direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D30.5C4, C30⋊7M4(2), C60.188C23, C3⋊C8⋊30D10, C5⋊2C8⋊30D6, C6⋊1(C8⋊D5), C20.74(C4×S3), C12.42(C4×D5), C10⋊3(C8⋊S3), C60.152(C2×C4), (C4×D15).14C4, D30.38(C2×C4), (C2×C20).332D6, C15⋊20(C2×M4(2)), (C2×C12).336D10, C20.185(C22×S3), C30.110(C22×C4), C4.23(D30.C2), (C2×C60).234C22, (C2×Dic15).23C4, Dic15.46(C2×C4), (C4×D15).66C22, (C22×D15).14C4, C12.185(C22×D5), C22.14(D30.C2), C5⋊5(C2×C8⋊S3), (C2×C3⋊C8)⋊12D5, C3⋊2(C2×C8⋊D5), (C10×C3⋊C8)⋊14C2, C6.42(C2×C4×D5), C10.75(S3×C2×C4), C4.158(C2×S3×D5), (C6×C5⋊2C8)⋊14C2, (C2×C5⋊2C8)⋊12S3, (C5×C3⋊C8)⋊37C22, (C2×C4×D15).21C2, (C2×C6).22(C4×D5), (C2×C10).46(C4×S3), (C2×C4).237(S3×D5), C2.7(C2×D30.C2), (C2×C30).107(C2×C4), (C3×C5⋊2C8)⋊37C22, SmallGroup(480,371)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D30.5C4
G = < a,b,c,d | a2=b30=c2=1, d4=b15, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b19, dcd-1=b3c >
Subgroups: 668 in 136 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60, D30, D30, C2×C30, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, C4×D15, C2×Dic15, C2×C60, C22×D15, C2×C8⋊D5, D30.5C4, C6×C5⋊2C8, C10×C3⋊C8, C2×C4×D15, C2×D30.5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, C8⋊S3, S3×C2×C4, S3×D5, C8⋊D5, C2×C4×D5, C2×C8⋊S3, D30.C2, C2×S3×D5, C2×C8⋊D5, D30.5C4, C2×D30.C2, C2×D30.5C4
►On 240 points
Generators in S240
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(121 204)(122 205)(123 206)(124 207)(125 208)(126 209)(127 210)(128 181)(129 182)(130 183)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)(141 194)(142 195)(143 196)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 72)(29 71)(30 70)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(121 186)(122 185)(123 184)(124 183)(125 182)(126 181)(127 210)(128 209)(129 208)(130 207)(131 206)(132 205)(133 204)(134 203)(135 202)(136 201)(137 200)(138 199)(139 198)(140 197)(141 196)(142 195)(143 194)(144 193)(145 192)(146 191)(147 190)(148 189)(149 188)(150 187)(151 225)(152 224)(153 223)(154 222)(155 221)(156 220)(157 219)(158 218)(159 217)(160 216)(161 215)(162 214)(163 213)(164 212)(165 211)(166 240)(167 239)(168 238)(169 237)(170 236)(171 235)(172 234)(173 233)(174 232)(175 231)(176 230)(177 229)(178 228)(179 227)(180 226)
(1 166 59 135 16 151 44 150)(2 155 60 124 17 170 45 139)(3 174 31 143 18 159 46 128)(4 163 32 132 19 178 47 147)(5 152 33 121 20 167 48 136)(6 171 34 140 21 156 49 125)(7 160 35 129 22 175 50 144)(8 179 36 148 23 164 51 133)(9 168 37 137 24 153 52 122)(10 157 38 126 25 172 53 141)(11 176 39 145 26 161 54 130)(12 165 40 134 27 180 55 149)(13 154 41 123 28 169 56 138)(14 173 42 142 29 158 57 127)(15 162 43 131 30 177 58 146)(61 235 102 197 76 220 117 182)(62 224 103 186 77 239 118 201)(63 213 104 205 78 228 119 190)(64 232 105 194 79 217 120 209)(65 221 106 183 80 236 91 198)(66 240 107 202 81 225 92 187)(67 229 108 191 82 214 93 206)(68 218 109 210 83 233 94 195)(69 237 110 199 84 222 95 184)(70 226 111 188 85 211 96 203)(71 215 112 207 86 230 97 192)(72 234 113 196 87 219 98 181)(73 223 114 185 88 238 99 200)(74 212 115 204 89 227 100 189)(75 231 116 193 90 216 101 208)
G:=sub<Sym(240)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,210)(128,209)(129,208)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)(151,225)(152,224)(153,223)(154,222)(155,221)(156,220)(157,219)(158,218)(159,217)(160,216)(161,215)(162,214)(163,213)(164,212)(165,211)(166,240)(167,239)(168,238)(169,237)(170,236)(171,235)(172,234)(173,233)(174,232)(175,231)(176,230)(177,229)(178,228)(179,227)(180,226), (1,166,59,135,16,151,44,150)(2,155,60,124,17,170,45,139)(3,174,31,143,18,159,46,128)(4,163,32,132,19,178,47,147)(5,152,33,121,20,167,48,136)(6,171,34,140,21,156,49,125)(7,160,35,129,22,175,50,144)(8,179,36,148,23,164,51,133)(9,168,37,137,24,153,52,122)(10,157,38,126,25,172,53,141)(11,176,39,145,26,161,54,130)(12,165,40,134,27,180,55,149)(13,154,41,123,28,169,56,138)(14,173,42,142,29,158,57,127)(15,162,43,131,30,177,58,146)(61,235,102,197,76,220,117,182)(62,224,103,186,77,239,118,201)(63,213,104,205,78,228,119,190)(64,232,105,194,79,217,120,209)(65,221,106,183,80,236,91,198)(66,240,107,202,81,225,92,187)(67,229,108,191,82,214,93,206)(68,218,109,210,83,233,94,195)(69,237,110,199,84,222,95,184)(70,226,111,188,85,211,96,203)(71,215,112,207,86,230,97,192)(72,234,113,196,87,219,98,181)(73,223,114,185,88,238,99,200)(74,212,115,204,89,227,100,189)(75,231,116,193,90,216,101,208)>;
G:=Group( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(121,204)(122,205)(123,206)(124,207)(125,208)(126,209)(127,210)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,210)(128,209)(129,208)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)(151,225)(152,224)(153,223)(154,222)(155,221)(156,220)(157,219)(158,218)(159,217)(160,216)(161,215)(162,214)(163,213)(164,212)(165,211)(166,240)(167,239)(168,238)(169,237)(170,236)(171,235)(172,234)(173,233)(174,232)(175,231)(176,230)(177,229)(178,228)(179,227)(180,226), (1,166,59,135,16,151,44,150)(2,155,60,124,17,170,45,139)(3,174,31,143,18,159,46,128)(4,163,32,132,19,178,47,147)(5,152,33,121,20,167,48,136)(6,171,34,140,21,156,49,125)(7,160,35,129,22,175,50,144)(8,179,36,148,23,164,51,133)(9,168,37,137,24,153,52,122)(10,157,38,126,25,172,53,141)(11,176,39,145,26,161,54,130)(12,165,40,134,27,180,55,149)(13,154,41,123,28,169,56,138)(14,173,42,142,29,158,57,127)(15,162,43,131,30,177,58,146)(61,235,102,197,76,220,117,182)(62,224,103,186,77,239,118,201)(63,213,104,205,78,228,119,190)(64,232,105,194,79,217,120,209)(65,221,106,183,80,236,91,198)(66,240,107,202,81,225,92,187)(67,229,108,191,82,214,93,206)(68,218,109,210,83,233,94,195)(69,237,110,199,84,222,95,184)(70,226,111,188,85,211,96,203)(71,215,112,207,86,230,97,192)(72,234,113,196,87,219,98,181)(73,223,114,185,88,238,99,200)(74,212,115,204,89,227,100,189)(75,231,116,193,90,216,101,208) );
G=PermutationGroup([[(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(121,204),(122,205),(123,206),(124,207),(125,208),(126,209),(127,210),(128,181),(129,182),(130,183),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,191),(139,192),(140,193),(141,194),(142,195),(143,196),(144,197),(145,198),(146,199),(147,200),(148,201),(149,202),(150,203),(151,211),(152,212),(153,213),(154,214),(155,215),(156,216),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,90),(11,89),(12,88),(13,87),(14,86),(15,85),(16,84),(17,83),(18,82),(19,81),(20,80),(21,79),(22,78),(23,77),(24,76),(25,75),(26,74),(27,73),(28,72),(29,71),(30,70),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109),(121,186),(122,185),(123,184),(124,183),(125,182),(126,181),(127,210),(128,209),(129,208),(130,207),(131,206),(132,205),(133,204),(134,203),(135,202),(136,201),(137,200),(138,199),(139,198),(140,197),(141,196),(142,195),(143,194),(144,193),(145,192),(146,191),(147,190),(148,189),(149,188),(150,187),(151,225),(152,224),(153,223),(154,222),(155,221),(156,220),(157,219),(158,218),(159,217),(160,216),(161,215),(162,214),(163,213),(164,212),(165,211),(166,240),(167,239),(168,238),(169,237),(170,236),(171,235),(172,234),(173,233),(174,232),(175,231),(176,230),(177,229),(178,228),(179,227),(180,226)], [(1,166,59,135,16,151,44,150),(2,155,60,124,17,170,45,139),(3,174,31,143,18,159,46,128),(4,163,32,132,19,178,47,147),(5,152,33,121,20,167,48,136),(6,171,34,140,21,156,49,125),(7,160,35,129,22,175,50,144),(8,179,36,148,23,164,51,133),(9,168,37,137,24,153,52,122),(10,157,38,126,25,172,53,141),(11,176,39,145,26,161,54,130),(12,165,40,134,27,180,55,149),(13,154,41,123,28,169,56,138),(14,173,42,142,29,158,57,127),(15,162,43,131,30,177,58,146),(61,235,102,197,76,220,117,182),(62,224,103,186,77,239,118,201),(63,213,104,205,78,228,119,190),(64,232,105,194,79,217,120,209),(65,221,106,183,80,236,91,198),(66,240,107,202,81,225,92,187),(67,229,108,191,82,214,93,206),(68,218,109,210,83,233,94,195),(69,237,110,199,84,222,95,184),(70,226,111,188,85,211,96,203),(71,215,112,207,86,230,97,192),(72,234,113,196,87,219,98,181),(73,223,114,185,88,238,99,200),(74,212,115,204,89,227,100,189),(75,231,116,193,90,216,101,208)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | M4(2) | D10 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | C8⋊S3 | C8⋊D5 | S3×D5 | D30.C2 | C2×S3×D5 | D30.C2 | D30.5C4 |
| kernel | C2×D30.5C4 | D30.5C4 | C6×C5⋊2C8 | C10×C3⋊C8 | C2×C4×D15 | C4×D15 | C2×Dic15 | C22×D15 | C2×C5⋊2C8 | C2×C3⋊C8 | C5⋊2C8 | C2×C20 | C30 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×D30.5C4 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 192 |
| 0 | 0 | 0 | 0 | 64 | 2 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 64 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 49 |
| 0 | 0 | 0 | 0 | 177 | 239 |
| 64 | 2 | 0 | 0 | 0 | 0 |
| 89 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 190 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 207 | 14 |
| 0 | 0 | 0 | 0 | 85 | 34 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,240,64,0,0,0,0,192,2],[240,64,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,177,0,0,0,0,49,239],[64,89,0,0,0,0,2,177,0,0,0,0,0,0,240,190,0,0,0,0,0,1,0,0,0,0,0,0,207,85,0,0,0,0,14,34] >;
C2×D30.5C4 in GAP, Magma, Sage, TeX
C_2\times D_{30}._5C_4 % in TeX
G:=Group("C2xD30.5C4"); // GroupNames label
G:=SmallGroup(480,371);
// by ID
G=gap.SmallGroup(480,371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,64,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^2=1,d^4=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^19,d*c*d^-1=b^3*c>;
// generators/relations