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## G = C4×C15⋊D4order 480 = 25·3·5

### Direct product of C4 and C15⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C4×C15⋊D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — C4×C15⋊D4
 Lower central C15 — C30 — C4×C15⋊D4
 Upper central C1 — C2×C4

Generators and relations for C4×C15⋊D4
G = < a,b,c,d | a4=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 796 in 188 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×4], C12 [×2], C12, D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×C3⋊D4, C15⋊D4 [×4], D5×C12 [×2], C6×Dic5, S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, C4×C5⋊D4, D10⋊Dic3, D6⋊Dic5, Dic155C4, C4×Dic15, C2×C15⋊D4, D5×C2×C12, S3×C2×C20, C4×C15⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C3⋊D4, C15⋊D4 [×2], C2×S3×D5, C4×C5⋊D4, D6.D10, C4×S3×D5, C2×C15⋊D4, C4×C15⋊D4

Smallest permutation representation of C4×C15⋊D4
On 240 points
Generators in S240
(1 104 42 68)(2 105 43 69)(3 91 44 70)(4 92 45 71)(5 93 31 72)(6 94 32 73)(7 95 33 74)(8 96 34 75)(9 97 35 61)(10 98 36 62)(11 99 37 63)(12 100 38 64)(13 101 39 65)(14 102 40 66)(15 103 41 67)(16 115 50 88)(17 116 51 89)(18 117 52 90)(19 118 53 76)(20 119 54 77)(21 120 55 78)(22 106 56 79)(23 107 57 80)(24 108 58 81)(25 109 59 82)(26 110 60 83)(27 111 46 84)(28 112 47 85)(29 113 48 86)(30 114 49 87)(121 215 155 185)(122 216 156 186)(123 217 157 187)(124 218 158 188)(125 219 159 189)(126 220 160 190)(127 221 161 191)(128 222 162 192)(129 223 163 193)(130 224 164 194)(131 225 165 195)(132 211 151 181)(133 212 152 182)(134 213 153 183)(135 214 154 184)(136 230 172 198)(137 231 173 199)(138 232 174 200)(139 233 175 201)(140 234 176 202)(141 235 177 203)(142 236 178 204)(143 237 179 205)(144 238 180 206)(145 239 166 207)(146 240 167 208)(147 226 168 209)(148 227 169 210)(149 228 170 196)(150 229 171 197)
(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)
(1 162 29 166)(2 161 30 180)(3 160 16 179)(4 159 17 178)(5 158 18 177)(6 157 19 176)(7 156 20 175)(8 155 21 174)(9 154 22 173)(10 153 23 172)(11 152 24 171)(12 151 25 170)(13 165 26 169)(14 164 27 168)(15 163 28 167)(31 124 52 141)(32 123 53 140)(33 122 54 139)(34 121 55 138)(35 135 56 137)(36 134 57 136)(37 133 58 150)(38 132 59 149)(39 131 60 148)(40 130 46 147)(41 129 47 146)(42 128 48 145)(43 127 49 144)(44 126 50 143)(45 125 51 142)(61 214 79 231)(62 213 80 230)(63 212 81 229)(64 211 82 228)(65 225 83 227)(66 224 84 226)(67 223 85 240)(68 222 86 239)(69 221 87 238)(70 220 88 237)(71 219 89 236)(72 218 90 235)(73 217 76 234)(74 216 77 233)(75 215 78 232)(91 190 115 205)(92 189 116 204)(93 188 117 203)(94 187 118 202)(95 186 119 201)(96 185 120 200)(97 184 106 199)(98 183 107 198)(99 182 108 197)(100 181 109 196)(101 195 110 210)(102 194 111 209)(103 193 112 208)(104 192 113 207)(105 191 114 206)
(1 42)(2 31)(3 35)(4 39)(5 43)(6 32)(7 36)(8 40)(9 44)(10 33)(11 37)(12 41)(13 45)(14 34)(15 38)(16 56)(17 60)(18 49)(19 53)(20 57)(21 46)(22 50)(23 54)(24 58)(25 47)(26 51)(27 55)(28 59)(29 48)(30 52)(61 91)(62 95)(63 99)(64 103)(65 92)(66 96)(67 100)(68 104)(69 93)(70 97)(71 101)(72 105)(73 94)(74 98)(75 102)(76 118)(77 107)(78 111)(79 115)(80 119)(81 108)(82 112)(83 116)(84 120)(85 109)(86 113)(87 117)(88 106)(89 110)(90 114)(121 168)(122 172)(123 176)(124 180)(125 169)(126 173)(127 177)(128 166)(129 170)(130 174)(131 178)(132 167)(133 171)(134 175)(135 179)(136 156)(137 160)(138 164)(139 153)(140 157)(141 161)(142 165)(143 154)(144 158)(145 162)(146 151)(147 155)(148 159)(149 163)(150 152)(181 240)(182 229)(183 233)(184 237)(185 226)(186 230)(187 234)(188 238)(189 227)(190 231)(191 235)(192 239)(193 228)(194 232)(195 236)(196 223)(197 212)(198 216)(199 220)(200 224)(201 213)(202 217)(203 221)(204 225)(205 214)(206 218)(207 222)(208 211)(209 215)(210 219)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 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 C2 C2 C2 C4 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 D10 C3⋊D4 C4×S3 C5⋊D4 C4×D5 C4○D12 C4○D20 S3×D5 C15⋊D4 C2×S3×D5 D6.D10 C4×S3×D5 kernel C4×C15⋊D4 D10⋊Dic3 D6⋊Dic5 Dic15⋊5C4 C4×Dic15 C2×C15⋊D4 D5×C2×C12 S3×C2×C20 C15⋊D4 C2×C4×D5 C60 S3×C2×C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 C20 D10 C12 D6 C10 C6 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 1 1 2 2 2 2 4 4 8 8 4 8 2 4 2 4 4

Matrix representation of C4×C15⋊D4 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 0 47 0 0 0 0 0 44 0 0 18 43
,
 0 1 0 0 60 0 0 0 0 0 53 47 0 0 9 8
,
 60 0 0 0 0 1 0 0 0 0 43 1 0 0 43 18
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,47,0,0,0,0,0,18,0,0,44,43],[0,60,0,0,1,0,0,0,0,0,53,9,0,0,47,8],[60,0,0,0,0,1,0,0,0,0,43,43,0,0,1,18] >;

C4×C15⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_{15}\rtimes D_4
% in TeX

G:=Group("C4xC15:D4");
// GroupNames label

G:=SmallGroup(480,515);
// by ID

G=gap.SmallGroup(480,515);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^15=c^4=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

׿
×
𝔽