direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D30⋊4C4, C10⋊2(D6⋊C4), D30⋊29(C2×C4), C6.66(C2×D20), (C2×C6).45D20, (C2×C30).71D4, C30⋊6(C22⋊C4), (C2×Dic5)⋊20D6, (C2×C10).45D12, C30.220(C2×D4), C10.65(C2×D12), (C22×D15)⋊9C4, C6⋊1(D10⋊C4), C23.63(S3×D5), (C2×Dic3)⋊20D10, (C22×Dic3)⋊3D5, (C22×Dic5)⋊6S3, (C23×D15).3C2, (C22×C6).85D10, (C2×C30).182C23, C30.139(C22×C4), (C6×Dic5)⋊24C22, (C22×C10).102D6, (C10×Dic3)⋊24C22, C22.24(C5⋊D12), C22.24(C3⋊D20), (C22×C30).44C22, C22.16(D30.C2), (C22×D15).112C22, C5⋊3(C2×D6⋊C4), C6.52(C2×C4×D5), C10.84(S3×C2×C4), (C2×C6×Dic5)⋊3C2, C3⋊2(C2×D10⋊C4), C15⋊10(C2×C22⋊C4), (Dic3×C2×C10)⋊3C2, (C2×C6).23(C4×D5), C6.19(C2×C5⋊D4), C2.3(C2×C3⋊D20), C2.3(C2×C5⋊D12), C22.79(C2×S3×D5), (C2×C10).48(C4×S3), C10.20(C2×C3⋊D4), (C2×C30).114(C2×C4), (C2×C6).36(C5⋊D4), C2.16(C2×D30.C2), (C2×C10).36(C3⋊D4), (C2×C6).194(C22×D5), (C2×C10).194(C22×S3), SmallGroup(480,616)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D30⋊4C4
G = < a,b,c,d | a2=b30=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b25c >
Subgroups: 1628 in 264 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, C30, C30, C2×C22⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C3×Dic5, D30, D30, C2×C30, C2×C30, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C2×D6⋊C4, C6×Dic5, C6×Dic5, C10×Dic3, C10×Dic3, C22×D15, C22×D15, C22×C30, C2×D10⋊C4, D30⋊4C4, C2×C6×Dic5, Dic3×C2×C10, C23×D15, C2×D30⋊4C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×D5, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D6⋊C4, D30.C2, C3⋊D20, C5⋊D12, C2×S3×D5, C2×D10⋊C4, D30⋊4C4, C2×D30.C2, C2×C3⋊D20, C2×C5⋊D12, C2×D30⋊4C4
►On 240 points
Generators in S240
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(121 234)(122 235)(123 236)(124 237)(125 238)(126 239)(127 240)(128 211)(129 212)(130 213)(131 214)(132 215)(133 216)(134 217)(135 218)(136 219)(137 220)(138 221)(139 222)(140 223)(141 224)(142 225)(143 226)(144 227)(145 228)(146 229)(147 230)(148 231)(149 232)(150 233)(151 189)(152 190)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)
(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 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(121 217)(122 216)(123 215)(124 214)(125 213)(126 212)(127 211)(128 240)(129 239)(130 238)(131 237)(132 236)(133 235)(134 234)(135 233)(136 232)(137 231)(138 230)(139 229)(140 228)(141 227)(142 226)(143 225)(144 224)(145 223)(146 222)(147 221)(148 220)(149 219)(150 218)(151 186)(152 185)(153 184)(154 183)(155 182)(156 181)(157 210)(158 209)(159 208)(160 207)(161 206)(162 205)(163 204)(164 203)(165 202)(166 201)(167 200)(168 199)(169 198)(170 197)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)
(1 211 36 203)(2 222 37 184)(3 233 38 195)(4 214 39 206)(5 225 40 187)(6 236 41 198)(7 217 42 209)(8 228 43 190)(9 239 44 201)(10 220 45 182)(11 231 46 193)(12 212 47 204)(13 223 48 185)(14 234 49 196)(15 215 50 207)(16 226 51 188)(17 237 52 199)(18 218 53 210)(19 229 54 191)(20 240 55 202)(21 221 56 183)(22 232 57 194)(23 213 58 205)(24 224 59 186)(25 235 60 197)(26 216 31 208)(27 227 32 189)(28 238 33 200)(29 219 34 181)(30 230 35 192)(61 174 94 137)(62 155 95 148)(63 166 96 129)(64 177 97 140)(65 158 98 121)(66 169 99 132)(67 180 100 143)(68 161 101 124)(69 172 102 135)(70 153 103 146)(71 164 104 127)(72 175 105 138)(73 156 106 149)(74 167 107 130)(75 178 108 141)(76 159 109 122)(77 170 110 133)(78 151 111 144)(79 162 112 125)(80 173 113 136)(81 154 114 147)(82 165 115 128)(83 176 116 139)(84 157 117 150)(85 168 118 131)(86 179 119 142)(87 160 120 123)(88 171 91 134)(89 152 92 145)(90 163 93 126)
G:=sub<Sym(240)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,211)(129,212)(130,213)(131,214)(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)(139,222)(140,223)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)(147,230)(148,231)(149,232)(150,233)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,120)(11,119)(12,118)(13,117)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,240)(129,239)(130,238)(131,237)(132,236)(133,235)(134,234)(135,233)(136,232)(137,231)(138,230)(139,229)(140,228)(141,227)(142,226)(143,225)(144,224)(145,223)(146,222)(147,221)(148,220)(149,219)(150,218)(151,186)(152,185)(153,184)(154,183)(155,182)(156,181)(157,210)(158,209)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187), (1,211,36,203)(2,222,37,184)(3,233,38,195)(4,214,39,206)(5,225,40,187)(6,236,41,198)(7,217,42,209)(8,228,43,190)(9,239,44,201)(10,220,45,182)(11,231,46,193)(12,212,47,204)(13,223,48,185)(14,234,49,196)(15,215,50,207)(16,226,51,188)(17,237,52,199)(18,218,53,210)(19,229,54,191)(20,240,55,202)(21,221,56,183)(22,232,57,194)(23,213,58,205)(24,224,59,186)(25,235,60,197)(26,216,31,208)(27,227,32,189)(28,238,33,200)(29,219,34,181)(30,230,35,192)(61,174,94,137)(62,155,95,148)(63,166,96,129)(64,177,97,140)(65,158,98,121)(66,169,99,132)(67,180,100,143)(68,161,101,124)(69,172,102,135)(70,153,103,146)(71,164,104,127)(72,175,105,138)(73,156,106,149)(74,167,107,130)(75,178,108,141)(76,159,109,122)(77,170,110,133)(78,151,111,144)(79,162,112,125)(80,173,113,136)(81,154,114,147)(82,165,115,128)(83,176,116,139)(84,157,117,150)(85,168,118,131)(86,179,119,142)(87,160,120,123)(88,171,91,134)(89,152,92,145)(90,163,93,126)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,211)(129,212)(130,213)(131,214)(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)(139,222)(140,223)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)(147,230)(148,231)(149,232)(150,233)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188), (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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,120)(11,119)(12,118)(13,117)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,240)(129,239)(130,238)(131,237)(132,236)(133,235)(134,234)(135,233)(136,232)(137,231)(138,230)(139,229)(140,228)(141,227)(142,226)(143,225)(144,224)(145,223)(146,222)(147,221)(148,220)(149,219)(150,218)(151,186)(152,185)(153,184)(154,183)(155,182)(156,181)(157,210)(158,209)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187), (1,211,36,203)(2,222,37,184)(3,233,38,195)(4,214,39,206)(5,225,40,187)(6,236,41,198)(7,217,42,209)(8,228,43,190)(9,239,44,201)(10,220,45,182)(11,231,46,193)(12,212,47,204)(13,223,48,185)(14,234,49,196)(15,215,50,207)(16,226,51,188)(17,237,52,199)(18,218,53,210)(19,229,54,191)(20,240,55,202)(21,221,56,183)(22,232,57,194)(23,213,58,205)(24,224,59,186)(25,235,60,197)(26,216,31,208)(27,227,32,189)(28,238,33,200)(29,219,34,181)(30,230,35,192)(61,174,94,137)(62,155,95,148)(63,166,96,129)(64,177,97,140)(65,158,98,121)(66,169,99,132)(67,180,100,143)(68,161,101,124)(69,172,102,135)(70,153,103,146)(71,164,104,127)(72,175,105,138)(73,156,106,149)(74,167,107,130)(75,178,108,141)(76,159,109,122)(77,170,110,133)(78,151,111,144)(79,162,112,125)(80,173,113,136)(81,154,114,147)(82,165,115,128)(83,176,116,139)(84,157,117,150)(85,168,118,131)(86,179,119,142)(87,160,120,123)(88,171,91,134)(89,152,92,145)(90,163,93,126) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(121,234),(122,235),(123,236),(124,237),(125,238),(126,239),(127,240),(128,211),(129,212),(130,213),(131,214),(132,215),(133,216),(134,217),(135,218),(136,219),(137,220),(138,221),(139,222),(140,223),(141,224),(142,225),(143,226),(144,227),(145,228),(146,229),(147,230),(148,231),(149,232),(150,233),(151,189),(152,190),(153,191),(154,192),(155,193),(156,194),(157,195),(158,196),(159,197),(160,198),(161,199),(162,200),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206),(169,207),(170,208),(171,209),(172,210),(173,181),(174,182),(175,183),(176,184),(177,185),(178,186),(179,187),(180,188)], [(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)], [(1,99),(2,98),(3,97),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,120),(11,119),(12,118),(13,117),(14,116),(15,115),(16,114),(17,113),(18,112),(19,111),(20,110),(21,109),(22,108),(23,107),(24,106),(25,105),(26,104),(27,103),(28,102),(29,101),(30,100),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(121,217),(122,216),(123,215),(124,214),(125,213),(126,212),(127,211),(128,240),(129,239),(130,238),(131,237),(132,236),(133,235),(134,234),(135,233),(136,232),(137,231),(138,230),(139,229),(140,228),(141,227),(142,226),(143,225),(144,224),(145,223),(146,222),(147,221),(148,220),(149,219),(150,218),(151,186),(152,185),(153,184),(154,183),(155,182),(156,181),(157,210),(158,209),(159,208),(160,207),(161,206),(162,205),(163,204),(164,203),(165,202),(166,201),(167,200),(168,199),(169,198),(170,197),(171,196),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187)], [(1,211,36,203),(2,222,37,184),(3,233,38,195),(4,214,39,206),(5,225,40,187),(6,236,41,198),(7,217,42,209),(8,228,43,190),(9,239,44,201),(10,220,45,182),(11,231,46,193),(12,212,47,204),(13,223,48,185),(14,234,49,196),(15,215,50,207),(16,226,51,188),(17,237,52,199),(18,218,53,210),(19,229,54,191),(20,240,55,202),(21,221,56,183),(22,232,57,194),(23,213,58,205),(24,224,59,186),(25,235,60,197),(26,216,31,208),(27,227,32,189),(28,238,33,200),(29,219,34,181),(30,230,35,192),(61,174,94,137),(62,155,95,148),(63,166,96,129),(64,177,97,140),(65,158,98,121),(66,169,99,132),(67,180,100,143),(68,161,101,124),(69,172,102,135),(70,153,103,146),(71,164,104,127),(72,175,105,138),(73,156,106,149),(74,167,107,130),(75,178,108,141),(76,159,109,122),(77,170,110,133),(78,151,111,144),(79,162,112,125),(80,173,113,136),(81,154,114,147),(82,165,115,128),(83,176,116,139),(84,157,117,150),(85,168,118,131),(86,179,119,142),(87,160,120,123),(88,171,91,134),(89,152,92,145),(90,163,93,126)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 30 | 30 | 30 | 30 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 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 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | C4×S3 | D12 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | S3×D5 | D30.C2 | C3⋊D20 | C5⋊D12 | C2×S3×D5 |
| kernel | C2×D30⋊4C4 | D30⋊4C4 | C2×C6×Dic5 | Dic3×C2×C10 | C23×D15 | C22×D15 | C22×Dic5 | C2×C30 | C22×Dic3 | C2×Dic5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C2×D30⋊4C4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 42 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 42 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 43 | 0 | 0 |
| 0 | 0 | 52 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,60,42,0,0,0,0,1,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,42,0,0,0,0,0,1],[11,0,0,0,0,0,0,60,0,0,0,0,0,0,52,52,0,0,0,0,43,9,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C2×D30⋊4C4 in GAP, Magma, Sage, TeX
C_2\times D_{30}\rtimes_4C_4 % in TeX
G:=Group("C2xD30:4C4"); // GroupNames label
G:=SmallGroup(480,616);
// by ID
G=gap.SmallGroup(480,616);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,176,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^25*c>;
// generators/relations