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G = (S3×C20)⋊7C4order 480 = 25·3·5

3rd semidirect product of S3×C20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (S3×C20)⋊7C4, (C4×Dic5)⋊9S3, (C4×S3)⋊3Dic5, C20.112(C4×S3), C60.154(C2×C4), C56(C422S3), (C12×Dic5)⋊8C2, (C2×C20).334D6, D6.5(C2×Dic5), C4.24(S3×Dic5), (C4×Dic15)⋊31C2, C6.26(C4○D20), C30.37(C4○D4), (C2×C12).338D10, D6⋊Dic5.15C2, (C2×C30).61C23, C6.Dic1042C2, C12.29(C2×Dic5), C1510(C42⋊C2), C10.30(C4○D12), C6.9(C22×Dic5), (C2×C60).236C22, C30.121(C22×C4), (C2×Dic5).162D6, Dic3.7(C2×Dic5), (C22×S3).68D10, (C2×Dic3).142D10, C2.2(D6.D10), C31(C23.21D10), (C6×Dic5).184C22, (C2×Dic15).190C22, (C10×Dic3).166C22, (S3×C2×C4).10D5, (S3×C2×C20).10C2, C10.117(S3×C2×C4), C2.11(C2×S3×Dic5), C22.34(C2×S3×D5), (C2×C4).239(S3×D5), (S3×C10).37(C2×C4), (S3×C2×C10).80C22, (C2×C6).73(C22×D5), (C2×C10).73(C22×S3), (C5×Dic3).45(C2×C4), SmallGroup(480,447)

Series: Derived Chief Lower central Upper central

C1C30 — (S3×C20)⋊7C4
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — (S3×C20)⋊7C4
C15C30 — (S3×C20)⋊7C4
C1C2×C4

Generators and relations for (S3×C20)⋊7C4
 G = < a,b,c,d | a20=b3=c2=d4=1, ab=ba, ac=ca, dad-1=a9, cbc=b-1, bd=db, dcd-1=a10c >

Subgroups: 556 in 152 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×9], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30, C30 [×2], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×5], C22×C10, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×2], C4×C12, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5, C4×Dic5, C4⋊Dic5 [×2], C23.D5 [×2], C22×C20, C422S3, C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.21D10, D6⋊Dic5 [×2], C6.Dic10 [×2], C12×Dic5, C4×Dic15, S3×C2×C20, (S3×C20)⋊7C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C2×Dic5 [×6], C22×D5, S3×C2×C4, C4○D12 [×2], S3×D5, C4○D20 [×2], C22×Dic5, C422S3, S3×Dic5 [×2], C2×S3×D5, C23.21D10, D6.D10 [×2], C2×S3×Dic5, (S3×C20)⋊7C4

Smallest permutation representation of (S3×C20)⋊7C4
On 240 points
Generators in S240
(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)
(1 107 221)(2 108 222)(3 109 223)(4 110 224)(5 111 225)(6 112 226)(7 113 227)(8 114 228)(9 115 229)(10 116 230)(11 117 231)(12 118 232)(13 119 233)(14 120 234)(15 101 235)(16 102 236)(17 103 237)(18 104 238)(19 105 239)(20 106 240)(21 62 204)(22 63 205)(23 64 206)(24 65 207)(25 66 208)(26 67 209)(27 68 210)(28 69 211)(29 70 212)(30 71 213)(31 72 214)(32 73 215)(33 74 216)(34 75 217)(35 76 218)(36 77 219)(37 78 220)(38 79 201)(39 80 202)(40 61 203)(41 128 150)(42 129 151)(43 130 152)(44 131 153)(45 132 154)(46 133 155)(47 134 156)(48 135 157)(49 136 158)(50 137 159)(51 138 160)(52 139 141)(53 140 142)(54 121 143)(55 122 144)(56 123 145)(57 124 146)(58 125 147)(59 126 148)(60 127 149)(81 176 195)(82 177 196)(83 178 197)(84 179 198)(85 180 199)(86 161 200)(87 162 181)(88 163 182)(89 164 183)(90 165 184)(91 166 185)(92 167 186)(93 168 187)(94 169 188)(95 170 189)(96 171 190)(97 172 191)(98 173 192)(99 174 193)(100 175 194)
(1 200)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 192)(14 193)(15 194)(16 195)(17 196)(18 197)(19 198)(20 199)(21 160)(22 141)(23 142)(24 143)(25 144)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(41 214)(42 215)(43 216)(44 217)(45 218)(46 219)(47 220)(48 201)(49 202)(50 203)(51 204)(52 205)(53 206)(54 207)(55 208)(56 209)(57 210)(58 211)(59 212)(60 213)(61 137)(62 138)(63 139)(64 140)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 236)(82 237)(83 238)(84 239)(85 240)(86 221)(87 222)(88 223)(89 224)(90 225)(91 226)(92 227)(93 228)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 175)(102 176)(103 177)(104 178)(105 179)(106 180)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)
(1 144 200 35)(2 153 181 24)(3 142 182 33)(4 151 183 22)(5 160 184 31)(6 149 185 40)(7 158 186 29)(8 147 187 38)(9 156 188 27)(10 145 189 36)(11 154 190 25)(12 143 191 34)(13 152 192 23)(14 141 193 32)(15 150 194 21)(16 159 195 30)(17 148 196 39)(18 157 197 28)(19 146 198 37)(20 155 199 26)(41 100 62 101)(42 89 63 110)(43 98 64 119)(44 87 65 108)(45 96 66 117)(46 85 67 106)(47 94 68 115)(48 83 69 104)(49 92 70 113)(50 81 71 102)(51 90 72 111)(52 99 73 120)(53 88 74 109)(54 97 75 118)(55 86 76 107)(56 95 77 116)(57 84 78 105)(58 93 79 114)(59 82 80 103)(60 91 61 112)(121 172 217 232)(122 161 218 221)(123 170 219 230)(124 179 220 239)(125 168 201 228)(126 177 202 237)(127 166 203 226)(128 175 204 235)(129 164 205 224)(130 173 206 233)(131 162 207 222)(132 171 208 231)(133 180 209 240)(134 169 210 229)(135 178 211 238)(136 167 212 227)(137 176 213 236)(138 165 214 225)(139 174 215 234)(140 163 216 223)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E···12L15A15B20A···20H20I···20P30A···30F60A···60H
order1222223444444444444445566610···1010···101212121212···12151520···2020···2030···3060···60
size11116621111661010101030303030222222···26···6222210···10442···26···64···44···4

84 irreducible representations

dim11111112222222222224444
type++++++++++-++++-+
imageC1C2C2C2C2C2C4S3D5D6D6C4○D4Dic5D10D10D10C4×S3C4○D12C4○D20S3×D5S3×Dic5C2×S3×D5D6.D10
kernel(S3×C20)⋊7C4D6⋊Dic5C6.Dic10C12×Dic5C4×Dic15S3×C2×C20S3×C20C4×Dic5S3×C2×C4C2×Dic5C2×C20C30C4×S3C2×Dic3C2×C12C22×S3C20C10C6C2×C4C4C22C2
# reps122111812214822248162428

Matrix representation of (S3×C20)⋊7C4 in GL5(𝔽61)

600000
060000
006000
000530
0001923
,
10000
014700
0225900
00010
00001
,
10000
01000
0226000
00010
0004660
,
500000
01000
00100
000533
000408

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,53,19,0,0,0,0,23],[1,0,0,0,0,0,1,22,0,0,0,47,59,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,22,0,0,0,0,60,0,0,0,0,0,1,46,0,0,0,0,60],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,40,0,0,0,3,8] >;

(S3×C20)⋊7C4 in GAP, Magma, Sage, TeX

(S_3\times C_{20})\rtimes_7C_4
% in TeX

G:=Group("(S3xC20):7C4");
// GroupNames label

G:=SmallGroup(480,447);
// by ID

G=gap.SmallGroup(480,447);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^3=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^10*c>;
// generators/relations

׿
×
𝔽