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## G = C2×D4.D15order 480 = 25·3·5

### Direct product of C2 and D4.D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×D4.D15
 Chief series C1 — C5 — C15 — C30 — C60 — Dic30 — C2×Dic30 — C2×D4.D15
 Lower central C15 — C30 — C60 — C2×D4.D15
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4.D15
G = < a,b,c,d,e | a2=b4=c2=d15=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 596 in 136 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30, C30 [×2], C30 [×2], C2×SD16, C52C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C3⋊C8, D4.S3 [×4], C2×Dic6, C6×D4, Dic15 [×2], C60 [×2], C2×C30, C2×C30 [×4], C2×C52C8, D4.D5 [×4], C2×Dic10, D4×C10, C2×D4.S3, C153C8 [×2], Dic30 [×2], Dic30, C2×Dic15, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C2×D4.D5, C2×C153C8, D4.D15 [×4], C2×Dic30, D4×C30, C2×D4.D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C2×SD16, C5⋊D4 [×2], C22×D5, D4.S3 [×2], C2×C3⋊D4, D30 [×3], D4.D5 [×2], C2×C5⋊D4, C2×D4.S3, C157D4 [×2], C22×D15, C2×D4.D5, D4.D15 [×2], C2×C157D4, C2×D4.D15

Smallest permutation representation of C2×D4.D15
On 240 points
Generators in S240
(1 75)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 106)(46 104)(47 105)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(121 192)(122 193)(123 194)(124 195)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(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)(166 219)(167 220)(168 221)(169 222)(170 223)(171 224)(172 225)(173 211)(174 212)(175 213)(176 214)(177 215)(178 216)(179 217)(180 218)
(1 54 24 34)(2 55 25 35)(3 56 26 36)(4 57 27 37)(5 58 28 38)(6 59 29 39)(7 60 30 40)(8 46 16 41)(9 47 17 42)(10 48 18 43)(11 49 19 44)(12 50 20 45)(13 51 21 31)(14 52 22 32)(15 53 23 33)(61 98 77 111)(62 99 78 112)(63 100 79 113)(64 101 80 114)(65 102 81 115)(66 103 82 116)(67 104 83 117)(68 105 84 118)(69 91 85 119)(70 92 86 120)(71 93 87 106)(72 94 88 107)(73 95 89 108)(74 96 90 109)(75 97 76 110)(121 159 144 177)(122 160 145 178)(123 161 146 179)(124 162 147 180)(125 163 148 166)(126 164 149 167)(127 165 150 168)(128 151 136 169)(129 152 137 170)(130 153 138 171)(131 154 139 172)(132 155 140 173)(133 156 141 174)(134 157 142 175)(135 158 143 176)(181 238 208 219)(182 239 209 220)(183 240 210 221)(184 226 196 222)(185 227 197 223)(186 228 198 224)(187 229 199 225)(188 230 200 211)(189 231 201 212)(190 232 202 213)(191 233 203 214)(192 234 204 215)(193 235 205 216)(194 236 206 217)(195 237 207 218)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 31)(14 32)(15 33)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 106)(72 107)(73 108)(74 109)(75 110)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 136)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(181 208)(182 209)(183 210)(184 196)(185 197)(186 198)(187 199)(188 200)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)
(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)
(1 147 24 124)(2 146 25 123)(3 145 26 122)(4 144 27 121)(5 143 28 135)(6 142 29 134)(7 141 30 133)(8 140 16 132)(9 139 17 131)(10 138 18 130)(11 137 19 129)(12 136 20 128)(13 150 21 127)(14 149 22 126)(15 148 23 125)(31 168 51 165)(32 167 52 164)(33 166 53 163)(34 180 54 162)(35 179 55 161)(36 178 56 160)(37 177 57 159)(38 176 58 158)(39 175 59 157)(40 174 60 156)(41 173 46 155)(42 172 47 154)(43 171 48 153)(44 170 49 152)(45 169 50 151)(61 206 77 194)(62 205 78 193)(63 204 79 192)(64 203 80 191)(65 202 81 190)(66 201 82 189)(67 200 83 188)(68 199 84 187)(69 198 85 186)(70 197 86 185)(71 196 87 184)(72 210 88 183)(73 209 89 182)(74 208 90 181)(75 207 76 195)(91 228 119 224)(92 227 120 223)(93 226 106 222)(94 240 107 221)(95 239 108 220)(96 238 109 219)(97 237 110 218)(98 236 111 217)(99 235 112 216)(100 234 113 215)(101 233 114 214)(102 232 115 213)(103 231 116 212)(104 230 117 211)(105 229 118 225)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 2 2 2 60 60 2 2 2 2 2 4 4 4 4 30 30 30 30 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 SD16 D10 D10 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 D30 C15⋊7D4 C15⋊7D4 D4.S3 D4.D5 D4.D15 kernel C2×D4.D15 C2×C15⋊3C8 D4.D15 C2×Dic30 D4×C30 D4×C10 C60 C2×C30 C6×D4 C2×C20 C5×D4 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C10 C6 C2 # reps 1 1 4 1 1 1 1 1 2 1 2 4 2 4 2 2 4 4 4 4 8 8 8 2 4 8

Matrix representation of C2×D4.D15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240
,
 240 0 0 0 0 240 0 0 0 0 0 1 0 0 240 0
,
 240 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 231 0 0 0 0 24 0 0 0 0 1 0 0 0 0 1
,
 0 24 0 0 231 0 0 0 0 0 19 222 0 0 222 222
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[231,0,0,0,0,24,0,0,0,0,1,0,0,0,0,1],[0,231,0,0,24,0,0,0,0,0,19,222,0,0,222,222] >;

C2×D4.D15 in GAP, Magma, Sage, TeX

C_2\times D_4.D_{15}
% in TeX

G:=Group("C2xD4.D15");
// GroupNames label

G:=SmallGroup(480,898);
// by ID

G=gap.SmallGroup(480,898);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,675,185,80,2693,18822]);
// Polycyclic

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

׿
×
𝔽