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## G = D4×C2×C30order 480 = 25·3·5

### Direct product of C2×C30 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C30
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15 — D4×C30 — D4×C2×C30
 Lower central C1 — C2 — D4×C2×C30
 Upper central C1 — C22×C30 — D4×C2×C30

Generators and relations for D4×C2×C30
G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 632 in 472 conjugacy classes, 312 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, C23, C10, C10, C10, C12, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C30, C30, C30, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C22×C12, C6×D4, C23×C6, C60, C2×C30, C2×C30, C22×C20, D4×C10, C23×C10, D4×C2×C6, C2×C60, D4×C15, C22×C30, C22×C30, C22×C30, D4×C2×C10, C22×C60, D4×C30, C23×C30, D4×C2×C30
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C24, C2×C10, C3×D4, C22×C6, C30, C22×D4, C5×D4, C22×C10, C6×D4, C23×C6, C2×C30, D4×C10, C23×C10, D4×C2×C6, D4×C15, C22×C30, D4×C2×C10, D4×C30, C23×C30, D4×C2×C30

Smallest permutation representation of D4×C2×C30
On 240 points
Generators in S240
(1 221)(2 222)(3 223)(4 224)(5 225)(6 226)(7 227)(8 228)(9 229)(10 230)(11 231)(12 232)(13 233)(14 234)(15 235)(16 236)(17 237)(18 238)(19 239)(20 240)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 121)(58 122)(59 123)(60 124)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 91)(90 92)(151 189)(152 190)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)
(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)
(1 123 90 151)(2 124 61 152)(3 125 62 153)(4 126 63 154)(5 127 64 155)(6 128 65 156)(7 129 66 157)(8 130 67 158)(9 131 68 159)(10 132 69 160)(11 133 70 161)(12 134 71 162)(13 135 72 163)(14 136 73 164)(15 137 74 165)(16 138 75 166)(17 139 76 167)(18 140 77 168)(19 141 78 169)(20 142 79 170)(21 143 80 171)(22 144 81 172)(23 145 82 173)(24 146 83 174)(25 147 84 175)(26 148 85 176)(27 149 86 177)(28 150 87 178)(29 121 88 179)(30 122 89 180)(31 94 191 223)(32 95 192 224)(33 96 193 225)(34 97 194 226)(35 98 195 227)(36 99 196 228)(37 100 197 229)(38 101 198 230)(39 102 199 231)(40 103 200 232)(41 104 201 233)(42 105 202 234)(43 106 203 235)(44 107 204 236)(45 108 205 237)(46 109 206 238)(47 110 207 239)(48 111 208 240)(49 112 209 211)(50 113 210 212)(51 114 181 213)(52 115 182 214)(53 116 183 215)(54 117 184 216)(55 118 185 217)(56 119 186 218)(57 120 187 219)(58 91 188 220)(59 92 189 221)(60 93 190 222)
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 140)(32 141)(33 142)(34 143)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(41 150)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 237)(62 238)(63 239)(64 240)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 217)(72 218)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 181)(159 182)(160 183)(161 184)(162 185)(163 186)(164 187)(165 188)(166 189)(167 190)(168 191)(169 192)(170 193)(171 194)(172 195)(173 196)(174 197)(175 198)(176 199)(177 200)(178 201)(179 202)(180 203)

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

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

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

300 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6N 6O ··· 6AD 10A ··· 10AB 10AC ··· 10BH 12A ··· 12H 15A ··· 15H 20A ··· 20P 30A ··· 30BD 30BE ··· 30DP 60A ··· 60AF order 1 2 ··· 2 2 ··· 2 3 3 4 4 4 4 5 5 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 ··· 2 1 1 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 D4 C3×D4 C5×D4 D4×C15 kernel D4×C2×C30 C22×C60 D4×C30 C23×C30 D4×C2×C10 D4×C2×C6 C22×C20 D4×C10 C23×C10 C22×C12 C6×D4 C23×C6 C22×D4 C22×C4 C2×D4 C24 C2×C30 C2×C10 C2×C6 C22 # reps 1 1 12 2 2 4 2 24 4 4 48 8 8 8 96 16 4 8 16 32

Matrix representation of D4×C2×C30 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 0 58 0 0 0 0 47 0 0 0 0 47
,
 60 0 0 0 0 1 0 0 0 0 51 2 0 0 41 10
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 51 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,58,0,0,0,0,47,0,0,0,0,47],[60,0,0,0,0,1,0,0,0,0,51,41,0,0,2,10],[1,0,0,0,0,1,0,0,0,0,60,51,0,0,0,1] >;

D4×C2×C30 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{30}
% in TeX

G:=Group("D4xC2xC30");
// GroupNames label

G:=SmallGroup(480,1181);
// by ID

G=gap.SmallGroup(480,1181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389]);
// Polycyclic

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

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