direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D30⋊3C4, C22.16D60, C23.37D30, (C2×C4)⋊8D30, (C2×C20)⋊32D6, C2.3(C2×D60), C10⋊3(D6⋊C4), D30⋊28(C2×C4), (C2×C12)⋊32D10, (C22×C20)⋊8S3, (C22×C60)⋊5C2, C6.44(C2×D20), (C2×C6).23D20, (C22×C12)⋊4D5, (C22×C4)⋊3D15, C30⋊7(C22⋊C4), (C2×C60)⋊42C22, C10.45(C2×D12), (C2×C30).143D4, (C2×C10).23D12, C30.272(C2×D4), (C22×D15)⋊8C4, C6⋊2(D10⋊C4), (C23×D15).2C2, C22.17(C4×D15), C30.170(C22×C4), (C2×C30).301C23, (C22×Dic15)⋊3C2, (C22×C6).118D10, (C22×C10).136D6, (C2×Dic15)⋊23C22, C22.20(C15⋊7D4), C22.23(C22×D15), (C22×C30).141C22, (C22×D15).85C22, C5⋊4(C2×D6⋊C4), C6.75(C2×C4×D5), C2.19(C2×C4×D15), C3⋊3(C2×D10⋊C4), C15⋊17(C2×C22⋊C4), C10.107(S3×C2×C4), (C2×C6).36(C4×D5), C2.2(C2×C15⋊7D4), C6.99(C2×C5⋊D4), (C2×C10).61(C4×S3), C10.99(C2×C3⋊D4), (C2×C30).142(C2×C4), (C2×C6).75(C5⋊D4), (C2×C10).75(C3⋊D4), (C2×C6).297(C22×D5), (C2×C10).296(C22×S3), SmallGroup(480,892)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D30⋊3C4
G = < a,b,c,d | a2=b30=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >
Subgroups: 1716 in 264 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, D15, C30, C30, C2×C22⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C2×D6⋊C4, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, C22×D15, C22×C30, C2×D10⋊C4, D30⋊3C4, C22×Dic15, C22×C60, C23×D15, C2×D30⋊3C4
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, D15, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, D30, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D6⋊C4, C4×D15, D60, C15⋊7D4, C22×D15, C2×D10⋊C4, D30⋊3C4, C2×C4×D15, C2×D60, C2×C15⋊7D4, C2×D30⋊3C4
►On 240 points
Generators in S240
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 171)(21 172)(22 173)(23 174)(24 175)(25 176)(26 177)(27 178)(28 179)(29 180)(30 151)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 240)(62 211)(63 212)(64 213)(65 214)(66 215)(67 216)(68 217)(69 218)(70 219)(71 220)(72 221)(73 222)(74 223)(75 224)(76 225)(77 226)(78 227)(79 228)(80 229)(81 230)(82 231)(83 232)(84 233)(85 234)(86 235)(87 236)(88 237)(89 238)(90 239)(91 205)(92 206)(93 207)(94 208)(95 209)(96 210)(97 181)(98 182)(99 183)(100 184)(101 185)(102 186)(103 187)(104 188)(105 189)(106 190)(107 191)(108 192)(109 193)(110 194)(111 195)(112 196)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)
(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)
(1 166)(2 165)(3 164)(4 163)(5 162)(6 161)(7 160)(8 159)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 180)(18 179)(19 178)(20 177)(21 176)(22 175)(23 174)(24 173)(25 172)(26 171)(27 170)(28 169)(29 168)(30 167)(31 149)(32 148)(33 147)(34 146)(35 145)(36 144)(37 143)(38 142)(39 141)(40 140)(41 139)(42 138)(43 137)(44 136)(45 135)(46 134)(47 133)(48 132)(49 131)(50 130)(51 129)(52 128)(53 127)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 150)(61 237)(62 236)(63 235)(64 234)(65 233)(66 232)(67 231)(68 230)(69 229)(70 228)(71 227)(72 226)(73 225)(74 224)(75 223)(76 222)(77 221)(78 220)(79 219)(80 218)(81 217)(82 216)(83 215)(84 214)(85 213)(86 212)(87 211)(88 240)(89 239)(90 238)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 192)(106 191)(107 190)(108 189)(109 188)(110 187)(111 186)(112 185)(113 184)(114 183)(115 182)(116 181)(117 210)(118 209)(119 208)(120 207)
(1 107 35 75)(2 108 36 76)(3 109 37 77)(4 110 38 78)(5 111 39 79)(6 112 40 80)(7 113 41 81)(8 114 42 82)(9 115 43 83)(10 116 44 84)(11 117 45 85)(12 118 46 86)(13 119 47 87)(14 120 48 88)(15 91 49 89)(16 92 50 90)(17 93 51 61)(18 94 52 62)(19 95 53 63)(20 96 54 64)(21 97 55 65)(22 98 56 66)(23 99 57 67)(24 100 58 68)(25 101 59 69)(26 102 60 70)(27 103 31 71)(28 104 32 72)(29 105 33 73)(30 106 34 74)(121 214 172 181)(122 215 173 182)(123 216 174 183)(124 217 175 184)(125 218 176 185)(126 219 177 186)(127 220 178 187)(128 221 179 188)(129 222 180 189)(130 223 151 190)(131 224 152 191)(132 225 153 192)(133 226 154 193)(134 227 155 194)(135 228 156 195)(136 229 157 196)(137 230 158 197)(138 231 159 198)(139 232 160 199)(140 233 161 200)(141 234 162 201)(142 235 163 202)(143 236 164 203)(144 237 165 204)(145 238 166 205)(146 239 167 206)(147 240 168 207)(148 211 169 208)(149 212 170 209)(150 213 171 210)
G:=sub<Sym(240)| (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,175)(25,176)(26,177)(27,178)(28,179)(29,180)(30,151)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,240)(62,211)(63,212)(64,213)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,225)(77,226)(78,227)(79,228)(80,229)(81,230)(82,231)(83,232)(84,233)(85,234)(86,235)(87,236)(88,237)(89,238)(90,239)(91,205)(92,206)(93,207)(94,208)(95,209)(96,210)(97,181)(98,182)(99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)(112,196)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204), (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), (1,166)(2,165)(3,164)(4,163)(5,162)(6,161)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,180)(18,179)(19,178)(20,177)(21,176)(22,175)(23,174)(24,173)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,149)(32,148)(33,147)(34,146)(35,145)(36,144)(37,143)(38,142)(39,141)(40,140)(41,139)(42,138)(43,137)(44,136)(45,135)(46,134)(47,133)(48,132)(49,131)(50,130)(51,129)(52,128)(53,127)(54,126)(55,125)(56,124)(57,123)(58,122)(59,121)(60,150)(61,237)(62,236)(63,235)(64,234)(65,233)(66,232)(67,231)(68,230)(69,229)(70,228)(71,227)(72,226)(73,225)(74,224)(75,223)(76,222)(77,221)(78,220)(79,219)(80,218)(81,217)(82,216)(83,215)(84,214)(85,213)(86,212)(87,211)(88,240)(89,239)(90,238)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,184)(114,183)(115,182)(116,181)(117,210)(118,209)(119,208)(120,207), (1,107,35,75)(2,108,36,76)(3,109,37,77)(4,110,38,78)(5,111,39,79)(6,112,40,80)(7,113,41,81)(8,114,42,82)(9,115,43,83)(10,116,44,84)(11,117,45,85)(12,118,46,86)(13,119,47,87)(14,120,48,88)(15,91,49,89)(16,92,50,90)(17,93,51,61)(18,94,52,62)(19,95,53,63)(20,96,54,64)(21,97,55,65)(22,98,56,66)(23,99,57,67)(24,100,58,68)(25,101,59,69)(26,102,60,70)(27,103,31,71)(28,104,32,72)(29,105,33,73)(30,106,34,74)(121,214,172,181)(122,215,173,182)(123,216,174,183)(124,217,175,184)(125,218,176,185)(126,219,177,186)(127,220,178,187)(128,221,179,188)(129,222,180,189)(130,223,151,190)(131,224,152,191)(132,225,153,192)(133,226,154,193)(134,227,155,194)(135,228,156,195)(136,229,157,196)(137,230,158,197)(138,231,159,198)(139,232,160,199)(140,233,161,200)(141,234,162,201)(142,235,163,202)(143,236,164,203)(144,237,165,204)(145,238,166,205)(146,239,167,206)(147,240,168,207)(148,211,169,208)(149,212,170,209)(150,213,171,210)>;
G:=Group( (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,175)(25,176)(26,177)(27,178)(28,179)(29,180)(30,151)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,240)(62,211)(63,212)(64,213)(65,214)(66,215)(67,216)(68,217)(69,218)(70,219)(71,220)(72,221)(73,222)(74,223)(75,224)(76,225)(77,226)(78,227)(79,228)(80,229)(81,230)(82,231)(83,232)(84,233)(85,234)(86,235)(87,236)(88,237)(89,238)(90,239)(91,205)(92,206)(93,207)(94,208)(95,209)(96,210)(97,181)(98,182)(99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)(112,196)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204), (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), (1,166)(2,165)(3,164)(4,163)(5,162)(6,161)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,180)(18,179)(19,178)(20,177)(21,176)(22,175)(23,174)(24,173)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,149)(32,148)(33,147)(34,146)(35,145)(36,144)(37,143)(38,142)(39,141)(40,140)(41,139)(42,138)(43,137)(44,136)(45,135)(46,134)(47,133)(48,132)(49,131)(50,130)(51,129)(52,128)(53,127)(54,126)(55,125)(56,124)(57,123)(58,122)(59,121)(60,150)(61,237)(62,236)(63,235)(64,234)(65,233)(66,232)(67,231)(68,230)(69,229)(70,228)(71,227)(72,226)(73,225)(74,224)(75,223)(76,222)(77,221)(78,220)(79,219)(80,218)(81,217)(82,216)(83,215)(84,214)(85,213)(86,212)(87,211)(88,240)(89,239)(90,238)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,184)(114,183)(115,182)(116,181)(117,210)(118,209)(119,208)(120,207), (1,107,35,75)(2,108,36,76)(3,109,37,77)(4,110,38,78)(5,111,39,79)(6,112,40,80)(7,113,41,81)(8,114,42,82)(9,115,43,83)(10,116,44,84)(11,117,45,85)(12,118,46,86)(13,119,47,87)(14,120,48,88)(15,91,49,89)(16,92,50,90)(17,93,51,61)(18,94,52,62)(19,95,53,63)(20,96,54,64)(21,97,55,65)(22,98,56,66)(23,99,57,67)(24,100,58,68)(25,101,59,69)(26,102,60,70)(27,103,31,71)(28,104,32,72)(29,105,33,73)(30,106,34,74)(121,214,172,181)(122,215,173,182)(123,216,174,183)(124,217,175,184)(125,218,176,185)(126,219,177,186)(127,220,178,187)(128,221,179,188)(129,222,180,189)(130,223,151,190)(131,224,152,191)(132,225,153,192)(133,226,154,193)(134,227,155,194)(135,228,156,195)(136,229,157,196)(137,230,158,197)(138,231,159,198)(139,232,160,199)(140,233,161,200)(141,234,162,201)(142,235,163,202)(143,236,164,203)(144,237,165,204)(145,238,166,205)(146,239,167,206)(147,240,168,207)(148,211,169,208)(149,212,170,209)(150,213,171,210) );
G=PermutationGroup([[(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,161),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,169),(19,170),(20,171),(21,172),(22,173),(23,174),(24,175),(25,176),(26,177),(27,178),(28,179),(29,180),(30,151),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,240),(62,211),(63,212),(64,213),(65,214),(66,215),(67,216),(68,217),(69,218),(70,219),(71,220),(72,221),(73,222),(74,223),(75,224),(76,225),(77,226),(78,227),(79,228),(80,229),(81,230),(82,231),(83,232),(84,233),(85,234),(86,235),(87,236),(88,237),(89,238),(90,239),(91,205),(92,206),(93,207),(94,208),(95,209),(96,210),(97,181),(98,182),(99,183),(100,184),(101,185),(102,186),(103,187),(104,188),(105,189),(106,190),(107,191),(108,192),(109,193),(110,194),(111,195),(112,196),(113,197),(114,198),(115,199),(116,200),(117,201),(118,202),(119,203),(120,204)], [(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)], [(1,166),(2,165),(3,164),(4,163),(5,162),(6,161),(7,160),(8,159),(9,158),(10,157),(11,156),(12,155),(13,154),(14,153),(15,152),(16,151),(17,180),(18,179),(19,178),(20,177),(21,176),(22,175),(23,174),(24,173),(25,172),(26,171),(27,170),(28,169),(29,168),(30,167),(31,149),(32,148),(33,147),(34,146),(35,145),(36,144),(37,143),(38,142),(39,141),(40,140),(41,139),(42,138),(43,137),(44,136),(45,135),(46,134),(47,133),(48,132),(49,131),(50,130),(51,129),(52,128),(53,127),(54,126),(55,125),(56,124),(57,123),(58,122),(59,121),(60,150),(61,237),(62,236),(63,235),(64,234),(65,233),(66,232),(67,231),(68,230),(69,229),(70,228),(71,227),(72,226),(73,225),(74,224),(75,223),(76,222),(77,221),(78,220),(79,219),(80,218),(81,217),(82,216),(83,215),(84,214),(85,213),(86,212),(87,211),(88,240),(89,239),(90,238),(91,206),(92,205),(93,204),(94,203),(95,202),(96,201),(97,200),(98,199),(99,198),(100,197),(101,196),(102,195),(103,194),(104,193),(105,192),(106,191),(107,190),(108,189),(109,188),(110,187),(111,186),(112,185),(113,184),(114,183),(115,182),(116,181),(117,210),(118,209),(119,208),(120,207)], [(1,107,35,75),(2,108,36,76),(3,109,37,77),(4,110,38,78),(5,111,39,79),(6,112,40,80),(7,113,41,81),(8,114,42,82),(9,115,43,83),(10,116,44,84),(11,117,45,85),(12,118,46,86),(13,119,47,87),(14,120,48,88),(15,91,49,89),(16,92,50,90),(17,93,51,61),(18,94,52,62),(19,95,53,63),(20,96,54,64),(21,97,55,65),(22,98,56,66),(23,99,57,67),(24,100,58,68),(25,101,59,69),(26,102,60,70),(27,103,31,71),(28,104,32,72),(29,105,33,73),(30,106,34,74),(121,214,172,181),(122,215,173,182),(123,216,174,183),(124,217,175,184),(125,218,176,185),(126,219,177,186),(127,220,178,187),(128,221,179,188),(129,222,180,189),(130,223,151,190),(131,224,152,191),(132,225,153,192),(133,226,154,193),(134,227,155,194),(135,228,156,195),(136,229,157,196),(137,230,158,197),(138,231,159,198),(139,232,160,199),(140,233,161,200),(141,234,162,201),(142,235,163,202),(143,236,164,203),(144,237,165,204),(145,238,166,205),(146,239,167,206),(147,240,168,207),(148,211,169,208),(149,212,170,209),(150,213,171,210)]])
132 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 | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| 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 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | C4×S3 | D12 | C3⋊D4 | D15 | C4×D5 | D20 | C5⋊D4 | D30 | D30 | C4×D15 | D60 | C15⋊7D4 |
| kernel | C2×D30⋊3C4 | D30⋊3C4 | C22×Dic15 | C22×C60 | C23×D15 | C22×D15 | C22×C20 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 16 | 16 | 16 |
Matrix representation of C2×D30⋊3C4 ►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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 8 | 0 | 0 |
| 0 | 0 | 6 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 9 |
| 0 | 0 | 0 | 0 | 22 | 14 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 45 | 0 | 0 |
| 0 | 0 | 39 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 16 |
| 0 | 0 | 0 | 0 | 6 | 24 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 4 |
| 0 | 0 | 0 | 0 | 3 | 29 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,37,6,0,0,0,0,8,31,0,0,0,0,0,0,36,22,0,0,0,0,9,14],[1,0,0,0,0,0,1,60,0,0,0,0,0,0,25,39,0,0,0,0,45,36,0,0,0,0,0,0,37,6,0,0,0,0,16,24],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,32,3,0,0,0,0,4,29] >;
C2×D30⋊3C4 in GAP, Magma, Sage, TeX
C_2\times D_{30}\rtimes_3C_4 % in TeX
G:=Group("C2xD30:3C4"); // GroupNames label
G:=SmallGroup(480,892);
// by ID
G=gap.SmallGroup(480,892);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,2693,18822]);
// 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,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations