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## G = C4×C5⋊D12order 480 = 25·3·5

### Direct product of C4 and C5⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C4×C5⋊D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×C5⋊D12 — C4×C5⋊D12
 Lower central C15 — C30 — C4×C5⋊D12
 Upper central C1 — C2×C4

Generators and relations for C4×C5⋊D12
G = < a,b,c,d | a4=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 940 in 188 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], C12 [×2], C12 [×3], D6 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5, C22×C10, C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15, C60 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×D12, C5⋊D12 [×4], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C4×C5⋊D4, D6⋊Dic5, D304C4, C30.Q8, C12×Dic5, C2×C5⋊D12, S3×C2×C20, C2×C4×D15, C4×C5⋊D12
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], D12 [×2], C22×S3, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×D12, C5⋊D12 [×2], C2×S3×D5, C4×C5⋊D4, D6.D10, C4×S3×D5, C2×C5⋊D12, C4×C5⋊D12

Smallest permutation representation of C4×C5⋊D12
On 240 points
Generators in S240
(1 170 15 164)(2 171 16 165)(3 172 17 166)(4 173 18 167)(5 174 19 168)(6 175 20 157)(7 176 21 158)(8 177 22 159)(9 178 23 160)(10 179 24 161)(11 180 13 162)(12 169 14 163)(25 222 191 87)(26 223 192 88)(27 224 181 89)(28 225 182 90)(29 226 183 91)(30 227 184 92)(31 228 185 93)(32 217 186 94)(33 218 187 95)(34 219 188 96)(35 220 189 85)(36 221 190 86)(37 141 59 195)(38 142 60 196)(39 143 49 197)(40 144 50 198)(41 133 51 199)(42 134 52 200)(43 135 53 201)(44 136 54 202)(45 137 55 203)(46 138 56 204)(47 139 57 193)(48 140 58 194)(61 80 123 147)(62 81 124 148)(63 82 125 149)(64 83 126 150)(65 84 127 151)(66 73 128 152)(67 74 129 153)(68 75 130 154)(69 76 131 155)(70 77 132 156)(71 78 121 145)(72 79 122 146)(97 207 116 232)(98 208 117 233)(99 209 118 234)(100 210 119 235)(101 211 120 236)(102 212 109 237)(103 213 110 238)(104 214 111 239)(105 215 112 240)(106 216 113 229)(107 205 114 230)(108 206 115 231)
(1 235 182 67 196)(2 197 68 183 236)(3 237 184 69 198)(4 199 70 185 238)(5 239 186 71 200)(6 201 72 187 240)(7 229 188 61 202)(8 203 62 189 230)(9 231 190 63 204)(10 193 64 191 232)(11 233 192 65 194)(12 195 66 181 234)(13 208 26 127 140)(14 141 128 27 209)(15 210 28 129 142)(16 143 130 29 211)(17 212 30 131 144)(18 133 132 31 213)(19 214 32 121 134)(20 135 122 33 215)(21 216 34 123 136)(22 137 124 35 205)(23 206 36 125 138)(24 139 126 25 207)(37 73 89 99 169)(38 170 100 90 74)(39 75 91 101 171)(40 172 102 92 76)(41 77 93 103 173)(42 174 104 94 78)(43 79 95 105 175)(44 176 106 96 80)(45 81 85 107 177)(46 178 108 86 82)(47 83 87 97 179)(48 180 98 88 84)(49 154 226 120 165)(50 166 109 227 155)(51 156 228 110 167)(52 168 111 217 145)(53 146 218 112 157)(54 158 113 219 147)(55 148 220 114 159)(56 160 115 221 149)(57 150 222 116 161)(58 162 117 223 151)(59 152 224 118 163)(60 164 119 225 153)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 120)(46 119)(47 118)(48 117)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 108)(73 222)(74 221)(75 220)(76 219)(77 218)(78 217)(79 228)(80 227)(81 226)(82 225)(83 224)(84 223)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 156)(96 155)(121 186)(122 185)(123 184)(124 183)(125 182)(126 181)(127 192)(128 191)(129 190)(130 189)(131 188)(132 187)(133 240)(134 239)(135 238)(136 237)(137 236)(138 235)(139 234)(140 233)(141 232)(142 231)(143 230)(144 229)(157 173)(158 172)(159 171)(160 170)(161 169)(162 180)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(193 209)(194 208)(195 207)(196 206)(197 205)(198 216)(199 215)(200 214)(201 213)(202 212)(203 211)(204 210)

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

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

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

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 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E ··· 12L 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 10 ··· 10 10 ··· 10 12 12 12 12 12 ··· 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 30 30 2 1 1 1 1 6 6 10 10 10 10 30 30 2 2 2 2 2 2 ··· 2 6 ··· 6 2 2 2 2 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 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 C4○D4 D10 D10 D10 C4×S3 D12 C5⋊D4 C4×D5 C4○D12 C4○D20 S3×D5 C5⋊D12 C2×S3×D5 D6.D10 C4×S3×D5 kernel C4×C5⋊D12 D6⋊Dic5 D30⋊4C4 C30.Q8 C12×Dic5 C2×C5⋊D12 S3×C2×C20 C2×C4×D15 C5⋊D12 C4×Dic5 C60 S3×C2×C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C22×S3 Dic5 C20 C12 D6 C10 C6 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 2 1 2 2 2 2 4 4 8 8 4 8 2 4 2 4 4

Matrix representation of C4×C5⋊D12 in GL4(𝔽61) generated by

 11 0 0 0 0 11 0 0 0 0 1 0 0 0 0 1
,
 60 16 0 0 1 44 0 0 0 0 1 0 0 0 0 1
,
 17 17 0 0 1 44 0 0 0 0 15 38 0 0 23 38
,
 44 44 0 0 60 17 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(61))| [11,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,16,44,0,0,0,0,1,0,0,0,0,1],[17,1,0,0,17,44,0,0,0,0,15,23,0,0,38,38],[44,60,0,0,44,17,0,0,0,0,0,1,0,0,1,0] >;

C4×C5⋊D12 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes D_{12}
% in TeX

G:=Group("C4xC5:D12");
// GroupNames label

G:=SmallGroup(480,521);
// by ID

G=gap.SmallGroup(480,521);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,58,1356,18822]);
// Polycyclic

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

׿
×
𝔽