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G = (C2×C30).D4order 480 = 25·3·5

67th non-split extension by C2×C30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C30).67D4, C30.216(C2×D4), C23.27(S3×D5), C6.80(C4○D20), Dic155C430C2, (C2×Dic5).55D6, (C22×Dic3)⋊2D5, (C22×D5).28D6, (C22×C6).25D10, C30.137(C4○D4), C30.38D418C2, D10⋊Dic327C2, (C2×C30).178C23, (C22×C10).100D6, C54(C23.23D6), C10.52(D42S3), (C2×Dic3).161D10, C34(C23.23D10), C22.11(C15⋊D4), C1516(C22.D4), (C22×C30).40C22, C2.25(Dic5.D6), (C6×Dic5).105C22, (C2×Dic15).126C22, (C10×Dic3).197C22, (Dic3×C2×C10)⋊2C2, (C6×C5⋊D4).3C2, (C2×C5⋊D4).3S3, C6.90(C2×C5⋊D4), C2.22(C2×C15⋊D4), C10.91(C2×C3⋊D4), (D5×C2×C6).46C22, C22.221(C2×S3×D5), (C2×C6).16(C5⋊D4), (C2×C10).57(C3⋊D4), (C2×C6).190(C22×D5), (C2×C10).190(C22×S3), SmallGroup(480,612)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C30).D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — (C2×C30).D4
C15C2×C30 — (C2×C30).D4
C1C22C23

Generators and relations for (C2×C30).D4
 G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, cac-1=dad=ab15, cbc-1=b-1, dbd=b19, dcd=b15c-1 >

Subgroups: 668 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, C6, C6 [×2], C6 [×3], C2×C4 [×7], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], Dic3 [×4], C12, C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×4], C22×D5, C22×C10, Dic3⋊C4 [×2], C6.D4 [×3], C22×Dic3, C6×D4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C6×D5 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C23.23D6, C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15 [×2], D5×C2×C6, C22×C30, C23.23D10, D10⋊Dic3 [×2], Dic155C4 [×2], C30.38D4, C6×C5⋊D4, Dic3×C2×C10, (C2×C30).D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C22.D4, C5⋊D4 [×2], C22×D5, D42S3 [×2], C2×C3⋊D4, S3×D5, C4○D20 [×2], C2×C5⋊D4, C23.23D6, C15⋊D4 [×2], C2×S3×D5, C23.23D10, Dic5.D6 [×2], C2×C15⋊D4, (C2×C30).D4

Smallest permutation representation of (C2×C30).D4
On 240 points
Generators in S240
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(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)(181 231)(182 232)(183 233)(184 234)(185 235)(186 236)(187 237)(188 238)(189 239)(190 240)(191 211)(192 212)(193 213)(194 214)(195 215)(196 216)(197 217)(198 218)(199 219)(200 220)(201 221)(202 222)(203 223)(204 224)(205 225)(206 226)(207 227)(208 228)(209 229)(210 230)
(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)
(1 211 32 191)(2 240 33 190)(3 239 34 189)(4 238 35 188)(5 237 36 187)(6 236 37 186)(7 235 38 185)(8 234 39 184)(9 233 40 183)(10 232 41 182)(11 231 42 181)(12 230 43 210)(13 229 44 209)(14 228 45 208)(15 227 46 207)(16 226 47 206)(17 225 48 205)(18 224 49 204)(19 223 50 203)(20 222 51 202)(21 221 52 201)(22 220 53 200)(23 219 54 199)(24 218 55 198)(25 217 56 197)(26 216 57 196)(27 215 58 195)(28 214 59 194)(29 213 60 193)(30 212 31 192)(61 178 113 143)(62 177 114 142)(63 176 115 141)(64 175 116 140)(65 174 117 139)(66 173 118 138)(67 172 119 137)(68 171 120 136)(69 170 91 135)(70 169 92 134)(71 168 93 133)(72 167 94 132)(73 166 95 131)(74 165 96 130)(75 164 97 129)(76 163 98 128)(77 162 99 127)(78 161 100 126)(79 160 101 125)(80 159 102 124)(81 158 103 123)(82 157 104 122)(83 156 105 121)(84 155 106 150)(85 154 107 149)(86 153 108 148)(87 152 109 147)(88 151 110 146)(89 180 111 145)(90 179 112 144)
(1 121)(2 140)(3 129)(4 148)(5 137)(6 126)(7 145)(8 134)(9 123)(10 142)(11 131)(12 150)(13 139)(14 128)(15 147)(16 136)(17 125)(18 144)(19 133)(20 122)(21 141)(22 130)(23 149)(24 138)(25 127)(26 146)(27 135)(28 124)(29 143)(30 132)(31 167)(32 156)(33 175)(34 164)(35 153)(36 172)(37 161)(38 180)(39 169)(40 158)(41 177)(42 166)(43 155)(44 174)(45 163)(46 152)(47 171)(48 160)(49 179)(50 168)(51 157)(52 176)(53 165)(54 154)(55 173)(56 162)(57 151)(58 170)(59 159)(60 178)(61 208)(62 197)(63 186)(64 205)(65 194)(66 183)(67 202)(68 191)(69 210)(70 199)(71 188)(72 207)(73 196)(74 185)(75 204)(76 193)(77 182)(78 201)(79 190)(80 209)(81 198)(82 187)(83 206)(84 195)(85 184)(86 203)(87 192)(88 181)(89 200)(90 189)(91 230)(92 219)(93 238)(94 227)(95 216)(96 235)(97 224)(98 213)(99 232)(100 221)(101 240)(102 229)(103 218)(104 237)(105 226)(106 215)(107 234)(108 223)(109 212)(110 231)(111 220)(112 239)(113 228)(114 217)(115 236)(116 225)(117 214)(118 233)(119 222)(120 211)

G:=sub<Sym(240)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(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)(181,231)(182,232)(183,233)(184,234)(185,235)(186,236)(187,237)(188,238)(189,239)(190,240)(191,211)(192,212)(193,213)(194,214)(195,215)(196,216)(197,217)(198,218)(199,219)(200,220)(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230), (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), (1,211,32,191)(2,240,33,190)(3,239,34,189)(4,238,35,188)(5,237,36,187)(6,236,37,186)(7,235,38,185)(8,234,39,184)(9,233,40,183)(10,232,41,182)(11,231,42,181)(12,230,43,210)(13,229,44,209)(14,228,45,208)(15,227,46,207)(16,226,47,206)(17,225,48,205)(18,224,49,204)(19,223,50,203)(20,222,51,202)(21,221,52,201)(22,220,53,200)(23,219,54,199)(24,218,55,198)(25,217,56,197)(26,216,57,196)(27,215,58,195)(28,214,59,194)(29,213,60,193)(30,212,31,192)(61,178,113,143)(62,177,114,142)(63,176,115,141)(64,175,116,140)(65,174,117,139)(66,173,118,138)(67,172,119,137)(68,171,120,136)(69,170,91,135)(70,169,92,134)(71,168,93,133)(72,167,94,132)(73,166,95,131)(74,165,96,130)(75,164,97,129)(76,163,98,128)(77,162,99,127)(78,161,100,126)(79,160,101,125)(80,159,102,124)(81,158,103,123)(82,157,104,122)(83,156,105,121)(84,155,106,150)(85,154,107,149)(86,153,108,148)(87,152,109,147)(88,151,110,146)(89,180,111,145)(90,179,112,144), (1,121)(2,140)(3,129)(4,148)(5,137)(6,126)(7,145)(8,134)(9,123)(10,142)(11,131)(12,150)(13,139)(14,128)(15,147)(16,136)(17,125)(18,144)(19,133)(20,122)(21,141)(22,130)(23,149)(24,138)(25,127)(26,146)(27,135)(28,124)(29,143)(30,132)(31,167)(32,156)(33,175)(34,164)(35,153)(36,172)(37,161)(38,180)(39,169)(40,158)(41,177)(42,166)(43,155)(44,174)(45,163)(46,152)(47,171)(48,160)(49,179)(50,168)(51,157)(52,176)(53,165)(54,154)(55,173)(56,162)(57,151)(58,170)(59,159)(60,178)(61,208)(62,197)(63,186)(64,205)(65,194)(66,183)(67,202)(68,191)(69,210)(70,199)(71,188)(72,207)(73,196)(74,185)(75,204)(76,193)(77,182)(78,201)(79,190)(80,209)(81,198)(82,187)(83,206)(84,195)(85,184)(86,203)(87,192)(88,181)(89,200)(90,189)(91,230)(92,219)(93,238)(94,227)(95,216)(96,235)(97,224)(98,213)(99,232)(100,221)(101,240)(102,229)(103,218)(104,237)(105,226)(106,215)(107,234)(108,223)(109,212)(110,231)(111,220)(112,239)(113,228)(114,217)(115,236)(116,225)(117,214)(118,233)(119,222)(120,211)>;

G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(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)(181,231)(182,232)(183,233)(184,234)(185,235)(186,236)(187,237)(188,238)(189,239)(190,240)(191,211)(192,212)(193,213)(194,214)(195,215)(196,216)(197,217)(198,218)(199,219)(200,220)(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230), (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), (1,211,32,191)(2,240,33,190)(3,239,34,189)(4,238,35,188)(5,237,36,187)(6,236,37,186)(7,235,38,185)(8,234,39,184)(9,233,40,183)(10,232,41,182)(11,231,42,181)(12,230,43,210)(13,229,44,209)(14,228,45,208)(15,227,46,207)(16,226,47,206)(17,225,48,205)(18,224,49,204)(19,223,50,203)(20,222,51,202)(21,221,52,201)(22,220,53,200)(23,219,54,199)(24,218,55,198)(25,217,56,197)(26,216,57,196)(27,215,58,195)(28,214,59,194)(29,213,60,193)(30,212,31,192)(61,178,113,143)(62,177,114,142)(63,176,115,141)(64,175,116,140)(65,174,117,139)(66,173,118,138)(67,172,119,137)(68,171,120,136)(69,170,91,135)(70,169,92,134)(71,168,93,133)(72,167,94,132)(73,166,95,131)(74,165,96,130)(75,164,97,129)(76,163,98,128)(77,162,99,127)(78,161,100,126)(79,160,101,125)(80,159,102,124)(81,158,103,123)(82,157,104,122)(83,156,105,121)(84,155,106,150)(85,154,107,149)(86,153,108,148)(87,152,109,147)(88,151,110,146)(89,180,111,145)(90,179,112,144), (1,121)(2,140)(3,129)(4,148)(5,137)(6,126)(7,145)(8,134)(9,123)(10,142)(11,131)(12,150)(13,139)(14,128)(15,147)(16,136)(17,125)(18,144)(19,133)(20,122)(21,141)(22,130)(23,149)(24,138)(25,127)(26,146)(27,135)(28,124)(29,143)(30,132)(31,167)(32,156)(33,175)(34,164)(35,153)(36,172)(37,161)(38,180)(39,169)(40,158)(41,177)(42,166)(43,155)(44,174)(45,163)(46,152)(47,171)(48,160)(49,179)(50,168)(51,157)(52,176)(53,165)(54,154)(55,173)(56,162)(57,151)(58,170)(59,159)(60,178)(61,208)(62,197)(63,186)(64,205)(65,194)(66,183)(67,202)(68,191)(69,210)(70,199)(71,188)(72,207)(73,196)(74,185)(75,204)(76,193)(77,182)(78,201)(79,190)(80,209)(81,198)(82,187)(83,206)(84,195)(85,184)(86,203)(87,192)(88,181)(89,200)(90,189)(91,230)(92,219)(93,238)(94,227)(95,216)(96,235)(97,224)(98,213)(99,232)(100,221)(101,240)(102,229)(103,218)(104,237)(105,226)(106,215)(107,234)(108,223)(109,212)(110,231)(111,220)(112,239)(113,228)(114,217)(115,236)(116,225)(117,214)(118,233)(119,222)(120,211) );

G=PermutationGroup([(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,91),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(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),(181,231),(182,232),(183,233),(184,234),(185,235),(186,236),(187,237),(188,238),(189,239),(190,240),(191,211),(192,212),(193,213),(194,214),(195,215),(196,216),(197,217),(198,218),(199,219),(200,220),(201,221),(202,222),(203,223),(204,224),(205,225),(206,226),(207,227),(208,228),(209,229),(210,230)], [(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)], [(1,211,32,191),(2,240,33,190),(3,239,34,189),(4,238,35,188),(5,237,36,187),(6,236,37,186),(7,235,38,185),(8,234,39,184),(9,233,40,183),(10,232,41,182),(11,231,42,181),(12,230,43,210),(13,229,44,209),(14,228,45,208),(15,227,46,207),(16,226,47,206),(17,225,48,205),(18,224,49,204),(19,223,50,203),(20,222,51,202),(21,221,52,201),(22,220,53,200),(23,219,54,199),(24,218,55,198),(25,217,56,197),(26,216,57,196),(27,215,58,195),(28,214,59,194),(29,213,60,193),(30,212,31,192),(61,178,113,143),(62,177,114,142),(63,176,115,141),(64,175,116,140),(65,174,117,139),(66,173,118,138),(67,172,119,137),(68,171,120,136),(69,170,91,135),(70,169,92,134),(71,168,93,133),(72,167,94,132),(73,166,95,131),(74,165,96,130),(75,164,97,129),(76,163,98,128),(77,162,99,127),(78,161,100,126),(79,160,101,125),(80,159,102,124),(81,158,103,123),(82,157,104,122),(83,156,105,121),(84,155,106,150),(85,154,107,149),(86,153,108,148),(87,152,109,147),(88,151,110,146),(89,180,111,145),(90,179,112,144)], [(1,121),(2,140),(3,129),(4,148),(5,137),(6,126),(7,145),(8,134),(9,123),(10,142),(11,131),(12,150),(13,139),(14,128),(15,147),(16,136),(17,125),(18,144),(19,133),(20,122),(21,141),(22,130),(23,149),(24,138),(25,127),(26,146),(27,135),(28,124),(29,143),(30,132),(31,167),(32,156),(33,175),(34,164),(35,153),(36,172),(37,161),(38,180),(39,169),(40,158),(41,177),(42,166),(43,155),(44,174),(45,163),(46,152),(47,171),(48,160),(49,179),(50,168),(51,157),(52,176),(53,165),(54,154),(55,173),(56,162),(57,151),(58,170),(59,159),(60,178),(61,208),(62,197),(63,186),(64,205),(65,194),(66,183),(67,202),(68,191),(69,210),(70,199),(71,188),(72,207),(73,196),(74,185),(75,204),(76,193),(77,182),(78,201),(79,190),(80,209),(81,198),(82,187),(83,206),(84,195),(85,184),(86,203),(87,192),(88,181),(89,200),(90,189),(91,230),(92,219),(93,238),(94,227),(95,216),(96,235),(97,224),(98,213),(99,232),(100,221),(101,240),(102,229),(103,218),(104,237),(105,226),(106,215),(107,234),(108,223),(109,212),(110,231),(111,220),(112,239),(113,228),(114,217),(115,236),(116,225),(117,214),(118,233),(119,222),(120,211)])

72 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E6F6G10A···10N12A12B15A15B20A···20P30A···30N
order12222223444444455666666610···101212151520···2030···30
size1111222026666206060222224420202···22020446···64···4

72 irreducible representations

dim11111122222222222244444
type++++++++++++++-+-+
imageC1C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10C3⋊D4C5⋊D4C4○D20D42S3S3×D5C15⋊D4C2×S3×D5Dic5.D6
kernel(C2×C30).D4D10⋊Dic3Dic155C4C30.38D4C6×C5⋊D4Dic3×C2×C10C2×C5⋊D4C2×C30C22×Dic3C2×Dic5C22×D5C22×C10C30C2×Dic3C22×C6C2×C10C2×C6C6C10C23C22C22C2
# reps122111122111442481622428

Matrix representation of (C2×C30).D4 in GL6(𝔽61)

6000000
010000
0060000
0006000
0000600
0000060
,
6000000
0600000
0034000
0013900
0000119
00004859
,
0500000
1100000
00221400
00223900
00001554
00004146
,
010000
100000
00394700
0042200
00002737
00001034

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,13,0,0,0,0,0,9,0,0,0,0,0,0,1,48,0,0,0,0,19,59],[0,11,0,0,0,0,50,0,0,0,0,0,0,0,22,22,0,0,0,0,14,39,0,0,0,0,0,0,15,41,0,0,0,0,54,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,39,4,0,0,0,0,47,22,0,0,0,0,0,0,27,10,0,0,0,0,37,34] >;

(C2×C30).D4 in GAP, Magma, Sage, TeX

(C_2\times C_{30}).D_4
% in TeX

G:=Group("(C2xC30).D4");
// GroupNames label

G:=SmallGroup(480,612);
// by ID

G=gap.SmallGroup(480,612);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^15,c*b*c^-1=b^-1,d*b*d=b^19,d*c*d=b^15*c^-1>;
// generators/relations

׿
×
𝔽