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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⋊Dic526C2, C30.219(C2×D4), C23.38(S3×D5), Dic155C431C2, (C22×Dic5)⋊5S3, (C22×C10).41D6, (C22×C6).84D10, C10.79(C4○D12), C30.139(C4○D4), C6.52(D42D5), C30.38D419C2, (C2×C30).181C23, (C2×Dic5).188D6, (C2×Dic3).56D10, (C22×S3).25D10, C54(C23.28D6), C22.6(C15⋊D4), C1518(C22.D4), C34(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

C1C2×C30 — C30.(C2×D4)
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C30.(C2×D4)
C15C2×C30 — C30.(C2×D4)
C1C22C23

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 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, S3, C6, C6 [×2], C6 [×2], C2×C4 [×7], D4 [×2], C23, C23, C10, C10 [×2], C10 [×3], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×4], C20, C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×4], C22×S3, C22×C6, C5×S3, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10, C22×C10, Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], C23.D5 [×3], C22×Dic5, D4×C10, C23.28D6, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15 [×2], S3×C2×C10, C22×C30, C23.18D10, D6⋊Dic5 [×2], Dic155C4 [×2], C30.38D4, C2×C6×Dic5, C10×C3⋊D4, C30.(C2×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, C4○D12 [×2], C2×C3⋊D4, S3×D5, D42D5 [×2], C2×C5⋊D4, C23.28D6, C15⋊D4 [×2], C2×S3×D5, C23.18D10, Dic3.D10 [×2], C2×C15⋊D4, C30.(C2×D4)

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A···6G10A···10F10G10H10I10J10K10L10M10N12A···12H15A15B20A20B20C20D30A···30N
order122222234444444556···610···10101010101010101012···1215152020202030···30
size11112212210101010126060222···22···244441212121210···1044121212124···4

66 irreducible representations

dim11111122222222222244444
type+++++++++++++++--+
imageC1C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C3⋊D4C5⋊D4C4○D12S3×D5D42D5C15⋊D4C2×S3×D5Dic3.D10
kernelC30.(C2×D4)D6⋊Dic5Dic155C4C30.38D4C2×C6×Dic5C10×C3⋊D4C22×Dic5C2×C30C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C22×S3C22×C6C2×C10C2×C6C10C23C6C22C22C2
# reps12211112221422248824428

Matrix representation of C30.(C2×D4) in GL4(𝔽61) generated by

0100
604300
00140
00048
,
60000
06000
00600
0001
,
345400
522700
00011
00500
,
475900
61400
00500
00050
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

׿
×
𝔽