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## G = C5×D12.C4order 480 = 25·3·5

### Direct product of C5 and D12.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D12.C4
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×C4○D12 — C5×D12.C4
 Lower central C3 — C6 — C5×D12.C4
 Upper central C1 — C20 — C5×M4(2)

Generators and relations for C5×D12.C4
G = < a,b,c,d | a5=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >

Subgroups: 228 in 124 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C10, C10 [×3], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C5×S3 [×2], C30, C30, C8○D4, C40 [×2], C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], C2×C30, C2×C40 [×3], C5×M4(2), C5×M4(2) [×2], C5×C4○D4, D12.C4, C5×C3⋊C8 [×2], C120 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C2×C60, C5×C8○D4, S3×C40 [×2], C5×C8⋊S3 [×2], C10×C3⋊C8, C15×M4(2), C5×C4○D12, C5×D12.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C8○D4, C2×C20 [×6], C22×C10, S3×C2×C4, S3×C10 [×3], C22×C20, D12.C4, S3×C20 [×2], S3×C2×C10, C5×C8○D4, S3×C2×C20, C5×D12.C4

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

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40P 40Q ··· 40AF 40AG ··· 40AN 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 40 ··· 40 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 6 6 2 1 1 2 6 6 1 1 1 1 2 4 2 2 2 2 3 3 3 3 6 6 1 1 1 1 2 2 2 2 6 ··· 6 2 2 4 2 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 4 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 4 4 4 4 ··· 4

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C10 C20 C20 C20 S3 D6 D6 C4×S3 C4×S3 C5×S3 C8○D4 S3×C10 S3×C10 S3×C20 S3×C20 C5×C8○D4 D12.C4 C5×D12.C4 kernel C5×D12.C4 S3×C40 C5×C8⋊S3 C10×C3⋊C8 C15×M4(2) C5×C4○D12 C5×Dic6 C5×D12 C5×C3⋊D4 D12.C4 S3×C8 C8⋊S3 C2×C3⋊C8 C3×M4(2) C4○D12 Dic6 D12 C3⋊D4 C5×M4(2) C40 C2×C20 C20 C2×C10 M4(2) C15 C8 C2×C4 C4 C22 C3 C5 C1 # reps 1 2 2 1 1 1 2 2 4 4 8 8 4 4 4 8 8 16 1 2 1 2 2 4 4 8 4 8 8 16 2 8

Matrix representation of C5×D12.C4 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 87 0 0 0 0 87
,
 0 64 0 0 64 0 0 0 0 0 240 1 0 0 240 0
,
 0 64 0 0 177 0 0 0 0 0 240 0 0 0 240 1
,
 233 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,87,0,0,0,0,87],[0,64,0,0,64,0,0,0,0,0,240,240,0,0,1,0],[0,177,0,0,64,0,0,0,0,0,240,240,0,0,0,1],[233,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1] >;

C5×D12.C4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xD12.C4");
// GroupNames label

G:=SmallGroup(480,786);
// by ID

G=gap.SmallGroup(480,786);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,891,226,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations

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