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## G = C5×C12⋊D4order 480 = 25·3·5

### Direct product of C5 and C12⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C2×C20 — C5×C12⋊D4
 Lower central C3 — C2×C6 — C5×C12⋊D4
 Upper central C1 — C2×C10 — C5×C4⋊C4

Generators and relations for C5×C12⋊D4
G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >

Subgroups: 532 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], C10 [×3], C10 [×4], Dic3, C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C20 [×2], C20 [×3], C2×C10, C2×C10 [×10], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3 [×2], C5×S3 [×4], C30 [×3], C4⋊D4, C2×C20, C2×C20 [×2], C2×C20 [×3], C5×D4 [×6], C22×C10 [×3], D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C5×Dic3, C60 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×8], C2×C30, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, D4×C10 [×3], C12⋊D4, S3×C20 [×2], C5×D12 [×6], C10×Dic3, C2×C60, C2×C60 [×2], S3×C2×C10, S3×C2×C10 [×2], C5×C4⋊D4, C5×D6⋊C4 [×2], C15×C4⋊C4, S3×C2×C20, C10×D12, C10×D12 [×2], C5×C12⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×4], C23, C10 [×7], D6 [×3], C2×D4 [×2], C4○D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C4⋊D4, C5×D4 [×4], C22×C10, C2×D12, S3×D4, Q83S3, S3×C10 [×3], D4×C10 [×2], C5×C4○D4, C12⋊D4, C5×D12 [×2], S3×C2×C10, C5×C4⋊D4, C10×D12, C5×S3×D4, C5×Q83S3, C5×C12⋊D4

Smallest permutation representation of C5×C12⋊D4
On 240 points
Generators in S240
(1 25 68 170 194)(2 26 69 171 195)(3 27 70 172 196)(4 28 71 173 197)(5 29 72 174 198)(6 30 61 175 199)(7 31 62 176 200)(8 32 63 177 201)(9 33 64 178 202)(10 34 65 179 203)(11 35 66 180 204)(12 36 67 169 193)(13 56 148 125 183)(14 57 149 126 184)(15 58 150 127 185)(16 59 151 128 186)(17 60 152 129 187)(18 49 153 130 188)(19 50 154 131 189)(20 51 155 132 190)(21 52 156 121 191)(22 53 145 122 192)(23 54 146 123 181)(24 55 147 124 182)(37 77 206 164 238)(38 78 207 165 239)(39 79 208 166 240)(40 80 209 167 229)(41 81 210 168 230)(42 82 211 157 231)(43 83 212 158 232)(44 84 213 159 233)(45 73 214 160 234)(46 74 215 161 235)(47 75 216 162 236)(48 76 205 163 237)(85 114 226 100 138)(86 115 227 101 139)(87 116 228 102 140)(88 117 217 103 141)(89 118 218 104 142)(90 119 219 105 143)(91 120 220 106 144)(92 109 221 107 133)(93 110 222 108 134)(94 111 223 97 135)(95 112 224 98 136)(96 113 225 99 137)
(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)
(1 238 115 24)(2 233 116 19)(3 240 117 14)(4 235 118 21)(5 230 119 16)(6 237 120 23)(7 232 109 18)(8 239 110 13)(9 234 111 20)(10 229 112 15)(11 236 113 22)(12 231 114 17)(25 37 227 55)(26 44 228 50)(27 39 217 57)(28 46 218 52)(29 41 219 59)(30 48 220 54)(31 43 221 49)(32 38 222 56)(33 45 223 51)(34 40 224 58)(35 47 225 53)(36 42 226 60)(61 76 106 146)(62 83 107 153)(63 78 108 148)(64 73 97 155)(65 80 98 150)(66 75 99 145)(67 82 100 152)(68 77 101 147)(69 84 102 154)(70 79 103 149)(71 74 104 156)(72 81 105 151)(85 187 193 157)(86 182 194 164)(87 189 195 159)(88 184 196 166)(89 191 197 161)(90 186 198 168)(91 181 199 163)(92 188 200 158)(93 183 201 165)(94 190 202 160)(95 185 203 167)(96 192 204 162)(121 173 215 142)(122 180 216 137)(123 175 205 144)(124 170 206 139)(125 177 207 134)(126 172 208 141)(127 179 209 136)(128 174 210 143)(129 169 211 138)(130 176 212 133)(131 171 213 140)(132 178 214 135)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 60)(33 59)(34 58)(35 57)(36 56)(37 227)(38 226)(39 225)(40 224)(41 223)(42 222)(43 221)(44 220)(45 219)(46 218)(47 217)(48 228)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 156)(72 155)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 108)(83 107)(84 106)(85 165)(86 164)(87 163)(88 162)(89 161)(90 160)(91 159)(92 158)(93 157)(94 168)(95 167)(96 166)(109 232)(110 231)(111 230)(112 229)(113 240)(114 239)(115 238)(116 237)(117 236)(118 235)(119 234)(120 233)(121 173)(122 172)(123 171)(124 170)(125 169)(126 180)(127 179)(128 178)(129 177)(130 176)(131 175)(132 174)(133 212)(134 211)(135 210)(136 209)(137 208)(138 207)(139 206)(140 205)(141 216)(142 215)(143 214)(144 213)(181 195)(182 194)(183 193)(184 204)(185 203)(186 202)(187 201)(188 200)(189 199)(190 198)(191 197)(192 196)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10T 10U ··· 10AB 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20X 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 12 12 2 2 2 4 4 6 6 1 1 1 1 2 2 2 1 ··· 1 6 ··· 6 12 ··· 12 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 6 ··· 6 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D4 D6 C4○D4 D12 C5×S3 C5×D4 C5×D4 S3×C10 C5×C4○D4 C5×D12 S3×D4 Q8⋊3S3 C5×S3×D4 C5×Q8⋊3S3 kernel C5×C12⋊D4 C5×D6⋊C4 C15×C4⋊C4 S3×C2×C20 C10×D12 C12⋊D4 D6⋊C4 C3×C4⋊C4 S3×C2×C4 C2×D12 C5×C4⋊C4 C60 S3×C10 C2×C20 C30 C20 C4⋊C4 C12 D6 C2×C4 C6 C4 C10 C10 C2 C2 # reps 1 2 1 1 3 4 8 4 4 12 1 2 2 3 2 4 4 8 8 12 8 16 1 1 4 4

Matrix representation of C5×C12⋊D4 in GL6(𝔽61)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 58 0 0 0 0 0 0 58 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 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 50 0 0 0 0 0 0 11
,
 53 59 0 0 0 0 2 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 0
,
 53 59 0 0 0 0 1 8 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[53,2,0,0,0,0,59,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[53,1,0,0,0,0,59,8,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C5×C12⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}\rtimes D_4
% in TeX

G:=Group("C5xC12:D4");
// GroupNames label

G:=SmallGroup(480,774);
// by ID

G=gap.SmallGroup(480,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,226,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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