G = C30.(C2×D4) order 480 = 25·3·5
149th non-split extension by C30 of C2×D4 acting via C2×D4/C22=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30).70D4, D6⋊Dic5⋊26C2, C30.219(C2×D4), C23.38(S3×D5), Dic15⋊5C4⋊31C2, (C22×Dic5)⋊5S3, (C22×C10).41D6, (C22×C6).84D10, C10.79(C4○D12), C30.139(C4○D4), C6.52(D4⋊2D5), C30.38D4⋊19C2, (C2×C30).181C23, (C2×Dic5).188D6, (C2×Dic3).56D10, (C22×S3).25D10, C5⋊4(C23.28D6), C22.6(C15⋊D4), C15⋊18(C22.D4), C3⋊4(C23.18D10), (C22×C30).43C22, C2.24(Dic3.D10), (C6×Dic5).217C22, (C2×Dic15).127C22, (C10×Dic3).106C22, (C2×C6×Dic5)⋊2C2, (C2×C3⋊D4).3D5, C6.92(C2×C5⋊D4), (C10×C3⋊D4).3C2, C2.23(C2×C15⋊D4), C10.93(C2×C3⋊D4), C22.223(C2×S3×D5), (C2×C6).58(C5⋊D4), (S3×C2×C10).45C22, (C2×C10).16(C3⋊D4), (C2×C6).193(C22×D5), (C2×C10).193(C22×S3), SmallGroup(480,615)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.(C2×D4)
G = < a,b,c,d | a30=b2=c4=1, d2=a15, ab=ba, cac-1=a-1, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 636 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, S3×C10, C2×C30, C2×C30, C2×C30, C10.D4, C23.D5, C22×Dic5, D4×C10, C23.28D6, C6×Dic5, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, S3×C2×C10, C22×C30, C23.18D10, D6⋊Dic5, Dic15⋊5C4, C30.38D4, C2×C6×Dic5, C10×C3⋊D4, C30.(C2×D4)
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C5⋊D4, C22×D5, C4○D12, C2×C3⋊D4, S3×D5, D4⋊2D5, C2×C5⋊D4, C23.28D6, C15⋊D4, C2×S3×D5, C23.18D10, Dic3.D10, C2×C15⋊D4, C30.(C2×D4)
►On 240 points
Generators in S240
(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 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 31)(30 32)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 179)(145 180)(146 151)(147 152)(148 153)(149 154)(150 155)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 235)(191 236)(192 237)(193 238)(194 239)(195 240)(196 211)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 218)(204 219)(205 220)(206 221)(207 222)(208 223)(209 224)(210 225)
(1 231 48 186)(2 230 49 185)(3 229 50 184)(4 228 51 183)(5 227 52 182)(6 226 53 181)(7 225 54 210)(8 224 55 209)(9 223 56 208)(10 222 57 207)(11 221 58 206)(12 220 59 205)(13 219 60 204)(14 218 31 203)(15 217 32 202)(16 216 33 201)(17 215 34 200)(18 214 35 199)(19 213 36 198)(20 212 37 197)(21 211 38 196)(22 240 39 195)(23 239 40 194)(24 238 41 193)(25 237 42 192)(26 236 43 191)(27 235 44 190)(28 234 45 189)(29 233 46 188)(30 232 47 187)(61 177 100 127)(62 176 101 126)(63 175 102 125)(64 174 103 124)(65 173 104 123)(66 172 105 122)(67 171 106 121)(68 170 107 150)(69 169 108 149)(70 168 109 148)(71 167 110 147)(72 166 111 146)(73 165 112 145)(74 164 113 144)(75 163 114 143)(76 162 115 142)(77 161 116 141)(78 160 117 140)(79 159 118 139)(80 158 119 138)(81 157 120 137)(82 156 91 136)(83 155 92 135)(84 154 93 134)(85 153 94 133)(86 152 95 132)(87 151 96 131)(88 180 97 130)(89 179 98 129)(90 178 99 128)
(1 121 16 136)(2 140 17 125)(3 129 18 144)(4 148 19 133)(5 137 20 122)(6 126 21 141)(7 145 22 130)(8 134 23 149)(9 123 24 138)(10 142 25 127)(11 131 26 146)(12 150 27 135)(13 139 28 124)(14 128 29 143)(15 147 30 132)(31 178 46 163)(32 167 47 152)(33 156 48 171)(34 175 49 160)(35 164 50 179)(36 153 51 168)(37 172 52 157)(38 161 53 176)(39 180 54 165)(40 169 55 154)(41 158 56 173)(42 177 57 162)(43 166 58 151)(44 155 59 170)(45 174 60 159)(61 207 76 192)(62 196 77 181)(63 185 78 200)(64 204 79 189)(65 193 80 208)(66 182 81 197)(67 201 82 186)(68 190 83 205)(69 209 84 194)(70 198 85 183)(71 187 86 202)(72 206 87 191)(73 195 88 210)(74 184 89 199)(75 203 90 188)(91 231 106 216)(92 220 107 235)(93 239 108 224)(94 228 109 213)(95 217 110 232)(96 236 111 221)(97 225 112 240)(98 214 113 229)(99 233 114 218)(100 222 115 237)(101 211 116 226)(102 230 117 215)(103 219 118 234)(104 238 119 223)(105 227 120 212)
G:=sub<Sym(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), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,31)(30,32)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,179)(145,180)(146,151)(147,152)(148,153)(149,154)(150,155)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (1,231,48,186)(2,230,49,185)(3,229,50,184)(4,228,51,183)(5,227,52,182)(6,226,53,181)(7,225,54,210)(8,224,55,209)(9,223,56,208)(10,222,57,207)(11,221,58,206)(12,220,59,205)(13,219,60,204)(14,218,31,203)(15,217,32,202)(16,216,33,201)(17,215,34,200)(18,214,35,199)(19,213,36,198)(20,212,37,197)(21,211,38,196)(22,240,39,195)(23,239,40,194)(24,238,41,193)(25,237,42,192)(26,236,43,191)(27,235,44,190)(28,234,45,189)(29,233,46,188)(30,232,47,187)(61,177,100,127)(62,176,101,126)(63,175,102,125)(64,174,103,124)(65,173,104,123)(66,172,105,122)(67,171,106,121)(68,170,107,150)(69,169,108,149)(70,168,109,148)(71,167,110,147)(72,166,111,146)(73,165,112,145)(74,164,113,144)(75,163,114,143)(76,162,115,142)(77,161,116,141)(78,160,117,140)(79,159,118,139)(80,158,119,138)(81,157,120,137)(82,156,91,136)(83,155,92,135)(84,154,93,134)(85,153,94,133)(86,152,95,132)(87,151,96,131)(88,180,97,130)(89,179,98,129)(90,178,99,128), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,178,46,163)(32,167,47,152)(33,156,48,171)(34,175,49,160)(35,164,50,179)(36,153,51,168)(37,172,52,157)(38,161,53,176)(39,180,54,165)(40,169,55,154)(41,158,56,173)(42,177,57,162)(43,166,58,151)(44,155,59,170)(45,174,60,159)(61,207,76,192)(62,196,77,181)(63,185,78,200)(64,204,79,189)(65,193,80,208)(66,182,81,197)(67,201,82,186)(68,190,83,205)(69,209,84,194)(70,198,85,183)(71,187,86,202)(72,206,87,191)(73,195,88,210)(74,184,89,199)(75,203,90,188)(91,231,106,216)(92,220,107,235)(93,239,108,224)(94,228,109,213)(95,217,110,232)(96,236,111,221)(97,225,112,240)(98,214,113,229)(99,233,114,218)(100,222,115,237)(101,211,116,226)(102,230,117,215)(103,219,118,234)(104,238,119,223)(105,227,120,212)>;
G:=Group( (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), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,31)(30,32)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,179)(145,180)(146,151)(147,152)(148,153)(149,154)(150,155)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (1,231,48,186)(2,230,49,185)(3,229,50,184)(4,228,51,183)(5,227,52,182)(6,226,53,181)(7,225,54,210)(8,224,55,209)(9,223,56,208)(10,222,57,207)(11,221,58,206)(12,220,59,205)(13,219,60,204)(14,218,31,203)(15,217,32,202)(16,216,33,201)(17,215,34,200)(18,214,35,199)(19,213,36,198)(20,212,37,197)(21,211,38,196)(22,240,39,195)(23,239,40,194)(24,238,41,193)(25,237,42,192)(26,236,43,191)(27,235,44,190)(28,234,45,189)(29,233,46,188)(30,232,47,187)(61,177,100,127)(62,176,101,126)(63,175,102,125)(64,174,103,124)(65,173,104,123)(66,172,105,122)(67,171,106,121)(68,170,107,150)(69,169,108,149)(70,168,109,148)(71,167,110,147)(72,166,111,146)(73,165,112,145)(74,164,113,144)(75,163,114,143)(76,162,115,142)(77,161,116,141)(78,160,117,140)(79,159,118,139)(80,158,119,138)(81,157,120,137)(82,156,91,136)(83,155,92,135)(84,154,93,134)(85,153,94,133)(86,152,95,132)(87,151,96,131)(88,180,97,130)(89,179,98,129)(90,178,99,128), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,178,46,163)(32,167,47,152)(33,156,48,171)(34,175,49,160)(35,164,50,179)(36,153,51,168)(37,172,52,157)(38,161,53,176)(39,180,54,165)(40,169,55,154)(41,158,56,173)(42,177,57,162)(43,166,58,151)(44,155,59,170)(45,174,60,159)(61,207,76,192)(62,196,77,181)(63,185,78,200)(64,204,79,189)(65,193,80,208)(66,182,81,197)(67,201,82,186)(68,190,83,205)(69,209,84,194)(70,198,85,183)(71,187,86,202)(72,206,87,191)(73,195,88,210)(74,184,89,199)(75,203,90,188)(91,231,106,216)(92,220,107,235)(93,239,108,224)(94,228,109,213)(95,217,110,232)(96,236,111,221)(97,225,112,240)(98,214,113,229)(99,233,114,218)(100,222,115,237)(101,211,116,226)(102,230,117,215)(103,219,118,234)(104,238,119,223)(105,227,120,212) );
G=PermutationGroup([[(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)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,31),(30,32),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,162),(128,163),(129,164),(130,165),(131,166),(132,167),(133,168),(134,169),(135,170),(136,171),(137,172),(138,173),(139,174),(140,175),(141,176),(142,177),(143,178),(144,179),(145,180),(146,151),(147,152),(148,153),(149,154),(150,155),(181,226),(182,227),(183,228),(184,229),(185,230),(186,231),(187,232),(188,233),(189,234),(190,235),(191,236),(192,237),(193,238),(194,239),(195,240),(196,211),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,218),(204,219),(205,220),(206,221),(207,222),(208,223),(209,224),(210,225)], [(1,231,48,186),(2,230,49,185),(3,229,50,184),(4,228,51,183),(5,227,52,182),(6,226,53,181),(7,225,54,210),(8,224,55,209),(9,223,56,208),(10,222,57,207),(11,221,58,206),(12,220,59,205),(13,219,60,204),(14,218,31,203),(15,217,32,202),(16,216,33,201),(17,215,34,200),(18,214,35,199),(19,213,36,198),(20,212,37,197),(21,211,38,196),(22,240,39,195),(23,239,40,194),(24,238,41,193),(25,237,42,192),(26,236,43,191),(27,235,44,190),(28,234,45,189),(29,233,46,188),(30,232,47,187),(61,177,100,127),(62,176,101,126),(63,175,102,125),(64,174,103,124),(65,173,104,123),(66,172,105,122),(67,171,106,121),(68,170,107,150),(69,169,108,149),(70,168,109,148),(71,167,110,147),(72,166,111,146),(73,165,112,145),(74,164,113,144),(75,163,114,143),(76,162,115,142),(77,161,116,141),(78,160,117,140),(79,159,118,139),(80,158,119,138),(81,157,120,137),(82,156,91,136),(83,155,92,135),(84,154,93,134),(85,153,94,133),(86,152,95,132),(87,151,96,131),(88,180,97,130),(89,179,98,129),(90,178,99,128)], [(1,121,16,136),(2,140,17,125),(3,129,18,144),(4,148,19,133),(5,137,20,122),(6,126,21,141),(7,145,22,130),(8,134,23,149),(9,123,24,138),(10,142,25,127),(11,131,26,146),(12,150,27,135),(13,139,28,124),(14,128,29,143),(15,147,30,132),(31,178,46,163),(32,167,47,152),(33,156,48,171),(34,175,49,160),(35,164,50,179),(36,153,51,168),(37,172,52,157),(38,161,53,176),(39,180,54,165),(40,169,55,154),(41,158,56,173),(42,177,57,162),(43,166,58,151),(44,155,59,170),(45,174,60,159),(61,207,76,192),(62,196,77,181),(63,185,78,200),(64,204,79,189),(65,193,80,208),(66,182,81,197),(67,201,82,186),(68,190,83,205),(69,209,84,194),(70,198,85,183),(71,187,86,202),(72,206,87,191),(73,195,88,210),(74,184,89,199),(75,203,90,188),(91,231,106,216),(92,220,107,235),(93,239,108,224),(94,228,109,213),(95,217,110,232),(96,236,111,221),(97,225,112,240),(98,214,113,229),(99,233,114,218),(100,222,115,237),(101,211,116,226),(102,230,117,215),(103,219,118,234),(104,238,119,223),(105,227,120,212)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | ··· | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 10 | 10 | 10 | 10 | 12 | 60 | 60 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | C4○D12 | S3×D5 | D4⋊2D5 | C15⋊D4 | C2×S3×D5 | Dic3.D10 |
| kernel | C30.(C2×D4) | D6⋊Dic5 | Dic15⋊5C4 | C30.38D4 | C2×C6×Dic5 | C10×C3⋊D4 | C22×Dic5 | C2×C30 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C30.(C2×D4) ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 48 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 54 | 0 | 0 |
| 52 | 27 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 50 | 0 |
| 47 | 59 | 0 | 0 |
| 6 | 14 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,43,0,0,0,0,14,0,0,0,0,48],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[34,52,0,0,54,27,0,0,0,0,0,50,0,0,11,0],[47,6,0,0,59,14,0,0,0,0,50,0,0,0,0,50] >;
C30.(C2×D4) in GAP, Magma, Sage, TeX
C_{30}.(C_2\times D_4) % in TeX
G:=Group("C30.(C2xD4)"); // GroupNames label
G:=SmallGroup(480,615);
// by ID
G=gap.SmallGroup(480,615);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^19,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations