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## G = C2×D4⋊2D15order 480 = 25·3·5

### Direct product of C2 and D4⋊2D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D4⋊2D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C4×D15 — C2×D4⋊2D15
 Lower central C15 — C30 — C2×D4⋊2D15
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D42D15
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 1556 in 328 conjugacy classes, 127 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, C2×D42S3, Dic30, C4×D15, C2×Dic15, C2×Dic15, C157D4, C2×C60, D4×C15, C22×D15, C22×C30, C2×D42D5, C2×Dic30, C2×C4×D15, D42D15, C22×Dic15, C2×C157D4, D4×C30, C2×D42D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, D15, C2×C4○D4, C22×D5, D42S3, S3×C23, D30, D42D5, C23×D5, C2×D42S3, C22×D15, C2×D42D5, D42D15, C23×D15, C2×D42D15

Smallest permutation representation of C2×D42D15
On 240 points
Generators in S240
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 31)(30 32)(61 118)(62 119)(63 120)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 91)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 166)(133 167)(134 168)(135 169)(136 162)(137 163)(138 164)(139 165)(140 151)(141 152)(142 153)(143 154)(144 155)(145 156)(146 157)(147 158)(148 159)(149 160)(150 161)(181 237)(182 238)(183 239)(184 240)(185 226)(186 227)(187 228)(188 229)(189 230)(190 231)(191 232)(192 233)(193 234)(194 235)(195 236)(196 214)(197 215)(198 216)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)(205 223)(206 224)(207 225)(208 211)(209 212)(210 213)
(1 87 29 71)(2 88 30 72)(3 89 16 73)(4 90 17 74)(5 76 18 75)(6 77 19 61)(7 78 20 62)(8 79 21 63)(9 80 22 64)(10 81 23 65)(11 82 24 66)(12 83 25 67)(13 84 26 68)(14 85 27 69)(15 86 28 70)(31 113 55 103)(32 114 56 104)(33 115 57 105)(34 116 58 91)(35 117 59 92)(36 118 60 93)(37 119 46 94)(38 120 47 95)(39 106 48 96)(40 107 49 97)(41 108 50 98)(42 109 51 99)(43 110 52 100)(44 111 53 101)(45 112 54 102)(121 204 140 185)(122 205 141 186)(123 206 142 187)(124 207 143 188)(125 208 144 189)(126 209 145 190)(127 210 146 191)(128 196 147 192)(129 197 148 193)(130 198 149 194)(131 199 150 195)(132 200 136 181)(133 201 137 182)(134 202 138 183)(135 203 139 184)(151 226 170 222)(152 227 171 223)(153 228 172 224)(154 229 173 225)(155 230 174 211)(156 231 175 212)(157 232 176 213)(158 233 177 214)(159 234 178 215)(160 235 179 216)(161 236 180 217)(162 237 166 218)(163 238 167 219)(164 239 168 220)(165 240 169 221)
(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 76)(91 116)(92 117)(93 118)(94 119)(95 120)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 136)(133 137)(134 138)(135 139)(151 170)(152 171)(153 172)(154 173)(155 174)(156 175)(157 176)(158 177)(159 178)(160 179)(161 180)(162 166)(163 167)(164 168)(165 169)
(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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 145)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 136)(26 150)(27 149)(28 148)(29 147)(30 146)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 165)(40 164)(41 163)(42 162)(43 161)(44 160)(45 159)(46 171)(47 170)(48 169)(49 168)(50 167)(51 166)(52 180)(53 179)(54 178)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(61 187)(62 186)(63 185)(64 184)(65 183)(66 182)(67 181)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 210)(89 209)(90 208)(91 211)(92 225)(93 224)(94 223)(95 222)(96 221)(97 220)(98 219)(99 218)(100 217)(101 216)(102 215)(103 214)(104 213)(105 212)(106 240)(107 239)(108 238)(109 237)(110 236)(111 235)(112 234)(113 233)(114 232)(115 231)(116 230)(117 229)(118 228)(119 227)(120 226)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 6E 6F 6G 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 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 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 2 2 2 2 30 30 2 2 2 15 15 15 15 30 30 30 30 2 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D15 D30 D30 D30 D4⋊2S3 D4⋊2D5 D4⋊2D15 kernel C2×D4⋊2D15 C2×Dic30 C2×C4×D15 D4⋊2D15 C22×Dic15 C2×C15⋊7D4 D4×C30 D4×C10 C6×D4 C2×C20 C5×D4 C22×C10 C30 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C10 C6 C2 # reps 1 1 1 8 2 2 1 1 2 1 4 2 4 2 8 4 4 4 16 8 2 4 8

Matrix representation of C2×D42D15 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 1 21 0 0 0 58 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 1 21 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 53 38 0 0 0 23 5
,
 60 0 0 0 0 0 50 13 0 0 0 33 11 0 0 0 0 0 6 47 0 0 0 33 55

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,58,0,0,0,21,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,21,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,23,0,0,0,38,5],[60,0,0,0,0,0,50,33,0,0,0,13,11,0,0,0,0,0,6,33,0,0,0,47,55] >;

C2×D42D15 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2D_{15}
% in TeX

G:=Group("C2xD4:2D15");
// GroupNames label

G:=SmallGroup(480,1170);
// by ID

G=gap.SmallGroup(480,1170);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,185,2693,18822]);
// Polycyclic

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

׿
×
𝔽