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G = C4×C3⋊D20order 480 = 25·3·5

Direct product of C4 and C3⋊D20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C3⋊D20, C6014D4, C128D20, C34(C4×D20), C1519(C4×D4), D109(C4×S3), D3024(C2×C4), C208(C3⋊D4), Dic34(C4×D5), C6.60(C2×D20), (C4×Dic3)⋊16D5, C30.143(C2×D4), (C2×C20).340D6, D304C439C2, (Dic3×C20)⋊12C2, C6.35(C4○D20), C30.78(C4○D4), (C2×C12).344D10, C30.62(C22×C4), C6.Dic1043C2, (C22×D5).87D6, C10.38(C4○D12), D10⋊Dic340C2, (C2×C60).242C22, (C2×C30).133C23, (C2×Dic5).179D6, (C2×Dic3).152D10, C2.5(D6.D10), (C6×Dic5).205C22, (C2×Dic15).211C22, (C10×Dic3).185C22, (C22×D15).106C22, C52(C4×C3⋊D4), (C2×C4×D5)⋊10S3, (D5×C2×C12)⋊8C2, C2.32(C4×S3×D5), C6.30(C2×C4×D5), (C2×C4×D15)⋊28C2, C10.63(S3×C2×C4), (C6×D5)⋊19(C2×C4), C2.1(C2×C3⋊D20), C22.65(C2×S3×D5), (C2×C4).245(S3×D5), C10.15(C2×C3⋊D4), (C5×Dic3)⋊16(C2×C4), (C2×C3⋊D20).11C2, (D5×C2×C6).103C22, (C2×C6).145(C22×D5), (C2×C10).145(C22×S3), SmallGroup(480,519)

Series: Derived Chief Lower central Upper central

C1C30 — C4×C3⋊D20
C1C5C15C30C2×C30D5×C2×C6C2×C3⋊D20 — C4×C3⋊D20
C15C30 — C4×C3⋊D20
C1C2×C4

Generators and relations for C4×C3⋊D20
 G = < a,b,c,d | a4=b3=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 988 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 [×4], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12, D6 [×4], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×3], D10 [×2], D10 [×6], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15, C60 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C4×D5, C2×C4×D5, C2×D20, C4×C3⋊D4, C3⋊D20 [×4], D5×C12 [×2], C6×Dic5, C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4×D20, D10⋊Dic3, D304C4, C6.Dic10, Dic3×C20, C2×C3⋊D20, D5×C2×C12, C2×C4×D15, C4×C3⋊D20
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], D20 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, C2×D20, C4○D20, C4×C3⋊D4, C3⋊D20 [×2], C2×S3×D5, C4×D20, D6.D10, C4×S3×D5, C2×C3⋊D20, C4×C3⋊D20

Smallest permutation representation of C4×C3⋊D20
On 240 points
Generators in S240
(1 128 105 93)(2 129 106 94)(3 130 107 95)(4 131 108 96)(5 132 109 97)(6 133 110 98)(7 134 111 99)(8 135 112 100)(9 136 113 81)(10 137 114 82)(11 138 115 83)(12 139 116 84)(13 140 117 85)(14 121 118 86)(15 122 119 87)(16 123 120 88)(17 124 101 89)(18 125 102 90)(19 126 103 91)(20 127 104 92)(21 219 148 64)(22 220 149 65)(23 201 150 66)(24 202 151 67)(25 203 152 68)(26 204 153 69)(27 205 154 70)(28 206 155 71)(29 207 156 72)(30 208 157 73)(31 209 158 74)(32 210 159 75)(33 211 160 76)(34 212 141 77)(35 213 142 78)(36 214 143 79)(37 215 144 80)(38 216 145 61)(39 217 146 62)(40 218 147 63)(41 230 165 186)(42 231 166 187)(43 232 167 188)(44 233 168 189)(45 234 169 190)(46 235 170 191)(47 236 171 192)(48 237 172 193)(49 238 173 194)(50 239 174 195)(51 240 175 196)(52 221 176 197)(53 222 177 198)(54 223 178 199)(55 224 179 200)(56 225 180 181)(57 226 161 182)(58 227 162 183)(59 228 163 184)(60 229 164 185)
(1 215 169)(2 170 216)(3 217 171)(4 172 218)(5 219 173)(6 174 220)(7 201 175)(8 176 202)(9 203 177)(10 178 204)(11 205 179)(12 180 206)(13 207 161)(14 162 208)(15 209 163)(16 164 210)(17 211 165)(18 166 212)(19 213 167)(20 168 214)(21 238 97)(22 98 239)(23 240 99)(24 100 221)(25 222 81)(26 82 223)(27 224 83)(28 84 225)(29 226 85)(30 86 227)(31 228 87)(32 88 229)(33 230 89)(34 90 231)(35 232 91)(36 92 233)(37 234 93)(38 94 235)(39 236 95)(40 96 237)(41 101 76)(42 77 102)(43 103 78)(44 79 104)(45 105 80)(46 61 106)(47 107 62)(48 63 108)(49 109 64)(50 65 110)(51 111 66)(52 67 112)(53 113 68)(54 69 114)(55 115 70)(56 71 116)(57 117 72)(58 73 118)(59 119 74)(60 75 120)(121 183 157)(122 158 184)(123 185 159)(124 160 186)(125 187 141)(126 142 188)(127 189 143)(128 144 190)(129 191 145)(130 146 192)(131 193 147)(132 148 194)(133 195 149)(134 150 196)(135 197 151)(136 152 198)(137 199 153)(138 154 200)(139 181 155)(140 156 182)
(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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 120)(21 200)(22 199)(23 198)(24 197)(25 196)(26 195)(27 194)(28 193)(29 192)(30 191)(31 190)(32 189)(33 188)(34 187)(35 186)(36 185)(37 184)(38 183)(39 182)(40 181)(41 213)(42 212)(43 211)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 220)(55 219)(56 218)(57 217)(58 216)(59 215)(60 214)(61 162)(62 161)(63 180)(64 179)(65 178)(66 177)(67 176)(68 175)(69 174)(70 173)(71 172)(72 171)(73 170)(74 169)(75 168)(76 167)(77 166)(78 165)(79 164)(80 163)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)(141 231)(142 230)(143 229)(144 228)(145 227)(146 226)(147 225)(148 224)(149 223)(150 222)(151 221)(152 240)(153 239)(154 238)(155 237)(156 236)(157 235)(158 234)(159 233)(160 232)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E6F6G10A···10F12A12B12C12D12E12F12G12H15A15B20A···20H20I···20X30A···30F60A···60H
order12222222344444444444455666666610···101212121212121212151520···2020···2030···3060···60
size1111101030302111166661010303022222101010102···2222210101010442···26···64···44···4

84 irreducible representations

dim11111111122222222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6D6C4○D4D10D10C3⋊D4C4×S3C4×D5D20C4○D12C4○D20S3×D5C3⋊D20C2×S3×D5D6.D10C4×S3×D5
kernelC4×C3⋊D20D10⋊Dic3D304C4C6.Dic10Dic3×C20C2×C3⋊D20D5×C2×C12C2×C4×D15C3⋊D20C2×C4×D5C60C4×Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C20D10Dic3C12C10C6C2×C4C4C22C2C2
# reps11111111812211124244884824244

Matrix representation of C4×C3⋊D20 in GL4(𝔽61) generated by

11000
01100
00110
00011
,
1000
0100
00060
00160
,
593200
295400
00060
00600
,
176000
444400
00060
00600
G:=sub<GL(4,GF(61))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[59,29,0,0,32,54,0,0,0,0,0,60,0,0,60,0],[17,44,0,0,60,44,0,0,0,0,0,60,0,0,60,0] >;

C4×C3⋊D20 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes D_{20}
% in TeX

G:=Group("C4xC3:D20");
// GroupNames label

G:=SmallGroup(480,519);
// by ID

G=gap.SmallGroup(480,519);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^20=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