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## G = C2×D30.5C4order 480 = 25·3·5

### Direct product of C2 and D30.5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D30.5C4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D30.5C4 — C2×D30.5C4
 Lower central C15 — C30 — C2×D30.5C4
 Upper central C1 — C2×C4

Generators and relations for C2×D30.5C4
G = < a,b,c,d | a2=b30=c2=1, d4=b15, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b19, dcd-1=b3c >

Subgroups: 668 in 136 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8⋊D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C2×C8⋊S3, C5×C3⋊C8 [×2], C3×C52C8 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×C8⋊D5, D30.5C4 [×4], C6×C52C8, C10×C3⋊C8, C2×C4×D15, C2×D30.5C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, C8⋊S3 [×2], S3×C2×C4, S3×D5, C8⋊D5 [×2], C2×C4×D5, C2×C8⋊S3, D30.C2 [×2], C2×S3×D5, C2×C8⋊D5, D30.5C4 [×2], C2×D30.C2, C2×D30.5C4

Smallest permutation representation of C2×D30.5C4
On 240 points
Generators in S240
(1 88)(2 89)(3 90)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 91)(57 92)(58 93)(59 94)(60 95)(121 203)(122 204)(123 205)(124 206)(125 207)(126 208)(127 209)(128 210)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 211)(167 212)(168 213)(169 214)(170 215)(171 216)(172 217)(173 218)(174 219)(175 220)(176 221)(177 222)(178 223)(179 224)(180 225)
(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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 90)(29 89)(30 88)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 120)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 210)(141 209)(142 208)(143 207)(144 206)(145 205)(146 204)(147 203)(148 202)(149 201)(150 200)(151 238)(152 237)(153 236)(154 235)(155 234)(156 233)(157 232)(158 231)(159 230)(160 229)(161 228)(162 227)(163 226)(164 225)(165 224)(166 223)(167 222)(168 221)(169 220)(170 219)(171 218)(172 217)(173 216)(174 215)(175 214)(176 213)(177 212)(178 211)(179 240)(180 239)
(1 180 58 142 16 165 43 127)(2 169 59 131 17 154 44 146)(3 158 60 150 18 173 45 135)(4 177 31 139 19 162 46 124)(5 166 32 128 20 151 47 143)(6 155 33 147 21 170 48 132)(7 174 34 136 22 159 49 121)(8 163 35 125 23 178 50 140)(9 152 36 144 24 167 51 129)(10 171 37 133 25 156 52 148)(11 160 38 122 26 175 53 137)(12 179 39 141 27 164 54 126)(13 168 40 130 28 153 55 145)(14 157 41 149 29 172 56 134)(15 176 42 138 30 161 57 123)(61 222 96 191 76 237 111 206)(62 211 97 210 77 226 112 195)(63 230 98 199 78 215 113 184)(64 219 99 188 79 234 114 203)(65 238 100 207 80 223 115 192)(66 227 101 196 81 212 116 181)(67 216 102 185 82 231 117 200)(68 235 103 204 83 220 118 189)(69 224 104 193 84 239 119 208)(70 213 105 182 85 228 120 197)(71 232 106 201 86 217 91 186)(72 221 107 190 87 236 92 205)(73 240 108 209 88 225 93 194)(74 229 109 198 89 214 94 183)(75 218 110 187 90 233 95 202)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A ··· 20H 24A ··· 24H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 12 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 30 30 2 1 1 1 1 30 30 2 2 2 2 2 6 6 6 6 10 10 10 10 2 ··· 2 2 2 2 2 4 4 2 ··· 2 10 ··· 10 4 ··· 4 6 ··· 6 4 ··· 4

84 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 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D5 D6 D6 M4(2) D10 D10 C4×S3 C4×S3 C4×D5 C4×D5 C8⋊S3 C8⋊D5 S3×D5 D30.C2 C2×S3×D5 D30.C2 D30.5C4 kernel C2×D30.5C4 D30.5C4 C6×C5⋊2C8 C10×C3⋊C8 C2×C4×D15 C4×D15 C2×Dic15 C22×D15 C2×C5⋊2C8 C2×C3⋊C8 C5⋊2C8 C2×C20 C30 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C10 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 1 2 2 1 4 4 2 2 2 4 4 8 16 2 2 2 2 8

Matrix representation of C2×D30.5C4 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 240 51 0 0 0 0 0 0 240 192 0 0 0 0 64 2
,
 240 0 0 0 0 0 64 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 49 0 0 0 0 177 239
,
 64 2 0 0 0 0 89 177 0 0 0 0 0 0 240 0 0 0 0 0 190 1 0 0 0 0 0 0 207 14 0 0 0 0 85 34

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,240,64,0,0,0,0,192,2],[240,64,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,177,0,0,0,0,49,239],[64,89,0,0,0,0,2,177,0,0,0,0,0,0,240,190,0,0,0,0,0,1,0,0,0,0,0,0,207,85,0,0,0,0,14,34] >;

C2×D30.5C4 in GAP, Magma, Sage, TeX

C_2\times D_{30}._5C_4
% in TeX

G:=Group("C2xD30.5C4");
// GroupNames label

G:=SmallGroup(480,371);
// by ID

G=gap.SmallGroup(480,371);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,64,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^2=1,d^4=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^19,d*c*d^-1=b^3*c>;
// generators/relations

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