direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.28D10, Dic6⋊24D10, D60⋊37C22, C30.13C24, D30.4C23, C60.137C23, (C4×D5)⋊15D6, (C2×D60)⋊28C2, C30⋊7(C4○D4), C10⋊1(C4○D12), C6⋊1(Q8⋊2D5), (C10×Dic6)⋊8C2, (C2×Dic6)⋊15D5, (C2×C20).169D6, C3⋊D20⋊9C22, C6.13(C23×D5), (C2×C12).311D10, (D5×C12)⋊17C22, C10.13(S3×C23), D30.C2⋊6C22, (C6×D5).40C23, (C22×D5).98D6, (C2×C60).155C22, C20.129(C22×S3), (C2×C30).232C23, (C2×Dic5).220D6, (C5×Dic6)⋊21C22, D10.42(C22×S3), C12.161(C22×D5), (C5×Dic3).7C23, Dic3.7(C22×D5), (C2×Dic3).131D10, Dic5.56(C22×S3), (C3×Dic5).42C23, (C6×Dic5).229C22, (C22×D15).73C22, (C10×Dic3).130C22, (C2×C4×D5)⋊4S3, (D5×C2×C12)⋊5C2, C5⋊1(C2×C4○D12), C15⋊7(C2×C4○D4), C4.87(C2×S3×D5), C3⋊1(C2×Q8⋊2D5), (C2×C3⋊D20)⋊18C2, C2.17(C22×S3×D5), (C2×C4).168(S3×D5), C22.101(C2×S3×D5), (C2×D30.C2)⋊20C2, (D5×C2×C6).117C22, (C2×C6).242(C22×D5), (C2×C10).242(C22×S3), SmallGroup(480,1085)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.28D10
G = < a,b,c,d | a2=b12=d2=1, c10=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b6c9 >
Subgroups: 1692 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C30, C2×C4○D4, C4×D5, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5, D30, D30, C2×C30, C2×C4×D5, C2×C4×D5, C2×D20, Q8⋊2D5, Q8×C10, C2×C4○D12, D30.C2, C3⋊D20, D5×C12, C6×Dic5, C5×Dic6, C10×Dic3, D60, C2×C60, D5×C2×C6, C22×D15, C2×Q8⋊2D5, C12.28D10, C2×D30.C2, C2×C3⋊D20, D5×C2×C12, C10×Dic6, C2×D60, C2×C12.28D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, C4○D12, S3×C23, S3×D5, Q8⋊2D5, C23×D5, C2×C4○D12, C2×S3×D5, C2×Q8⋊2D5, C12.28D10, C22×S3×D5, C2×C12.28D10
►On 240 points
Generators in S240
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 213)(42 214)(43 215)(44 216)(45 217)(46 218)(47 219)(48 220)(49 201)(50 202)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 209)(58 210)(59 211)(60 212)(61 221)(62 222)(63 223)(64 224)(65 225)(66 226)(67 227)(68 228)(69 229)(70 230)(71 231)(72 232)(73 233)(74 234)(75 235)(76 236)(77 237)(78 238)(79 239)(80 240)(81 198)(82 199)(83 200)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 193)(97 194)(98 195)(99 196)(100 197)(141 173)(142 174)(143 175)(144 176)(145 177)(146 178)(147 179)(148 180)(149 161)(150 162)(151 163)(152 164)(153 165)(154 166)(155 167)(156 168)(157 169)(158 170)(159 171)(160 172)
(1 85 143 67 108 210 11 95 153 77 118 220)(2 201 119 78 154 96 12 211 109 68 144 86)(3 87 145 69 110 212 13 97 155 79 120 202)(4 203 101 80 156 98 14 213 111 70 146 88)(5 89 147 71 112 214 15 99 157 61 102 204)(6 205 103 62 158 100 16 215 113 72 148 90)(7 91 149 73 114 216 17 81 159 63 104 206)(8 207 105 64 160 82 18 217 115 74 150 92)(9 93 151 75 116 218 19 83 141 65 106 208)(10 209 107 66 142 84 20 219 117 76 152 94)(21 44 131 198 171 223 31 54 121 188 161 233)(22 234 162 189 122 55 32 224 172 199 132 45)(23 46 133 200 173 225 33 56 123 190 163 235)(24 236 164 191 124 57 34 226 174 181 134 47)(25 48 135 182 175 227 35 58 125 192 165 237)(26 238 166 193 126 59 36 228 176 183 136 49)(27 50 137 184 177 229 37 60 127 194 167 239)(28 240 168 195 128 41 38 230 178 185 138 51)(29 52 139 186 179 231 39 42 129 196 169 221)(30 222 170 197 130 43 40 232 180 187 140 53)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 140)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(179 180)(181 192)(182 191)(183 190)(184 189)(185 188)(186 187)(193 200)(194 199)(195 198)(196 197)(201 208)(202 207)(203 206)(204 205)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)
G:=sub<Sym(240)| (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,213)(42,214)(43,215)(44,216)(45,217)(46,218)(47,219)(48,220)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,209)(58,210)(59,211)(60,212)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,231)(72,232)(73,233)(74,234)(75,235)(76,236)(77,237)(78,238)(79,239)(80,240)(81,198)(82,199)(83,200)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(141,173)(142,174)(143,175)(144,176)(145,177)(146,178)(147,179)(148,180)(149,161)(150,162)(151,163)(152,164)(153,165)(154,166)(155,167)(156,168)(157,169)(158,170)(159,171)(160,172), (1,85,143,67,108,210,11,95,153,77,118,220)(2,201,119,78,154,96,12,211,109,68,144,86)(3,87,145,69,110,212,13,97,155,79,120,202)(4,203,101,80,156,98,14,213,111,70,146,88)(5,89,147,71,112,214,15,99,157,61,102,204)(6,205,103,62,158,100,16,215,113,72,148,90)(7,91,149,73,114,216,17,81,159,63,104,206)(8,207,105,64,160,82,18,217,115,74,150,92)(9,93,151,75,116,218,19,83,141,65,106,208)(10,209,107,66,142,84,20,219,117,76,152,94)(21,44,131,198,171,223,31,54,121,188,161,233)(22,234,162,189,122,55,32,224,172,199,132,45)(23,46,133,200,173,225,33,56,123,190,163,235)(24,236,164,191,124,57,34,226,174,181,134,47)(25,48,135,182,175,227,35,58,125,192,165,237)(26,238,166,193,126,59,36,228,176,183,136,49)(27,50,137,184,177,229,37,60,127,194,167,239)(28,240,168,195,128,41,38,230,178,185,138,51)(29,52,139,186,179,231,39,42,129,196,169,221)(30,222,170,197,130,43,40,232,180,187,140,53), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,44)(42,43)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(179,180)(181,192)(182,191)(183,190)(184,189)(185,188)(186,187)(193,200)(194,199)(195,198)(196,197)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232)>;
G:=Group( (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,213)(42,214)(43,215)(44,216)(45,217)(46,218)(47,219)(48,220)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,209)(58,210)(59,211)(60,212)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,231)(72,232)(73,233)(74,234)(75,235)(76,236)(77,237)(78,238)(79,239)(80,240)(81,198)(82,199)(83,200)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(141,173)(142,174)(143,175)(144,176)(145,177)(146,178)(147,179)(148,180)(149,161)(150,162)(151,163)(152,164)(153,165)(154,166)(155,167)(156,168)(157,169)(158,170)(159,171)(160,172), (1,85,143,67,108,210,11,95,153,77,118,220)(2,201,119,78,154,96,12,211,109,68,144,86)(3,87,145,69,110,212,13,97,155,79,120,202)(4,203,101,80,156,98,14,213,111,70,146,88)(5,89,147,71,112,214,15,99,157,61,102,204)(6,205,103,62,158,100,16,215,113,72,148,90)(7,91,149,73,114,216,17,81,159,63,104,206)(8,207,105,64,160,82,18,217,115,74,150,92)(9,93,151,75,116,218,19,83,141,65,106,208)(10,209,107,66,142,84,20,219,117,76,152,94)(21,44,131,198,171,223,31,54,121,188,161,233)(22,234,162,189,122,55,32,224,172,199,132,45)(23,46,133,200,173,225,33,56,123,190,163,235)(24,236,164,191,124,57,34,226,174,181,134,47)(25,48,135,182,175,227,35,58,125,192,165,237)(26,238,166,193,126,59,36,228,176,183,136,49)(27,50,137,184,177,229,37,60,127,194,167,239)(28,240,168,195,128,41,38,230,178,185,138,51)(29,52,139,186,179,231,39,42,129,196,169,221)(30,222,170,197,130,43,40,232,180,187,140,53), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,44)(42,43)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(179,180)(181,192)(182,191)(183,190)(184,189)(185,188)(186,187)(193,200)(194,199)(195,198)(196,197)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232) );
G=PermutationGroup([[(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(41,213),(42,214),(43,215),(44,216),(45,217),(46,218),(47,219),(48,220),(49,201),(50,202),(51,203),(52,204),(53,205),(54,206),(55,207),(56,208),(57,209),(58,210),(59,211),(60,212),(61,221),(62,222),(63,223),(64,224),(65,225),(66,226),(67,227),(68,228),(69,229),(70,230),(71,231),(72,232),(73,233),(74,234),(75,235),(76,236),(77,237),(78,238),(79,239),(80,240),(81,198),(82,199),(83,200),(84,181),(85,182),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,193),(97,194),(98,195),(99,196),(100,197),(141,173),(142,174),(143,175),(144,176),(145,177),(146,178),(147,179),(148,180),(149,161),(150,162),(151,163),(152,164),(153,165),(154,166),(155,167),(156,168),(157,169),(158,170),(159,171),(160,172)], [(1,85,143,67,108,210,11,95,153,77,118,220),(2,201,119,78,154,96,12,211,109,68,144,86),(3,87,145,69,110,212,13,97,155,79,120,202),(4,203,101,80,156,98,14,213,111,70,146,88),(5,89,147,71,112,214,15,99,157,61,102,204),(6,205,103,62,158,100,16,215,113,72,148,90),(7,91,149,73,114,216,17,81,159,63,104,206),(8,207,105,64,160,82,18,217,115,74,150,92),(9,93,151,75,116,218,19,83,141,65,106,208),(10,209,107,66,142,84,20,219,117,76,152,94),(21,44,131,198,171,223,31,54,121,188,161,233),(22,234,162,189,122,55,32,224,172,199,132,45),(23,46,133,200,173,225,33,56,123,190,163,235),(24,236,164,191,124,57,34,226,174,181,134,47),(25,48,135,182,175,227,35,58,125,192,165,237),(26,238,166,193,126,59,36,228,176,183,136,49),(27,50,137,184,177,229,37,60,127,194,167,239),(28,240,168,195,128,41,38,230,178,185,138,51),(29,52,139,186,179,231,39,42,129,196,169,221),(30,222,170,197,130,43,40,232,180,187,140,53)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,44),(42,43),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(99,100),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(139,140),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148),(155,160),(156,159),(157,158),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(179,180),(181,192),(182,191),(183,190),(184,189),(185,188),(186,187),(193,200),(194,199),(195,198),(196,197),(201,208),(202,207),(203,206),(204,205),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215),(221,222),(223,240),(224,239),(225,238),(226,237),(227,236),(228,235),(229,234),(230,233),(231,232)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | C2×S3×D5 | C12.28D10 |
| kernel | C2×C12.28D10 | C12.28D10 | C2×D30.C2 | C2×C3⋊D20 | D5×C2×C12 | C10×Dic6 | C2×D60 | C2×C4×D5 | C2×Dic6 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | Dic6 | C2×Dic3 | C2×C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 8 | 4 | 2 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C2×C12.28D10 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 19 | 29 |
| 1 | 44 | 0 | 0 |
| 17 | 17 | 0 | 0 |
| 0 | 0 | 32 | 20 |
| 0 | 0 | 25 | 29 |
| 1 | 44 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 29 | 41 |
| 0 | 0 | 42 | 32 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,40,19,0,0,0,29],[1,17,0,0,44,17,0,0,0,0,32,25,0,0,20,29],[1,0,0,0,44,60,0,0,0,0,29,42,0,0,41,32] >;
C2×C12.28D10 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{28}D_{10} % in TeX
G:=Group("C2xC12.28D10"); // GroupNames label
G:=SmallGroup(480,1085);
// by ID
G=gap.SmallGroup(480,1085);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=d^2=1,c^10=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^6*c^9>;
// generators/relations