direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C4○D20, C30.72C24, C60.282C23, (C6×D20)⋊30C2, D20⋊11(C2×C6), (C2×D20)⋊14C6, (C2×C12)⋊37D10, C30⋊12(C4○D4), (C22×C20)⋊12C6, (C2×C60)⋊48C22, (C22×C60)⋊17C2, (C22×C12)⋊13D5, C10.4(C23×C6), C23.31(C6×D5), C6.72(C23×D5), (C6×Dic10)⋊31C2, (C2×Dic10)⋊15C6, Dic10⋊11(C2×C6), (D5×C12)⋊25C22, (C3×D20)⋊42C22, C20.42(C22×C6), D10.1(C22×C6), (C6×D5).52C23, (C2×C30).382C23, C12.240(C22×D5), (C22×C6).111D10, Dic5.2(C22×C6), (C3×Dic10)⋊38C22, (C3×Dic5).54C23, (C22×C30).167C22, (C6×Dic5).257C22, C5⋊1(C6×C4○D4), (C2×C4×D5)⋊15C6, C4.43(D5×C2×C6), (D5×C2×C12)⋊31C2, (C4×D5)⋊6(C2×C6), C10⋊1(C3×C4○D4), (C2×C4)⋊10(C6×D5), C15⋊21(C2×C4○D4), C5⋊D4⋊6(C2×C6), (C2×C20)⋊13(C2×C6), (C6×C5⋊D4)⋊27C2, (C2×C5⋊D4)⋊12C6, C2.5(D5×C22×C6), C22.5(D5×C2×C6), (C22×C4)⋊8(C3×D5), (C3×C5⋊D4)⋊21C22, (D5×C2×C6).138C22, (C2×C10).65(C22×C6), (C22×C10).54(C2×C6), (C2×Dic5).45(C2×C6), (C22×D5).33(C2×C6), (C2×C6).259(C22×D5), SmallGroup(480,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 912 in 328 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, C6, C6 [×2], C6 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×10], C15, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×10], C3×D4 [×12], C3×Q8 [×4], C22×C6, C22×C6 [×2], C3×D5 [×4], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C22×C12, C22×C12 [×2], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic5 [×4], C60 [×4], C6×D5 [×4], C6×D5 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C6×C4○D4, C3×Dic10 [×4], D5×C12 [×8], C3×D20 [×4], C6×Dic5 [×2], C3×C5⋊D4 [×8], C2×C60 [×2], C2×C60 [×4], D5×C2×C6 [×2], C22×C30, C2×C4○D20, C6×Dic10, D5×C2×C12 [×2], C6×D20, C3×C4○D20 [×8], C6×C5⋊D4 [×2], C22×C60, C6×C4○D20
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C4○D4 [×2], C24, D10 [×7], C22×C6 [×15], C3×D5, C2×C4○D4, C22×D5 [×7], C3×C4○D4 [×2], C23×C6, C6×D5 [×7], C4○D20 [×2], C23×D5, C6×C4○D4, D5×C2×C6 [×7], C2×C4○D20, C3×C4○D20 [×2], D5×C22×C6, C6×C4○D20
Generators and relations
G = < a,b,c,d | a6=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
►On 240 points
Generators in S240
(1 28 169 206 58 118)(2 29 170 207 59 119)(3 30 171 208 60 120)(4 31 172 209 41 101)(5 32 173 210 42 102)(6 33 174 211 43 103)(7 34 175 212 44 104)(8 35 176 213 45 105)(9 36 177 214 46 106)(10 37 178 215 47 107)(11 38 179 216 48 108)(12 39 180 217 49 109)(13 40 161 218 50 110)(14 21 162 219 51 111)(15 22 163 220 52 112)(16 23 164 201 53 113)(17 24 165 202 54 114)(18 25 166 203 55 115)(19 26 167 204 56 116)(20 27 168 205 57 117)(61 126 238 94 149 196)(62 127 239 95 150 197)(63 128 240 96 151 198)(64 129 221 97 152 199)(65 130 222 98 153 200)(66 131 223 99 154 181)(67 132 224 100 155 182)(68 133 225 81 156 183)(69 134 226 82 157 184)(70 135 227 83 158 185)(71 136 228 84 159 186)(72 137 229 85 160 187)(73 138 230 86 141 188)(74 139 231 87 142 189)(75 140 232 88 143 190)(76 121 233 89 144 191)(77 122 234 90 145 192)(78 123 235 91 146 193)(79 124 236 92 147 194)(80 125 237 93 148 195)
(1 228 11 238)(2 229 12 239)(3 230 13 240)(4 231 14 221)(5 232 15 222)(6 233 16 223)(7 234 17 224)(8 235 18 225)(9 236 19 226)(10 237 20 227)(21 97 31 87)(22 98 32 88)(23 99 33 89)(24 100 34 90)(25 81 35 91)(26 82 36 92)(27 83 37 93)(28 84 38 94)(29 85 39 95)(30 86 40 96)(41 74 51 64)(42 75 52 65)(43 76 53 66)(44 77 54 67)(45 78 55 68)(46 79 56 69)(47 80 57 70)(48 61 58 71)(49 62 59 72)(50 63 60 73)(101 139 111 129)(102 140 112 130)(103 121 113 131)(104 122 114 132)(105 123 115 133)(106 124 116 134)(107 125 117 135)(108 126 118 136)(109 127 119 137)(110 128 120 138)(141 161 151 171)(142 162 152 172)(143 163 153 173)(144 164 154 174)(145 165 155 175)(146 166 156 176)(147 167 157 177)(148 168 158 178)(149 169 159 179)(150 170 160 180)(181 211 191 201)(182 212 192 202)(183 213 193 203)(184 214 194 204)(185 215 195 205)(186 216 196 206)(187 217 197 207)(188 218 198 208)(189 219 199 209)(190 220 200 210)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)(177 180)(178 179)(181 190)(182 189)(183 188)(184 187)(185 186)(191 200)(192 199)(193 198)(194 197)(195 196)(201 210)(202 209)(203 208)(204 207)(205 206)(211 220)(212 219)(213 218)(214 217)(215 216)(221 234)(222 233)(223 232)(224 231)(225 230)(226 229)(227 228)(235 240)(236 239)(237 238)
G:=sub<Sym(240)| (1,28,169,206,58,118)(2,29,170,207,59,119)(3,30,171,208,60,120)(4,31,172,209,41,101)(5,32,173,210,42,102)(6,33,174,211,43,103)(7,34,175,212,44,104)(8,35,176,213,45,105)(9,36,177,214,46,106)(10,37,178,215,47,107)(11,38,179,216,48,108)(12,39,180,217,49,109)(13,40,161,218,50,110)(14,21,162,219,51,111)(15,22,163,220,52,112)(16,23,164,201,53,113)(17,24,165,202,54,114)(18,25,166,203,55,115)(19,26,167,204,56,116)(20,27,168,205,57,117)(61,126,238,94,149,196)(62,127,239,95,150,197)(63,128,240,96,151,198)(64,129,221,97,152,199)(65,130,222,98,153,200)(66,131,223,99,154,181)(67,132,224,100,155,182)(68,133,225,81,156,183)(69,134,226,82,157,184)(70,135,227,83,158,185)(71,136,228,84,159,186)(72,137,229,85,160,187)(73,138,230,86,141,188)(74,139,231,87,142,189)(75,140,232,88,143,190)(76,121,233,89,144,191)(77,122,234,90,145,192)(78,123,235,91,146,193)(79,124,236,92,147,194)(80,125,237,93,148,195), (1,228,11,238)(2,229,12,239)(3,230,13,240)(4,231,14,221)(5,232,15,222)(6,233,16,223)(7,234,17,224)(8,235,18,225)(9,236,19,226)(10,237,20,227)(21,97,31,87)(22,98,32,88)(23,99,33,89)(24,100,34,90)(25,81,35,91)(26,82,36,92)(27,83,37,93)(28,84,38,94)(29,85,39,95)(30,86,40,96)(41,74,51,64)(42,75,52,65)(43,76,53,66)(44,77,54,67)(45,78,55,68)(46,79,56,69)(47,80,57,70)(48,61,58,71)(49,62,59,72)(50,63,60,73)(101,139,111,129)(102,140,112,130)(103,121,113,131)(104,122,114,132)(105,123,115,133)(106,124,116,134)(107,125,117,135)(108,126,118,136)(109,127,119,137)(110,128,120,138)(141,161,151,171)(142,162,152,172)(143,163,153,173)(144,164,154,174)(145,165,155,175)(146,166,156,176)(147,167,157,177)(148,168,158,178)(149,169,159,179)(150,170,160,180)(181,211,191,201)(182,212,192,202)(183,213,193,203)(184,214,194,204)(185,215,195,205)(186,216,196,206)(187,217,197,207)(188,218,198,208)(189,219,199,209)(190,220,200,210), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)(177,180)(178,179)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,210)(202,209)(203,208)(204,207)(205,206)(211,220)(212,219)(213,218)(214,217)(215,216)(221,234)(222,233)(223,232)(224,231)(225,230)(226,229)(227,228)(235,240)(236,239)(237,238)>;
G:=Group( (1,28,169,206,58,118)(2,29,170,207,59,119)(3,30,171,208,60,120)(4,31,172,209,41,101)(5,32,173,210,42,102)(6,33,174,211,43,103)(7,34,175,212,44,104)(8,35,176,213,45,105)(9,36,177,214,46,106)(10,37,178,215,47,107)(11,38,179,216,48,108)(12,39,180,217,49,109)(13,40,161,218,50,110)(14,21,162,219,51,111)(15,22,163,220,52,112)(16,23,164,201,53,113)(17,24,165,202,54,114)(18,25,166,203,55,115)(19,26,167,204,56,116)(20,27,168,205,57,117)(61,126,238,94,149,196)(62,127,239,95,150,197)(63,128,240,96,151,198)(64,129,221,97,152,199)(65,130,222,98,153,200)(66,131,223,99,154,181)(67,132,224,100,155,182)(68,133,225,81,156,183)(69,134,226,82,157,184)(70,135,227,83,158,185)(71,136,228,84,159,186)(72,137,229,85,160,187)(73,138,230,86,141,188)(74,139,231,87,142,189)(75,140,232,88,143,190)(76,121,233,89,144,191)(77,122,234,90,145,192)(78,123,235,91,146,193)(79,124,236,92,147,194)(80,125,237,93,148,195), (1,228,11,238)(2,229,12,239)(3,230,13,240)(4,231,14,221)(5,232,15,222)(6,233,16,223)(7,234,17,224)(8,235,18,225)(9,236,19,226)(10,237,20,227)(21,97,31,87)(22,98,32,88)(23,99,33,89)(24,100,34,90)(25,81,35,91)(26,82,36,92)(27,83,37,93)(28,84,38,94)(29,85,39,95)(30,86,40,96)(41,74,51,64)(42,75,52,65)(43,76,53,66)(44,77,54,67)(45,78,55,68)(46,79,56,69)(47,80,57,70)(48,61,58,71)(49,62,59,72)(50,63,60,73)(101,139,111,129)(102,140,112,130)(103,121,113,131)(104,122,114,132)(105,123,115,133)(106,124,116,134)(107,125,117,135)(108,126,118,136)(109,127,119,137)(110,128,120,138)(141,161,151,171)(142,162,152,172)(143,163,153,173)(144,164,154,174)(145,165,155,175)(146,166,156,176)(147,167,157,177)(148,168,158,178)(149,169,159,179)(150,170,160,180)(181,211,191,201)(182,212,192,202)(183,213,193,203)(184,214,194,204)(185,215,195,205)(186,216,196,206)(187,217,197,207)(188,218,198,208)(189,219,199,209)(190,220,200,210), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,160)(158,159)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)(177,180)(178,179)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,210)(202,209)(203,208)(204,207)(205,206)(211,220)(212,219)(213,218)(214,217)(215,216)(221,234)(222,233)(223,232)(224,231)(225,230)(226,229)(227,228)(235,240)(236,239)(237,238) );
G=PermutationGroup([(1,28,169,206,58,118),(2,29,170,207,59,119),(3,30,171,208,60,120),(4,31,172,209,41,101),(5,32,173,210,42,102),(6,33,174,211,43,103),(7,34,175,212,44,104),(8,35,176,213,45,105),(9,36,177,214,46,106),(10,37,178,215,47,107),(11,38,179,216,48,108),(12,39,180,217,49,109),(13,40,161,218,50,110),(14,21,162,219,51,111),(15,22,163,220,52,112),(16,23,164,201,53,113),(17,24,165,202,54,114),(18,25,166,203,55,115),(19,26,167,204,56,116),(20,27,168,205,57,117),(61,126,238,94,149,196),(62,127,239,95,150,197),(63,128,240,96,151,198),(64,129,221,97,152,199),(65,130,222,98,153,200),(66,131,223,99,154,181),(67,132,224,100,155,182),(68,133,225,81,156,183),(69,134,226,82,157,184),(70,135,227,83,158,185),(71,136,228,84,159,186),(72,137,229,85,160,187),(73,138,230,86,141,188),(74,139,231,87,142,189),(75,140,232,88,143,190),(76,121,233,89,144,191),(77,122,234,90,145,192),(78,123,235,91,146,193),(79,124,236,92,147,194),(80,125,237,93,148,195)], [(1,228,11,238),(2,229,12,239),(3,230,13,240),(4,231,14,221),(5,232,15,222),(6,233,16,223),(7,234,17,224),(8,235,18,225),(9,236,19,226),(10,237,20,227),(21,97,31,87),(22,98,32,88),(23,99,33,89),(24,100,34,90),(25,81,35,91),(26,82,36,92),(27,83,37,93),(28,84,38,94),(29,85,39,95),(30,86,40,96),(41,74,51,64),(42,75,52,65),(43,76,53,66),(44,77,54,67),(45,78,55,68),(46,79,56,69),(47,80,57,70),(48,61,58,71),(49,62,59,72),(50,63,60,73),(101,139,111,129),(102,140,112,130),(103,121,113,131),(104,122,114,132),(105,123,115,133),(106,124,116,134),(107,125,117,135),(108,126,118,136),(109,127,119,137),(110,128,120,138),(141,161,151,171),(142,162,152,172),(143,163,153,173),(144,164,154,174),(145,165,155,175),(146,166,156,176),(147,167,157,177),(148,168,158,178),(149,169,159,179),(150,170,160,180),(181,211,191,201),(182,212,192,202),(183,213,193,203),(184,214,194,204),(185,215,195,205),(186,216,196,206),(187,217,197,207),(188,218,198,208),(189,219,199,209),(190,220,200,210)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,160),(158,159),(161,176),(162,175),(163,174),(164,173),(165,172),(166,171),(167,170),(168,169),(177,180),(178,179),(181,190),(182,189),(183,188),(184,187),(185,186),(191,200),(192,199),(193,198),(194,197),(195,196),(201,210),(202,209),(203,208),(204,207),(205,206),(211,220),(212,219),(213,218),(214,217),(215,216),(221,234),(222,233),(223,232),(224,231),(225,230),(226,229),(227,228),(235,240),(236,239),(237,238)])
Matrix representation ►G ⊆ GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 60 | 0 | 0 |
| 0 | 2 | 32 |
| 0 | 29 | 7 |
| 1 | 0 | 0 |
| 0 | 7 | 29 |
| 0 | 32 | 54 |
G:=sub<GL(3,GF(61))| [48,0,0,0,1,0,0,0,1],[60,0,0,0,11,0,0,0,11],[60,0,0,0,2,29,0,32,7],[1,0,0,0,7,32,0,29,54] >;
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 10A | ··· | 10N | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 |
| kernel | C6×C4○D20 | C6×Dic10 | D5×C2×C12 | C6×D20 | C3×C4○D20 | C6×C5⋊D4 | C22×C60 | C2×C4○D20 | C2×Dic10 | C2×C4×D5 | C2×D20 | C4○D20 | C2×C5⋊D4 | C22×C20 | C22×C12 | C30 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 2 | 4 | 2 | 16 | 4 | 2 | 2 | 4 | 12 | 2 | 4 | 8 | 24 | 4 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_6\times C_4\circ D_{20} % in TeX
G:=Group("C6xC4oD20"); // GroupNames label
G:=SmallGroup(480,1138);
// by ID
G=gap.SmallGroup(480,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀