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## G = C4×S3×Dic5order 480 = 25·3·5

### Direct product of C4, S3 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×S3×Dic5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — C4×S3×Dic5
 Lower central C15 — C4×S3×Dic5
 Upper central C1 — C2×C4

Generators and relations for C4×S3×Dic5
G = < a,b,c,d,e | a4=b3=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 748 in 216 conjugacy classes, 104 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C42, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C30, C2×C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, C4×C12, S3×C2×C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C4×Dic5, C22×Dic5, C22×C20, S3×C42, S3×Dic5, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C2×C4×Dic5, Dic3×Dic5, C12×Dic5, C4×Dic15, C2×S3×Dic5, S3×C2×C20, C4×S3×Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C42, C22×C4, Dic5, D10, C4×S3, C22×S3, C2×C42, C4×D5, C2×Dic5, C22×D5, S3×C2×C4, S3×D5, C4×Dic5, C2×C4×D5, C22×Dic5, S3×C42, S3×Dic5, C2×S3×D5, C2×C4×Dic5, C4×S3×D5, C2×S3×Dic5, C4×S3×Dic5

Smallest permutation representation of C4×S3×Dic5
On 240 points
Generators in S240
(1 119 59 80)(2 120 60 71)(3 111 51 72)(4 112 52 73)(5 113 53 74)(6 114 54 75)(7 115 55 76)(8 116 56 77)(9 117 57 78)(10 118 58 79)(11 82 22 100)(12 83 23 91)(13 84 24 92)(14 85 25 93)(15 86 26 94)(16 87 27 95)(17 88 28 96)(18 89 29 97)(19 90 30 98)(20 81 21 99)(31 182 224 164)(32 183 225 165)(33 184 226 166)(34 185 227 167)(35 186 228 168)(36 187 229 169)(37 188 230 170)(38 189 221 161)(39 190 222 162)(40 181 223 163)(41 125 68 101)(42 126 69 102)(43 127 70 103)(44 128 61 104)(45 129 62 105)(46 130 63 106)(47 121 64 107)(48 122 65 108)(49 123 66 109)(50 124 67 110)(131 237 177 191)(132 238 178 192)(133 239 179 193)(134 240 180 194)(135 231 171 195)(136 232 172 196)(137 233 173 197)(138 234 174 198)(139 235 175 199)(140 236 176 200)(141 218 158 201)(142 219 159 202)(143 220 160 203)(144 211 151 204)(145 212 152 205)(146 213 153 206)(147 214 154 207)(148 215 155 208)(149 216 156 209)(150 217 157 210)
(1 28 41)(2 29 42)(3 30 43)(4 21 44)(5 22 45)(6 23 46)(7 24 47)(8 25 48)(9 26 49)(10 27 50)(11 62 53)(12 63 54)(13 64 55)(14 65 56)(15 66 57)(16 67 58)(17 68 59)(18 69 60)(19 70 51)(20 61 52)(31 236 219)(32 237 220)(33 238 211)(34 239 212)(35 240 213)(36 231 214)(37 232 215)(38 233 216)(39 234 217)(40 235 218)(71 89 102)(72 90 103)(73 81 104)(74 82 105)(75 83 106)(76 84 107)(77 85 108)(78 86 109)(79 87 110)(80 88 101)(91 130 114)(92 121 115)(93 122 116)(94 123 117)(95 124 118)(96 125 119)(97 126 120)(98 127 111)(99 128 112)(100 129 113)(131 143 165)(132 144 166)(133 145 167)(134 146 168)(135 147 169)(136 148 170)(137 149 161)(138 150 162)(139 141 163)(140 142 164)(151 184 178)(152 185 179)(153 186 180)(154 187 171)(155 188 172)(156 189 173)(157 190 174)(158 181 175)(159 182 176)(160 183 177)(191 203 225)(192 204 226)(193 205 227)(194 206 228)(195 207 229)(196 208 230)(197 209 221)(198 210 222)(199 201 223)(200 202 224)
(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 61)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 41)(29 42)(30 43)(31 219)(32 220)(33 211)(34 212)(35 213)(36 214)(37 215)(38 216)(39 217)(40 218)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 101)(89 102)(90 103)(91 130)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 169)(148 170)(149 161)(150 162)(151 184)(152 185)(153 186)(154 187)(155 188)(156 189)(157 190)(158 181)(159 182)(160 183)(201 223)(202 224)(203 225)(204 226)(205 227)(206 228)(207 229)(208 230)(209 221)(210 222)
(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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 148 16 143)(12 147 17 142)(13 146 18 141)(14 145 19 150)(15 144 20 149)(21 156 26 151)(22 155 27 160)(23 154 28 159)(24 153 29 158)(25 152 30 157)(31 106 36 101)(32 105 37 110)(33 104 38 109)(34 103 39 108)(35 102 40 107)(41 182 46 187)(42 181 47 186)(43 190 48 185)(44 189 49 184)(45 188 50 183)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 135 59 140)(55 134 60 139)(61 161 66 166)(62 170 67 165)(63 169 68 164)(64 168 69 163)(65 167 70 162)(71 235 76 240)(72 234 77 239)(73 233 78 238)(74 232 79 237)(75 231 80 236)(81 216 86 211)(82 215 87 220)(83 214 88 219)(84 213 89 218)(85 212 90 217)(91 207 96 202)(92 206 97 201)(93 205 98 210)(94 204 99 209)(95 203 100 208)(111 198 116 193)(112 197 117 192)(113 196 118 191)(114 195 119 200)(115 194 120 199)(121 228 126 223)(122 227 127 222)(123 226 128 221)(124 225 129 230)(125 224 130 229)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q ··· 4X 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 3 3 3 3 2 1 1 1 1 3 3 3 3 5 ··· 5 15 ··· 15 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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D5 D6 D6 Dic5 D10 D10 D10 C4×S3 C4×S3 C4×D5 S3×D5 S3×Dic5 C2×S3×D5 C4×S3×D5 kernel C4×S3×Dic5 Dic3×Dic5 C12×Dic5 C4×Dic15 C2×S3×Dic5 S3×C2×C20 S3×Dic5 S3×C20 C4×Dic5 S3×C2×C4 C2×Dic5 C2×C20 C4×S3 C2×Dic3 C2×C12 C22×S3 Dic5 C20 D6 C2×C4 C4 C22 C2 # reps 1 2 1 1 2 1 16 8 1 2 2 1 8 2 2 2 8 4 16 2 4 2 8

Matrix representation of C4×S3×Dic5 in GL5(𝔽61)

 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 1 0
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 60 60
,
 1 0 0 0 0 0 0 60 0 0 0 1 44 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 41 3 0 0 0 29 20 0 0 0 0 0 11 0 0 0 0 0 11

G:=sub<GL(5,GF(61))| [11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,1,0,0,0,60,0],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,60,44,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,41,29,0,0,0,3,20,0,0,0,0,0,11,0,0,0,0,0,11] >;

C4×S3×Dic5 in GAP, Magma, Sage, TeX

C_4\times S_3\times {\rm Dic}_5
% in TeX

G:=Group("C4xS3xDic5");
// GroupNames label

G:=SmallGroup(480,473);
// by ID

G=gap.SmallGroup(480,473);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^10=1,e^2=d^5,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽