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## G = C2×D30⋊4C4order 480 = 25·3·5

### Direct product of C2 and D30⋊4C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D30⋊4C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D30⋊4C4 — C2×D30⋊4C4
 Lower central C15 — C30 — C2×D30⋊4C4
 Upper central C1 — C23

Generators and relations for C2×D304C4
G = < a,b,c,d | a2=b30=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b25c >

Subgroups: 1628 in 264 conjugacy classes, 92 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], C5, S3 [×4], C6 [×3], C6 [×4], C2×C4 [×8], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], Dic3 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×4], C22×S3 [×10], C22×C6, D15 [×4], C30 [×3], C30 [×4], C2×C22⋊C4, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×4], C22×D5 [×10], C22×C10, D6⋊C4 [×4], C22×Dic3, C22×C12, S3×C23, C5×Dic3 [×2], C3×Dic5 [×2], D30 [×4], D30 [×12], C2×C30, C2×C30 [×6], D10⋊C4 [×4], C22×Dic5, C22×C20, C23×D5, C2×D6⋊C4, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C22×D15 [×6], C22×D15 [×4], C22×C30, C2×D10⋊C4, D304C4 [×4], C2×C6×Dic5, Dic3×C2×C10, C23×D15, C2×D304C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, S3×D5, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D6⋊C4, D30.C2 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C2×S3×D5, C2×D10⋊C4, D304C4 [×4], C2×D30.C2, C2×C3⋊D20, C2×C5⋊D12, C2×D304C4

Smallest permutation representation of C2×D304C4
On 240 points
Generators in S240
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(121 218)(122 219)(123 220)(124 221)(125 222)(126 223)(127 224)(128 225)(129 226)(130 227)(131 228)(132 229)(133 230)(134 231)(135 232)(136 233)(137 234)(138 235)(139 236)(140 237)(141 238)(142 239)(143 240)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 195)(152 196)(153 197)(154 198)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)(161 205)(162 206)(163 207)(164 208)(165 209)(166 210)(167 181)(168 182)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)
(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 92)(2 91)(3 120)(4 119)(5 118)(6 117)(7 116)(8 115)(9 114)(10 113)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 94)(30 93)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(121 229)(122 228)(123 227)(124 226)(125 225)(126 224)(127 223)(128 222)(129 221)(130 220)(131 219)(132 218)(133 217)(134 216)(135 215)(136 214)(137 213)(138 212)(139 211)(140 240)(141 239)(142 238)(143 237)(144 236)(145 235)(146 234)(147 233)(148 232)(149 231)(150 230)(151 192)(152 191)(153 190)(154 189)(155 188)(156 187)(157 186)(158 185)(159 184)(160 183)(161 182)(162 181)(163 210)(164 209)(165 208)(166 207)(167 206)(168 205)(169 204)(170 203)(171 202)(172 201)(173 200)(174 199)(175 198)(176 197)(177 196)(178 195)(179 194)(180 193)
(1 224 37 194)(2 235 38 205)(3 216 39 186)(4 227 40 197)(5 238 41 208)(6 219 42 189)(7 230 43 200)(8 211 44 181)(9 222 45 192)(10 233 46 203)(11 214 47 184)(12 225 48 195)(13 236 49 206)(14 217 50 187)(15 228 51 198)(16 239 52 209)(17 220 53 190)(18 231 54 201)(19 212 55 182)(20 223 56 193)(21 234 57 204)(22 215 58 185)(23 226 59 196)(24 237 60 207)(25 218 31 188)(26 229 32 199)(27 240 33 210)(28 221 34 191)(29 232 35 202)(30 213 36 183)(61 170 118 147)(62 151 119 128)(63 162 120 139)(64 173 91 150)(65 154 92 131)(66 165 93 142)(67 176 94 123)(68 157 95 134)(69 168 96 145)(70 179 97 126)(71 160 98 137)(72 171 99 148)(73 152 100 129)(74 163 101 140)(75 174 102 121)(76 155 103 132)(77 166 104 143)(78 177 105 124)(79 158 106 135)(80 169 107 146)(81 180 108 127)(82 161 109 138)(83 172 110 149)(84 153 111 130)(85 164 112 141)(86 175 113 122)(87 156 114 133)(88 167 115 144)(89 178 116 125)(90 159 117 136)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 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 S3 D4 D5 D6 D6 D10 D10 C4×S3 D12 C3⋊D4 C4×D5 D20 C5⋊D4 S3×D5 D30.C2 C3⋊D20 C5⋊D12 C2×S3×D5 kernel C2×D30⋊4C4 D30⋊4C4 C2×C6×Dic5 Dic3×C2×C10 C23×D15 C22×D15 C22×Dic5 C2×C30 C22×Dic3 C2×Dic5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C10 C2×C10 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 4 2 2 1 4 2 4 4 4 8 8 8 2 4 4 4 2

Matrix representation of C2×D304C4 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 60 1 0 0 0 0 42 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 42 1
,
 11 0 0 0 0 0 0 60 0 0 0 0 0 0 52 43 0 0 0 0 52 9 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,60,42,0,0,0,0,1,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,42,0,0,0,0,0,1],[11,0,0,0,0,0,0,60,0,0,0,0,0,0,52,52,0,0,0,0,43,9,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C2×D304C4 in GAP, Magma, Sage, TeX

C_2\times D_{30}\rtimes_4C_4
% in TeX

G:=Group("C2xD30:4C4");
// GroupNames label

G:=SmallGroup(480,616);
// by ID

G=gap.SmallGroup(480,616);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,176,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^25*c>;
// generators/relations

׿
×
𝔽