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G = C6.(C2×D20)  order 480 = 25·3·5

20th non-split extension by C6 of C2×D20 acting via C2×D20/C22×D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C6).8D20, C6.65(C2×D20), (C2×C30).68D4, C6.D46D5, C30.217(C2×D4), C23.49(S3×D5), C6.Dic1030C2, (C2×Dic5).56D6, (C22×D5).29D6, (C22×C6).26D10, (C22×C10).40D6, C30.138(C4○D4), C6.79(D42D5), D10⋊Dic328C2, (C2×C30).179C23, (C2×Dic3).55D10, C35(C22.D20), C52(C23.23D6), C10.79(D42S3), (C22×Dic15)⋊11C2, C22.11(C3⋊D20), C1517(C22.D4), (C22×C30).41C22, C2.24(C30.C23), (C6×Dic5).106C22, (C2×Dic15).222C22, (C10×Dic3).105C22, (C6×C5⋊D4).4C2, (C2×C5⋊D4).4S3, C10.19(C2×C3⋊D4), C2.22(C2×C3⋊D20), (D5×C2×C6).47C22, C22.222(C2×S3×D5), (C5×C6.D4)⋊6C2, (C2×C10).15(C3⋊D4), (C2×C6).191(C22×D5), (C2×C10).191(C22×S3), SmallGroup(480,613)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C6.(C2×D20)
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — C6.(C2×D20)
C15C2×C30 — C6.(C2×D20)
C1C22C23

Generators and relations for C6.(C2×D20)
 G = < a,b,c,d | a6=b2=c20=1, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=a3b, bd=db, dcd-1=a3c-1 >

Subgroups: 732 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, C6, C6 [×2], C6 [×3], C2×C4 [×7], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], Dic3 [×4], C12, C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, Dic3⋊C4 [×2], C6.D4, C6.D4 [×2], C22×Dic3, C6×D4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C6×D5 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C23.23D6, C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×Dic15 [×2], D5×C2×C6, C22×C30, C22.D20, D10⋊Dic3 [×2], C6.Dic10 [×2], C5×C6.D4, C6×C5⋊D4, C22×Dic15, C6.(C2×D20)
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C22.D4, D20 [×2], C22×D5, D42S3 [×2], C2×C3⋊D4, S3×D5, C2×D20, D42D5 [×2], C23.23D6, C3⋊D20 [×2], C2×S3×D5, C22.D20, C30.C23 [×2], C2×C3⋊D20, C6.(C2×D20)

Smallest permutation representation of C6.(C2×D20)
On 240 points
Generators in S240
(1 36 133 213 153 169)(2 170 154 214 134 37)(3 38 135 215 155 171)(4 172 156 216 136 39)(5 40 137 217 157 173)(6 174 158 218 138 21)(7 22 139 219 159 175)(8 176 160 220 140 23)(9 24 121 201 141 177)(10 178 142 202 122 25)(11 26 123 203 143 179)(12 180 144 204 124 27)(13 28 125 205 145 161)(14 162 146 206 126 29)(15 30 127 207 147 163)(16 164 148 208 128 31)(17 32 129 209 149 165)(18 166 150 210 130 33)(19 34 131 211 151 167)(20 168 152 212 132 35)(41 65 112 189 238 88)(42 89 239 190 113 66)(43 67 114 191 240 90)(44 91 221 192 115 68)(45 69 116 193 222 92)(46 93 223 194 117 70)(47 71 118 195 224 94)(48 95 225 196 119 72)(49 73 120 197 226 96)(50 97 227 198 101 74)(51 75 102 199 228 98)(52 99 229 200 103 76)(53 77 104 181 230 100)(54 81 231 182 105 78)(55 79 106 183 232 82)(56 83 233 184 107 80)(57 61 108 185 234 84)(58 85 235 186 109 62)(59 63 110 187 236 86)(60 87 237 188 111 64)
(1 203)(2 12)(3 205)(4 14)(5 207)(6 16)(7 209)(8 18)(9 211)(10 20)(11 213)(13 215)(15 217)(17 219)(19 201)(21 31)(22 149)(23 33)(24 151)(25 35)(26 153)(27 37)(28 155)(29 39)(30 157)(32 159)(34 141)(36 143)(38 145)(40 147)(41 199)(42 52)(43 181)(44 54)(45 183)(46 56)(47 185)(48 58)(49 187)(50 60)(51 189)(53 191)(55 193)(57 195)(59 197)(61 224)(62 72)(63 226)(64 74)(65 228)(66 76)(67 230)(68 78)(69 232)(70 80)(71 234)(73 236)(75 238)(77 240)(79 222)(81 91)(82 116)(83 93)(84 118)(85 95)(86 120)(87 97)(88 102)(89 99)(90 104)(92 106)(94 108)(96 110)(98 112)(100 114)(101 111)(103 113)(105 115)(107 117)(109 119)(121 167)(122 132)(123 169)(124 134)(125 171)(126 136)(127 173)(128 138)(129 175)(130 140)(131 177)(133 179)(135 161)(137 163)(139 165)(142 152)(144 154)(146 156)(148 158)(150 160)(162 172)(164 174)(166 176)(168 178)(170 180)(182 192)(184 194)(186 196)(188 198)(190 200)(202 212)(204 214)(206 216)(208 218)(210 220)(221 231)(223 233)(225 235)(227 237)(229 239)
(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)
(1 183 213 55)(2 54 214 182)(3 181 215 53)(4 52 216 200)(5 199 217 51)(6 50 218 198)(7 197 219 49)(8 48 220 196)(9 195 201 47)(10 46 202 194)(11 193 203 45)(12 44 204 192)(13 191 205 43)(14 42 206 190)(15 189 207 41)(16 60 208 188)(17 187 209 59)(18 58 210 186)(19 185 211 57)(20 56 212 184)(21 97 158 101)(22 120 159 96)(23 95 160 119)(24 118 141 94)(25 93 142 117)(26 116 143 92)(27 91 144 115)(28 114 145 90)(29 89 146 113)(30 112 147 88)(31 87 148 111)(32 110 149 86)(33 85 150 109)(34 108 151 84)(35 83 152 107)(36 106 153 82)(37 81 154 105)(38 104 155 100)(39 99 156 103)(40 102 157 98)(61 167 234 131)(62 130 235 166)(63 165 236 129)(64 128 237 164)(65 163 238 127)(66 126 239 162)(67 161 240 125)(68 124 221 180)(69 179 222 123)(70 122 223 178)(71 177 224 121)(72 140 225 176)(73 175 226 139)(74 138 227 174)(75 173 228 137)(76 136 229 172)(77 171 230 135)(78 134 231 170)(79 169 232 133)(80 132 233 168)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J12A12B15A15B20A···20H30A···30N
order12222223444444455666666610···10101010101212151520···2030···30
size11112220212122030303030222224420202···2444420204412···124···4

60 irreducible representations

dim11111122222222222444444
type+++++++++++++++-+-++-
imageC1C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C3⋊D4D20D42S3S3×D5D42D5C3⋊D20C2×S3×D5C30.C23
kernelC6.(C2×D20)D10⋊Dic3C6.Dic10C5×C6.D4C6×C5⋊D4C22×Dic15C2×C5⋊D4C2×C30C6.D4C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6C2×C10C2×C6C10C23C6C22C22C2
# reps12211112211144248224428

Matrix representation of C6.(C2×D20) in GL6(𝔽61)

100000
010000
0060000
0006000
00003956
00005621
,
100000
010000
001000
00256000
0000600
0000060
,
4100000
4930000
00354600
00372600
00004855
0000813
,
2470000
22590000
0050000
0005000
0000207
0000441

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,39,56,0,0,0,0,56,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[41,49,0,0,0,0,0,3,0,0,0,0,0,0,35,37,0,0,0,0,46,26,0,0,0,0,0,0,48,8,0,0,0,0,55,13],[2,22,0,0,0,0,47,59,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,20,4,0,0,0,0,7,41] >;

C6.(C2×D20) in GAP, Magma, Sage, TeX

C_6.(C_2\times D_{20})
% in TeX

G:=Group("C6.(C2xD20)");
// GroupNames label

G:=SmallGroup(480,613);
// by ID

G=gap.SmallGroup(480,613);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,422,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations

׿
×
𝔽