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G = D4×Dic15order 480 = 25·3·5

Direct product of D4 and Dic15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×Dic15, C23.21D30, C1543(C4×D4), C35(D4×Dic5), C56(D4×Dic3), C6013(C2×C4), (D4×C15)⋊9C4, (C6×D4).4D5, C2.5(D4×D15), (C3×D4)⋊3Dic5, (C5×D4)⋊6Dic3, C605C419C2, (D4×C30).4C2, (C2×D4).7D15, (D4×C10).4S3, C41(C2×Dic15), C123(C2×Dic5), C206(C2×Dic3), (C2×C4).49D30, C6.118(D4×D5), (C4×Dic15)⋊4C2, C10.120(S3×D4), C30.325(C2×D4), (C2×C20).147D6, C30.38D47C2, (C2×C12).146D10, (C2×C60).73C22, C222(C2×Dic15), (C22×C6).62D10, (C22×C10).77D6, C30.222(C4○D4), C2.5(D42D15), C30.215(C22×C4), (C2×C30).305C23, (C22×Dic15)⋊4C2, C6.101(D42D5), C2.6(C22×Dic15), C6.26(C22×Dic5), (C22×C30).18C22, C10.101(D42S3), C10.39(C22×Dic3), C22.25(C22×D15), (C2×Dic15).171C22, (C2×C30)⋊13(C2×C4), (C2×C6)⋊3(C2×Dic5), (C2×C10)⋊10(C2×Dic3), (C2×C6).301(C22×D5), (C2×C10).300(C22×S3), SmallGroup(480,899)

Series: Derived Chief Lower central Upper central

C1C30 — D4×Dic15
C1C5C15C30C2×C30C2×Dic15C22×Dic15 — D4×Dic15
C15C30 — D4×Dic15
C1C22C2×D4

Generators and relations for D4×Dic15
 G = < a,b,c,d | a4=b2=c30=1, d2=c15, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 788 in 188 conjugacy classes, 89 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×5], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×Dic3 [×8], C2×C12, C3×D4 [×4], C22×C6 [×2], C30 [×3], C30 [×4], C4×D4, C2×Dic5 [×8], C2×C20, C5×D4 [×4], C22×C10 [×2], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, Dic15 [×2], Dic15 [×3], C60 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5 [×2], D4×C10, D4×Dic3, C2×Dic15 [×2], C2×Dic15 [×2], C2×Dic15 [×4], C2×C60, D4×C15 [×4], C22×C30 [×2], D4×Dic5, C4×Dic15, C605C4, C30.38D4 [×2], C22×Dic15 [×2], D4×C30, D4×Dic15
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C2×Dic3 [×6], C22×S3, D15, C4×D4, C2×Dic5 [×6], C22×D5, S3×D4, D42S3, C22×Dic3, Dic15 [×4], D30 [×3], D4×D5, D42D5, C22×Dic5, D4×Dic3, C2×Dic15 [×6], C22×D15, D4×Dic5, D4×D15, D42D15, C22×Dic15, D4×Dic15

Smallest permutation representation of D4×Dic15
On 240 points
Generators in S240
(1 74 204 135)(2 75 205 136)(3 76 206 137)(4 77 207 138)(5 78 208 139)(6 79 209 140)(7 80 210 141)(8 81 181 142)(9 82 182 143)(10 83 183 144)(11 84 184 145)(12 85 185 146)(13 86 186 147)(14 87 187 148)(15 88 188 149)(16 89 189 150)(17 90 190 121)(18 61 191 122)(19 62 192 123)(20 63 193 124)(21 64 194 125)(22 65 195 126)(23 66 196 127)(24 67 197 128)(25 68 198 129)(26 69 199 130)(27 70 200 131)(28 71 201 132)(29 72 202 133)(30 73 203 134)(31 166 104 239)(32 167 105 240)(33 168 106 211)(34 169 107 212)(35 170 108 213)(36 171 109 214)(37 172 110 215)(38 173 111 216)(39 174 112 217)(40 175 113 218)(41 176 114 219)(42 177 115 220)(43 178 116 221)(44 179 117 222)(45 180 118 223)(46 151 119 224)(47 152 120 225)(48 153 91 226)(49 154 92 227)(50 155 93 228)(51 156 94 229)(52 157 95 230)(53 158 96 231)(54 159 97 232)(55 160 98 233)(56 161 99 234)(57 162 100 235)(58 163 101 236)(59 164 102 237)(60 165 103 238)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(151 239)(152 240)(153 211)(154 212)(155 213)(156 214)(157 215)(158 216)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 225)(168 226)(169 227)(170 228)(171 229)(172 230)(173 231)(174 232)(175 233)(176 234)(177 235)(178 236)(179 237)(180 238)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)
(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)
(1 119 16 104)(2 118 17 103)(3 117 18 102)(4 116 19 101)(5 115 20 100)(6 114 21 99)(7 113 22 98)(8 112 23 97)(9 111 24 96)(10 110 25 95)(11 109 26 94)(12 108 27 93)(13 107 28 92)(14 106 29 91)(15 105 30 120)(31 204 46 189)(32 203 47 188)(33 202 48 187)(34 201 49 186)(35 200 50 185)(36 199 51 184)(37 198 52 183)(38 197 53 182)(39 196 54 181)(40 195 55 210)(41 194 56 209)(42 193 57 208)(43 192 58 207)(44 191 59 206)(45 190 60 205)(61 237 76 222)(62 236 77 221)(63 235 78 220)(64 234 79 219)(65 233 80 218)(66 232 81 217)(67 231 82 216)(68 230 83 215)(69 229 84 214)(70 228 85 213)(71 227 86 212)(72 226 87 211)(73 225 88 240)(74 224 89 239)(75 223 90 238)(121 165 136 180)(122 164 137 179)(123 163 138 178)(124 162 139 177)(125 161 140 176)(126 160 141 175)(127 159 142 174)(128 158 143 173)(129 157 144 172)(130 156 145 171)(131 155 146 170)(132 154 147 169)(133 153 148 168)(134 152 149 167)(135 151 150 166)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G···4L5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order1222222234444444···455666666610···1010···101212151515152020202030···3030···3060···60
size111122222221515151530···302222244442···24···444222244442···24···44···4

90 irreducible representations

dim111111122222222222222444444
type++++++++++-++-+++-++-+-+-
imageC1C2C2C2C2C2C4S3D4D5D6Dic3D6C4○D4D10Dic5D10D15D30Dic15D30S3×D4D42S3D4×D5D42D5D4×D15D42D15
kernelD4×Dic15C4×Dic15C605C4C30.38D4C22×Dic15D4×C30D4×C15D4×C10Dic15C6×D4C2×C20C5×D4C22×C10C30C2×C12C3×D4C22×C6C2×D4C2×C4D4C23C10C10C6C6C2C2
# reps1112218122142228444168112244

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

60000
06000
0012
006060
,
60000
06000
0010
006060
,
523700
292300
0010
0001
,
304500
223100
00600
00060
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[52,29,0,0,37,23,0,0,0,0,1,0,0,0,0,1],[30,22,0,0,45,31,0,0,0,0,60,0,0,0,0,60] >;

D4×Dic15 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_{15}
% in TeX

G:=Group("D4xDic15");
// GroupNames label

G:=SmallGroup(480,899);
// by ID

G=gap.SmallGroup(480,899);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,2693,18822]);
// Polycyclic

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

׿
×
𝔽