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## G = C2×C15⋊D8order 480 = 25·3·5

### Direct product of C2 and C15⋊D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C15⋊D8
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C15⋊D8 — C2×C15⋊D8
 Lower central C15 — C30 — C60 — C2×C15⋊D8
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C15⋊D8
G = < a,b,c,d | a2=b15=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 764 in 152 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C10, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C20, D10, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, C30, C30, C2×D8, C52C8, D20, D20, C2×C20, C5×D4, C22×D5, C22×C10, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C60, C6×D5, S3×C10, C2×C30, C2×C52C8, D4⋊D5, C2×D20, D4×C10, C2×D4⋊S3, C153C8, C3×D20, C3×D20, C5×D12, C5×D12, C2×C60, D5×C2×C6, S3×C2×C10, C2×D4⋊D5, C15⋊D8, C2×C153C8, C6×D20, C10×D12, C2×C15⋊D8
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C3⋊D4, C22×S3, C2×D8, C5⋊D4, C22×D5, D4⋊S3, C2×C3⋊D4, S3×D5, D4⋊D5, C2×C5⋊D4, C2×D4⋊S3, C15⋊D4, C2×S3×D5, C2×D4⋊D5, C15⋊D8, C2×C15⋊D4, C2×C15⋊D8

Smallest permutation representation of C2×C15⋊D8
On 240 points
Generators in S240
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 61)(14 62)(15 63)(16 86)(17 87)(18 88)(19 89)(20 90)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 118)(47 119)(48 120)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(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)(151 224)(152 225)(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)(166 235)(167 236)(168 237)(169 238)(170 239)(171 240)(172 226)(173 227)(174 228)(175 229)(176 230)(177 231)(178 232)(179 233)(180 234)
(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)
(1 190 35 233 19 196 59 218)(2 189 36 232 20 210 60 217)(3 188 37 231 21 209 46 216)(4 187 38 230 22 208 47 215)(5 186 39 229 23 207 48 214)(6 185 40 228 24 206 49 213)(7 184 41 227 25 205 50 212)(8 183 42 226 26 204 51 211)(9 182 43 240 27 203 52 225)(10 181 44 239 28 202 53 224)(11 195 45 238 29 201 54 223)(12 194 31 237 30 200 55 222)(13 193 32 236 16 199 56 221)(14 192 33 235 17 198 57 220)(15 191 34 234 18 197 58 219)(61 122 98 167 86 139 113 163)(62 121 99 166 87 138 114 162)(63 135 100 180 88 137 115 161)(64 134 101 179 89 136 116 160)(65 133 102 178 90 150 117 159)(66 132 103 177 76 149 118 158)(67 131 104 176 77 148 119 157)(68 130 105 175 78 147 120 156)(69 129 91 174 79 146 106 155)(70 128 92 173 80 145 107 154)(71 127 93 172 81 144 108 153)(72 126 94 171 82 143 109 152)(73 125 95 170 83 142 110 151)(74 124 96 169 84 141 111 165)(75 123 97 168 85 140 112 164)
(1 64)(2 68)(3 72)(4 61)(5 65)(6 69)(7 73)(8 62)(9 66)(10 70)(11 74)(12 63)(13 67)(14 71)(15 75)(16 77)(17 81)(18 85)(19 89)(20 78)(21 82)(22 86)(23 90)(24 79)(25 83)(26 87)(27 76)(28 80)(29 84)(30 88)(31 115)(32 119)(33 108)(34 112)(35 116)(36 120)(37 109)(38 113)(39 117)(40 106)(41 110)(42 114)(43 118)(44 107)(45 111)(46 94)(47 98)(48 102)(49 91)(50 95)(51 99)(52 103)(53 92)(54 96)(55 100)(56 104)(57 93)(58 97)(59 101)(60 105)(121 211)(122 215)(123 219)(124 223)(125 212)(126 216)(127 220)(128 224)(129 213)(130 217)(131 221)(132 225)(133 214)(134 218)(135 222)(136 233)(137 237)(138 226)(139 230)(140 234)(141 238)(142 227)(143 231)(144 235)(145 239)(146 228)(147 232)(148 236)(149 240)(150 229)(151 184)(152 188)(153 192)(154 181)(155 185)(156 189)(157 193)(158 182)(159 186)(160 190)(161 194)(162 183)(163 187)(164 191)(165 195)(166 204)(167 208)(168 197)(169 201)(170 205)(171 209)(172 198)(173 202)(174 206)(175 210)(176 199)(177 203)(178 207)(179 196)(180 200)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 C3⋊D4 C3⋊D4 C5⋊D4 C5⋊D4 D4⋊S3 S3×D5 D4⋊D5 C15⋊D4 C2×S3×D5 C15⋊D4 C15⋊D8 kernel C2×C15⋊D8 C15⋊D8 C2×C15⋊3C8 C6×D20 C10×D12 C2×D20 C60 C2×C30 C2×D12 D20 C2×C20 C30 D12 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 2 2 4 2 2 2 8

Matrix representation of C2×C15⋊D8 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 190 0 0 0 0 51 190 0 0 0 0 0 0 15 0 0 0 0 0 172 225
,
 230 11 0 0 0 0 230 230 0 0 0 0 0 0 165 172 0 0 0 0 192 76 0 0 0 0 0 0 2 193 0 0 0 0 226 239
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 51 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,51,0,0,0,0,190,190,0,0,0,0,0,0,15,172,0,0,0,0,0,225],[230,230,0,0,0,0,11,230,0,0,0,0,0,0,165,192,0,0,0,0,172,76,0,0,0,0,0,0,2,226,0,0,0,0,193,239],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,51,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

C2×C15⋊D8 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes D_8
% in TeX

G:=Group("C2xC15:D8");
// GroupNames label

G:=SmallGroup(480,372);
// by ID

G=gap.SmallGroup(480,372);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^15=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;
// generators/relations

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