216)(186 220 201 235)(187 239 202 224)(188 228 203 213)(189 217 204 232)(190 236 205 221)(191 225 206 240)(192 214 207 229)(193 233 208 218)(194 222 209 237)(195 211 210 226)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A···20H30A···30N
order122222344444444556666610···101010101012121212151520···2030···30
size111146026610101220303022222442···24444202020204412···124···4

60 irreducible representations

dim11111112222222222444444444
type+++++++++++++++-++-++
imageC1C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C4○D12C4○D20S3×D4D42S3S3×D5D4×D5D42D5C2×S3×D5Dic5.D6Dic3.D10D10⋊D6
kernelDic15.19D4Dic3×Dic5D304C4C3×C23.D5C5×C6.D4C2×C15⋊Q8C2×C157D4C23.D5Dic15C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6C10C6C10C10C23C6C6C22C2C2C2
# reps11211111222144248112222444

Matrix representation of Dic15.19D4 in GL8(𝔽61)

600000000
060000000
006010000
006000000
000018100
0000426000
000000600
000000060
,
609000000
541000000
000600000
006000000
0000181700
0000424300
0000005360
00000048
,
1123000000
1650000000
00100000
00010000
0000434400
0000191800
000000110
000000011
,
500000000
4511000000
006000000
000600000
0000181700
0000424300
0000003450
0000002227

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,54,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,17,43,0,0,0,0,0,0,0,0,53,4,0,0,0,0,0,0,60,8],[11,16,0,0,0,0,0,0,23,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,19,0,0,0,0,0,0,44,18,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[50,45,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,17,43,0,0,0,0,0,0,0,0,34,22,0,0,0,0,0,0,50,27] >;

Dic15.19D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._{19}D_4
% in TeX

G:=Group("Dic15.19D4");
// GroupNames label

G:=SmallGroup(480,602);
// by ID

G=gap.SmallGroup(480,602);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

׿
×
𝔽