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## G = Dic15.10D4order 480 = 25·3·5

### 10th non-split extension by Dic15 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15.10D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — Dic15.10D4
 Lower central C15 — C2×C30 — Dic15.10D4
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic15.10D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, bc=cb, dbd=a15b, dcd=a15c-1 >

Subgroups: 812 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, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], 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, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30 [×3], C4.4D4, Dic10 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×3], S3×C10 [×3], C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.11D6, C15⋊D4 [×2], C6×Dic5, C10×Dic3, Dic30 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, Dic5.5D4, Dic3×Dic5, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C2×C15⋊D4, C2×Dic30, Dic15.10D4
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, D205S3, D125D5, D10⋊D6, Dic15.10D4

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 15 15 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 12 20 2 4 6 6 10 10 30 30 60 2 2 2 2 2 20 20 2 ··· 2 12 12 12 12 4 4 20 20 4 4 4 4 4 4 12 12 12 12 4 ··· 4 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 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + + - + - - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 D10 C4○D12 C4○D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 D4⋊2D5 C2×S3×D5 D20⋊5S3 D12⋊5D5 D10⋊D6 kernel Dic15.10D4 Dic3×Dic5 D10⋊Dic3 D6⋊Dic5 C3×D10⋊C4 C5×D6⋊C4 C2×C15⋊D4 C2×Dic30 D10⋊C4 Dic15 D6⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 2 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic15.10D4 in GL4(𝔽61) generated by

 0 60 0 0 1 18 0 0 0 0 14 0 0 0 2 48
,
 30 44 0 0 53 31 0 0 0 0 32 56 0 0 22 29
,
 36 4 0 0 27 25 0 0 0 0 50 0 0 0 0 50
,
 60 0 0 0 18 1 0 0 0 0 1 0 0 0 25 60
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,14,2,0,0,0,48],[30,53,0,0,44,31,0,0,0,0,32,22,0,0,56,29],[36,27,0,0,4,25,0,0,0,0,50,0,0,0,0,50],[60,18,0,0,0,1,0,0,0,0,1,25,0,0,0,60] >;

Dic15.10D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._{10}D_4
% in TeX

G:=Group("Dic15.10D4");
// GroupNames label

G:=SmallGroup(480,538);
// by ID

G=gap.SmallGroup(480,538);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,303,58,1356,18822]);
// 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^19,b*c=c*b,d*b*d=a^15*b,d*c*d=a^15*c^-1>;
// generators/relations

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