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## G = Dic15⋊5D4order 480 = 25·3·5

### 5th semidirect product of Dic15 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊5D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — Dic15⋊5D4
 Lower central C15 — C2×C30 — Dic15⋊5D4
 Upper central C1 — C22 — C23

Generators and relations for Dic155D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a11, bc=cb, dbd=a15b, dcd=c-1 >

Subgroups: 1308 in 216 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], D4 [×12], C23, C23 [×3], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C42, C2×D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, D15, C30 [×3], C30, C41D4, D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×2], C22×D5, C22×D5, C22×C10, C22×C10, C4×Dic3, C2×D12, C2×C3⋊D4, C2×C3⋊D4 [×3], C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C6×D5 [×3], S3×C10 [×3], D30 [×3], C2×C30, C2×C30 [×3], C4×Dic5, C2×D20, C2×C5⋊D4, C2×C5⋊D4 [×3], D4×C10, C123D4, C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15, C157D4 [×2], D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, C20⋊D4, Dic3×Dic5, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C6×C5⋊D4, C10×C3⋊D4, C2×C157D4, Dic155D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C41D4, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, D4×D5 [×2], C2×C5⋊D4, C123D4, C2×S3×D5, C20⋊D4, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, Dic155D4

Smallest permutation representation of Dic155D4
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)
(1 105 16 120)(2 104 17 119)(3 103 18 118)(4 102 19 117)(5 101 20 116)(6 100 21 115)(7 99 22 114)(8 98 23 113)(9 97 24 112)(10 96 25 111)(11 95 26 110)(12 94 27 109)(13 93 28 108)(14 92 29 107)(15 91 30 106)(31 83 46 68)(32 82 47 67)(33 81 48 66)(34 80 49 65)(35 79 50 64)(36 78 51 63)(37 77 52 62)(38 76 53 61)(39 75 54 90)(40 74 55 89)(41 73 56 88)(42 72 57 87)(43 71 58 86)(44 70 59 85)(45 69 60 84)(121 219 136 234)(122 218 137 233)(123 217 138 232)(124 216 139 231)(125 215 140 230)(126 214 141 229)(127 213 142 228)(128 212 143 227)(129 211 144 226)(130 240 145 225)(131 239 146 224)(132 238 147 223)(133 237 148 222)(134 236 149 221)(135 235 150 220)(151 193 166 208)(152 192 167 207)(153 191 168 206)(154 190 169 205)(155 189 170 204)(156 188 171 203)(157 187 172 202)(158 186 173 201)(159 185 174 200)(160 184 175 199)(161 183 176 198)(162 182 177 197)(163 181 178 196)(164 210 179 195)(165 209 180 194)
(1 219 31 188)(2 230 32 199)(3 211 33 210)(4 222 34 191)(5 233 35 202)(6 214 36 183)(7 225 37 194)(8 236 38 205)(9 217 39 186)(10 228 40 197)(11 239 41 208)(12 220 42 189)(13 231 43 200)(14 212 44 181)(15 223 45 192)(16 234 46 203)(17 215 47 184)(18 226 48 195)(19 237 49 206)(20 218 50 187)(21 229 51 198)(22 240 52 209)(23 221 53 190)(24 232 54 201)(25 213 55 182)(26 224 56 193)(27 235 57 204)(28 216 58 185)(29 227 59 196)(30 238 60 207)(61 169 113 134)(62 180 114 145)(63 161 115 126)(64 172 116 137)(65 153 117 148)(66 164 118 129)(67 175 119 140)(68 156 120 121)(69 167 91 132)(70 178 92 143)(71 159 93 124)(72 170 94 135)(73 151 95 146)(74 162 96 127)(75 173 97 138)(76 154 98 149)(77 165 99 130)(78 176 100 141)(79 157 101 122)(80 168 102 133)(81 179 103 144)(82 160 104 125)(83 171 105 136)(84 152 106 147)(85 163 107 128)(86 174 108 139)(87 155 109 150)(88 166 110 131)(89 177 111 142)(90 158 112 123)
(1 121)(2 132)(3 143)(4 124)(5 135)(6 146)(7 127)(8 138)(9 149)(10 130)(11 141)(12 122)(13 133)(14 144)(15 125)(16 136)(17 147)(18 128)(19 139)(20 150)(21 131)(22 142)(23 123)(24 134)(25 145)(26 126)(27 137)(28 148)(29 129)(30 140)(31 156)(32 167)(33 178)(34 159)(35 170)(36 151)(37 162)(38 173)(39 154)(40 165)(41 176)(42 157)(43 168)(44 179)(45 160)(46 171)(47 152)(48 163)(49 174)(50 155)(51 166)(52 177)(53 158)(54 169)(55 180)(56 161)(57 172)(58 153)(59 164)(60 175)(61 201)(62 182)(63 193)(64 204)(65 185)(66 196)(67 207)(68 188)(69 199)(70 210)(71 191)(72 202)(73 183)(74 194)(75 205)(76 186)(77 197)(78 208)(79 189)(80 200)(81 181)(82 192)(83 203)(84 184)(85 195)(86 206)(87 187)(88 198)(89 209)(90 190)(91 230)(92 211)(93 222)(94 233)(95 214)(96 225)(97 236)(98 217)(99 228)(100 239)(101 220)(102 231)(103 212)(104 223)(105 234)(106 215)(107 226)(108 237)(109 218)(110 229)(111 240)(112 221)(113 232)(114 213)(115 224)(116 235)(117 216)(118 227)(119 238)(120 219)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A ··· 30N order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 30 ··· 30 size 1 1 1 1 4 12 20 60 2 6 6 10 10 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 4 4 4 4 12 12 12 12 20 20 4 4 12 12 12 12 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C5⋊D4 S3×D4 S3×D5 D4×D5 C2×S3×D5 D5×C3⋊D4 S3×C5⋊D4 D10⋊D6 kernel Dic15⋊5D4 Dic3×Dic5 C2×C15⋊D4 C2×C3⋊D20 C2×C5⋊D12 C6×C5⋊D4 C10×C3⋊D4 C2×C15⋊7D4 C2×C5⋊D4 C5×Dic3 C3×Dic5 Dic15 C2×C3⋊D4 C2×Dic5 C22×D5 C22×C10 C2×Dic3 C22×S3 C22×C6 Dic5 Dic3 C10 C23 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 2 2 2 4 8 2 2 4 2 4 4 4

Matrix representation of Dic155D4 in GL4(𝔽61) generated by

 60 9 0 0 20 2 0 0 0 0 1 60 0 0 19 43
,
 27 21 0 0 3 34 0 0 0 0 8 14 0 0 52 53
,
 34 40 0 0 58 27 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 41 60 0 0 0 0 30 45 0 0 60 31
G:=sub<GL(4,GF(61))| [60,20,0,0,9,2,0,0,0,0,1,19,0,0,60,43],[27,3,0,0,21,34,0,0,0,0,8,52,0,0,14,53],[34,58,0,0,40,27,0,0,0,0,1,0,0,0,0,1],[1,41,0,0,0,60,0,0,0,0,30,60,0,0,45,31] >;

Dic155D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_5D_4
% in TeX

G:=Group("Dic15:5D4");
// GroupNames label

G:=SmallGroup(480,643);
// by ID

G=gap.SmallGroup(480,643);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,303,1356,18822]);
// 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,b*c=c*b,d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽