direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20⋊2D4, C60⋊17D4, C20⋊2(C3×D4), (C6×D4)⋊11D5, (C6×D5)⋊15D4, (D4×C10)⋊3C6, D10⋊3(C3×D4), (D4×C30)⋊10C2, C4⋊Dic5⋊14C6, C10.39(C6×D4), C6.193(D4×D5), C15⋊36(C4⋊D4), C12⋊12(C5⋊D4), C23.8(C6×D5), C30.351(C2×D4), C23.D5⋊11C6, (C22×C6).9D10, (C2×C12).363D10, C30.240(C4○D4), (C2×C30).370C23, (C2×C60).294C22, C6.122(D4⋊2D5), (C22×C30).106C22, (C6×Dic5).164C22, (C2×C4×D5)⋊2C6, C5⋊4(C3×C4⋊D4), C4⋊2(C3×C5⋊D4), C2.26(C3×D4×D5), (D5×C2×C12)⋊12C2, (C2×D4)⋊4(C3×D5), (C2×C5⋊D4)⋊5C6, (C6×C5⋊D4)⋊20C2, (C2×C4).50(C6×D5), C2.14(C6×C5⋊D4), C22.60(D5×C2×C6), (C2×C20).31(C2×C6), (C3×C4⋊Dic5)⋊32C2, C10.30(C3×C4○D4), C6.135(C2×C5⋊D4), C2.17(C3×D4⋊2D5), (C3×C23.D5)⋊27C2, (D5×C2×C6).136C22, (C2×C10).53(C22×C6), (C22×C10).25(C2×C6), (C2×Dic5).16(C2×C6), (C22×D5).31(C2×C6), (C2×C6).366(C22×D5), SmallGroup(480,731)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20⋊2D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >
Subgroups: 608 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C3×C4⋊D4, D5×C12, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, C22×C30, C20⋊2D4, C3×C4⋊Dic5, C3×C23.D5, D5×C2×C12, C6×C5⋊D4, D4×C30, C3×C20⋊2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4⋊D4, C5⋊D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C3×C4⋊D4, C3×C5⋊D4, D5×C2×C6, C20⋊2D4, C3×D4×D5, C3×D4⋊2D5, C6×C5⋊D4, C3×C20⋊2D4
►On 240 points
Generators in S240
(1 153 111)(2 154 112)(3 155 113)(4 156 114)(5 157 115)(6 158 116)(7 159 117)(8 160 118)(9 141 119)(10 142 120)(11 143 101)(12 144 102)(13 145 103)(14 146 104)(15 147 105)(16 148 106)(17 149 107)(18 150 108)(19 151 109)(20 152 110)(21 230 80)(22 231 61)(23 232 62)(24 233 63)(25 234 64)(26 235 65)(27 236 66)(28 237 67)(29 238 68)(30 239 69)(31 240 70)(32 221 71)(33 222 72)(34 223 73)(35 224 74)(36 225 75)(37 226 76)(38 227 77)(39 228 78)(40 229 79)(41 209 163)(42 210 164)(43 211 165)(44 212 166)(45 213 167)(46 214 168)(47 215 169)(48 216 170)(49 217 171)(50 218 172)(51 219 173)(52 220 174)(53 201 175)(54 202 176)(55 203 177)(56 204 178)(57 205 179)(58 206 180)(59 207 161)(60 208 162)(81 124 187)(82 125 188)(83 126 189)(84 127 190)(85 128 191)(86 129 192)(87 130 193)(88 131 194)(89 132 195)(90 133 196)(91 134 197)(92 135 198)(93 136 199)(94 137 200)(95 138 181)(96 139 182)(97 140 183)(98 121 184)(99 122 185)(100 123 186)
(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)
(1 235 168 98)(2 234 169 97)(3 233 170 96)(4 232 171 95)(5 231 172 94)(6 230 173 93)(7 229 174 92)(8 228 175 91)(9 227 176 90)(10 226 177 89)(11 225 178 88)(12 224 179 87)(13 223 180 86)(14 222 161 85)(15 221 162 84)(16 240 163 83)(17 239 164 82)(18 238 165 81)(19 237 166 100)(20 236 167 99)(21 219 199 116)(22 218 200 115)(23 217 181 114)(24 216 182 113)(25 215 183 112)(26 214 184 111)(27 213 185 110)(28 212 186 109)(29 211 187 108)(30 210 188 107)(31 209 189 106)(32 208 190 105)(33 207 191 104)(34 206 192 103)(35 205 193 102)(36 204 194 101)(37 203 195 120)(38 202 196 119)(39 201 197 118)(40 220 198 117)(41 126 148 70)(42 125 149 69)(43 124 150 68)(44 123 151 67)(45 122 152 66)(46 121 153 65)(47 140 154 64)(48 139 155 63)(49 138 156 62)(50 137 157 61)(51 136 158 80)(52 135 159 79)(53 134 160 78)(54 133 141 77)(55 132 142 76)(56 131 143 75)(57 130 144 74)(58 129 145 73)(59 128 146 72)(60 127 147 71)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 199)(22 188)(23 197)(24 186)(25 195)(26 184)(27 193)(28 182)(29 191)(30 200)(31 189)(32 198)(33 187)(34 196)(35 185)(36 194)(37 183)(38 192)(39 181)(40 190)(42 50)(43 59)(44 48)(45 57)(47 55)(49 53)(52 60)(54 58)(61 125)(62 134)(63 123)(64 132)(65 121)(66 130)(67 139)(68 128)(69 137)(70 126)(71 135)(72 124)(73 133)(74 122)(75 131)(76 140)(77 129)(78 138)(79 127)(80 136)(81 222)(82 231)(83 240)(84 229)(85 238)(86 227)(87 236)(88 225)(89 234)(90 223)(91 232)(92 221)(93 230)(94 239)(95 228)(96 237)(97 226)(98 235)(99 224)(100 233)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)(141 145)(142 154)(144 152)(146 150)(147 159)(149 157)(151 155)(156 160)(161 165)(162 174)(164 172)(166 170)(167 179)(169 177)(171 175)(176 180)(201 217)(202 206)(203 215)(205 213)(207 211)(208 220)(210 218)(212 216)
G:=sub<Sym(240)| (1,153,111)(2,154,112)(3,155,113)(4,156,114)(5,157,115)(6,158,116)(7,159,117)(8,160,118)(9,141,119)(10,142,120)(11,143,101)(12,144,102)(13,145,103)(14,146,104)(15,147,105)(16,148,106)(17,149,107)(18,150,108)(19,151,109)(20,152,110)(21,230,80)(22,231,61)(23,232,62)(24,233,63)(25,234,64)(26,235,65)(27,236,66)(28,237,67)(29,238,68)(30,239,69)(31,240,70)(32,221,71)(33,222,72)(34,223,73)(35,224,74)(36,225,75)(37,226,76)(38,227,77)(39,228,78)(40,229,79)(41,209,163)(42,210,164)(43,211,165)(44,212,166)(45,213,167)(46,214,168)(47,215,169)(48,216,170)(49,217,171)(50,218,172)(51,219,173)(52,220,174)(53,201,175)(54,202,176)(55,203,177)(56,204,178)(57,205,179)(58,206,180)(59,207,161)(60,208,162)(81,124,187)(82,125,188)(83,126,189)(84,127,190)(85,128,191)(86,129,192)(87,130,193)(88,131,194)(89,132,195)(90,133,196)(91,134,197)(92,135,198)(93,136,199)(94,137,200)(95,138,181)(96,139,182)(97,140,183)(98,121,184)(99,122,185)(100,123,186), (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), (1,235,168,98)(2,234,169,97)(3,233,170,96)(4,232,171,95)(5,231,172,94)(6,230,173,93)(7,229,174,92)(8,228,175,91)(9,227,176,90)(10,226,177,89)(11,225,178,88)(12,224,179,87)(13,223,180,86)(14,222,161,85)(15,221,162,84)(16,240,163,83)(17,239,164,82)(18,238,165,81)(19,237,166,100)(20,236,167,99)(21,219,199,116)(22,218,200,115)(23,217,181,114)(24,216,182,113)(25,215,183,112)(26,214,184,111)(27,213,185,110)(28,212,186,109)(29,211,187,108)(30,210,188,107)(31,209,189,106)(32,208,190,105)(33,207,191,104)(34,206,192,103)(35,205,193,102)(36,204,194,101)(37,203,195,120)(38,202,196,119)(39,201,197,118)(40,220,198,117)(41,126,148,70)(42,125,149,69)(43,124,150,68)(44,123,151,67)(45,122,152,66)(46,121,153,65)(47,140,154,64)(48,139,155,63)(49,138,156,62)(50,137,157,61)(51,136,158,80)(52,135,159,79)(53,134,160,78)(54,133,141,77)(55,132,142,76)(56,131,143,75)(57,130,144,74)(58,129,145,73)(59,128,146,72)(60,127,147,71), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,199)(22,188)(23,197)(24,186)(25,195)(26,184)(27,193)(28,182)(29,191)(30,200)(31,189)(32,198)(33,187)(34,196)(35,185)(36,194)(37,183)(38,192)(39,181)(40,190)(42,50)(43,59)(44,48)(45,57)(47,55)(49,53)(52,60)(54,58)(61,125)(62,134)(63,123)(64,132)(65,121)(66,130)(67,139)(68,128)(69,137)(70,126)(71,135)(72,124)(73,133)(74,122)(75,131)(76,140)(77,129)(78,138)(79,127)(80,136)(81,222)(82,231)(83,240)(84,229)(85,238)(86,227)(87,236)(88,225)(89,234)(90,223)(91,232)(92,221)(93,230)(94,239)(95,228)(96,237)(97,226)(98,235)(99,224)(100,233)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(141,145)(142,154)(144,152)(146,150)(147,159)(149,157)(151,155)(156,160)(161,165)(162,174)(164,172)(166,170)(167,179)(169,177)(171,175)(176,180)(201,217)(202,206)(203,215)(205,213)(207,211)(208,220)(210,218)(212,216)>;
G:=Group( (1,153,111)(2,154,112)(3,155,113)(4,156,114)(5,157,115)(6,158,116)(7,159,117)(8,160,118)(9,141,119)(10,142,120)(11,143,101)(12,144,102)(13,145,103)(14,146,104)(15,147,105)(16,148,106)(17,149,107)(18,150,108)(19,151,109)(20,152,110)(21,230,80)(22,231,61)(23,232,62)(24,233,63)(25,234,64)(26,235,65)(27,236,66)(28,237,67)(29,238,68)(30,239,69)(31,240,70)(32,221,71)(33,222,72)(34,223,73)(35,224,74)(36,225,75)(37,226,76)(38,227,77)(39,228,78)(40,229,79)(41,209,163)(42,210,164)(43,211,165)(44,212,166)(45,213,167)(46,214,168)(47,215,169)(48,216,170)(49,217,171)(50,218,172)(51,219,173)(52,220,174)(53,201,175)(54,202,176)(55,203,177)(56,204,178)(57,205,179)(58,206,180)(59,207,161)(60,208,162)(81,124,187)(82,125,188)(83,126,189)(84,127,190)(85,128,191)(86,129,192)(87,130,193)(88,131,194)(89,132,195)(90,133,196)(91,134,197)(92,135,198)(93,136,199)(94,137,200)(95,138,181)(96,139,182)(97,140,183)(98,121,184)(99,122,185)(100,123,186), (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), (1,235,168,98)(2,234,169,97)(3,233,170,96)(4,232,171,95)(5,231,172,94)(6,230,173,93)(7,229,174,92)(8,228,175,91)(9,227,176,90)(10,226,177,89)(11,225,178,88)(12,224,179,87)(13,223,180,86)(14,222,161,85)(15,221,162,84)(16,240,163,83)(17,239,164,82)(18,238,165,81)(19,237,166,100)(20,236,167,99)(21,219,199,116)(22,218,200,115)(23,217,181,114)(24,216,182,113)(25,215,183,112)(26,214,184,111)(27,213,185,110)(28,212,186,109)(29,211,187,108)(30,210,188,107)(31,209,189,106)(32,208,190,105)(33,207,191,104)(34,206,192,103)(35,205,193,102)(36,204,194,101)(37,203,195,120)(38,202,196,119)(39,201,197,118)(40,220,198,117)(41,126,148,70)(42,125,149,69)(43,124,150,68)(44,123,151,67)(45,122,152,66)(46,121,153,65)(47,140,154,64)(48,139,155,63)(49,138,156,62)(50,137,157,61)(51,136,158,80)(52,135,159,79)(53,134,160,78)(54,133,141,77)(55,132,142,76)(56,131,143,75)(57,130,144,74)(58,129,145,73)(59,128,146,72)(60,127,147,71), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,199)(22,188)(23,197)(24,186)(25,195)(26,184)(27,193)(28,182)(29,191)(30,200)(31,189)(32,198)(33,187)(34,196)(35,185)(36,194)(37,183)(38,192)(39,181)(40,190)(42,50)(43,59)(44,48)(45,57)(47,55)(49,53)(52,60)(54,58)(61,125)(62,134)(63,123)(64,132)(65,121)(66,130)(67,139)(68,128)(69,137)(70,126)(71,135)(72,124)(73,133)(74,122)(75,131)(76,140)(77,129)(78,138)(79,127)(80,136)(81,222)(82,231)(83,240)(84,229)(85,238)(86,227)(87,236)(88,225)(89,234)(90,223)(91,232)(92,221)(93,230)(94,239)(95,228)(96,237)(97,226)(98,235)(99,224)(100,233)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(141,145)(142,154)(144,152)(146,150)(147,159)(149,157)(151,155)(156,160)(161,165)(162,174)(164,172)(166,170)(167,179)(169,177)(171,175)(176,180)(201,217)(202,206)(203,215)(205,213)(207,211)(208,220)(210,218)(212,216) );
G=PermutationGroup([[(1,153,111),(2,154,112),(3,155,113),(4,156,114),(5,157,115),(6,158,116),(7,159,117),(8,160,118),(9,141,119),(10,142,120),(11,143,101),(12,144,102),(13,145,103),(14,146,104),(15,147,105),(16,148,106),(17,149,107),(18,150,108),(19,151,109),(20,152,110),(21,230,80),(22,231,61),(23,232,62),(24,233,63),(25,234,64),(26,235,65),(27,236,66),(28,237,67),(29,238,68),(30,239,69),(31,240,70),(32,221,71),(33,222,72),(34,223,73),(35,224,74),(36,225,75),(37,226,76),(38,227,77),(39,228,78),(40,229,79),(41,209,163),(42,210,164),(43,211,165),(44,212,166),(45,213,167),(46,214,168),(47,215,169),(48,216,170),(49,217,171),(50,218,172),(51,219,173),(52,220,174),(53,201,175),(54,202,176),(55,203,177),(56,204,178),(57,205,179),(58,206,180),(59,207,161),(60,208,162),(81,124,187),(82,125,188),(83,126,189),(84,127,190),(85,128,191),(86,129,192),(87,130,193),(88,131,194),(89,132,195),(90,133,196),(91,134,197),(92,135,198),(93,136,199),(94,137,200),(95,138,181),(96,139,182),(97,140,183),(98,121,184),(99,122,185),(100,123,186)], [(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)], [(1,235,168,98),(2,234,169,97),(3,233,170,96),(4,232,171,95),(5,231,172,94),(6,230,173,93),(7,229,174,92),(8,228,175,91),(9,227,176,90),(10,226,177,89),(11,225,178,88),(12,224,179,87),(13,223,180,86),(14,222,161,85),(15,221,162,84),(16,240,163,83),(17,239,164,82),(18,238,165,81),(19,237,166,100),(20,236,167,99),(21,219,199,116),(22,218,200,115),(23,217,181,114),(24,216,182,113),(25,215,183,112),(26,214,184,111),(27,213,185,110),(28,212,186,109),(29,211,187,108),(30,210,188,107),(31,209,189,106),(32,208,190,105),(33,207,191,104),(34,206,192,103),(35,205,193,102),(36,204,194,101),(37,203,195,120),(38,202,196,119),(39,201,197,118),(40,220,198,117),(41,126,148,70),(42,125,149,69),(43,124,150,68),(44,123,151,67),(45,122,152,66),(46,121,153,65),(47,140,154,64),(48,139,155,63),(49,138,156,62),(50,137,157,61),(51,136,158,80),(52,135,159,79),(53,134,160,78),(54,133,141,77),(55,132,142,76),(56,131,143,75),(57,130,144,74),(58,129,145,73),(59,128,146,72),(60,127,147,71)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,199),(22,188),(23,197),(24,186),(25,195),(26,184),(27,193),(28,182),(29,191),(30,200),(31,189),(32,198),(33,187),(34,196),(35,185),(36,194),(37,183),(38,192),(39,181),(40,190),(42,50),(43,59),(44,48),(45,57),(47,55),(49,53),(52,60),(54,58),(61,125),(62,134),(63,123),(64,132),(65,121),(66,130),(67,139),(68,128),(69,137),(70,126),(71,135),(72,124),(73,133),(74,122),(75,131),(76,140),(77,129),(78,138),(79,127),(80,136),(81,222),(82,231),(83,240),(84,229),(85,238),(86,227),(87,236),(88,225),(89,234),(90,223),(91,232),(92,221),(93,230),(94,239),(95,228),(96,237),(97,226),(98,235),(99,224),(100,233),(102,110),(103,119),(104,108),(105,117),(107,115),(109,113),(112,120),(114,118),(141,145),(142,154),(144,152),(146,150),(147,159),(149,157),(151,155),(156,160),(161,165),(162,174),(164,172),(166,170),(167,179),(169,177),(171,175),(176,180),(201,217),(202,206),(203,215),(205,213),(207,211),(208,220),(210,218),(212,216)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×C20⋊2D4 | C3×C4⋊Dic5 | C3×C23.D5 | D5×C2×C12 | C6×C5⋊D4 | D4×C30 | C20⋊2D4 | C4⋊Dic5 | C23.D5 | C2×C4×D5 | C2×C5⋊D4 | D4×C10 | C60 | C6×D5 | C6×D4 | C30 | C2×C12 | C22×C6 | C20 | D10 | C2×D4 | C12 | C10 | C2×C4 | C23 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C20⋊2D4 ►in GL6(𝔽61)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 34 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 8 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,60,0,0,0,0,1,17,0,0,0,0,0,0,50,34,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,44,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,15,60,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,8,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;
C3×C20⋊2D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}\rtimes_2D_4 % in TeX
G:=Group("C3xC20:2D4"); // GroupNames label
G:=SmallGroup(480,731);
// by ID
G=gap.SmallGroup(480,731);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations