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G = C10.(C2×D12)  order 480 = 25·3·5

21st non-split extension by C10 of C2×D12 acting via C2×D12/C22×S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C30).73D4, C23.D55S3, C10.67(C2×D12), (C2×C10).47D12, C30.222(C2×D4), D304C426C2, C23.28(S3×D5), C6.81(C4○D20), C30.Q831C2, (C2×Dic5).57D6, (C22×Dic3)⋊4D5, (C22×C6).27D10, C30.140(C4○D4), (C2×C30).184C23, (C22×C10).104D6, C54(C23.21D6), C22.6(C5⋊D12), C10.53(D42S3), (C2×Dic3).163D10, C33(C23.23D10), C1519(C22.D4), (C22×C30).46C22, C2.26(Dic5.D6), (C6×Dic5).107C22, (C22×D15).60C22, (C2×Dic15).128C22, (C10×Dic3).199C22, (Dic3×C2×C10)⋊4C2, C6.21(C2×C5⋊D4), (C2×C157D4).9C2, C2.22(C2×C5⋊D12), (C3×C23.D5)⋊5C2, C22.224(C2×S3×D5), (C2×C6).17(C5⋊D4), (C2×C6).196(C22×D5), (C2×C10).196(C22×S3), SmallGroup(480,618)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C10.(C2×D12)
C1C5C15C30C2×C30C6×Dic5C30.Q8 — C10.(C2×D12)
C15C2×C30 — C10.(C2×D12)
C1C22C23

Generators and relations for C10.(C2×D12)
 G = < a,b,c,d | a10=b2=c12=1, d2=a5, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 812 in 156 conjugacy classes, 52 normal (24 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, D5, C10, C10 [×2], C10 [×2], 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 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, D15, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×4], C22×D5, C22×C10, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C23.21D6, C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], C22×D15, C22×C30, C23.23D10, D304C4 [×2], C30.Q8 [×2], C3×C23.D5, Dic3×C2×C10, C2×C157D4, C10.(C2×D12)
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], D12 [×2], C22×S3, C22.D4, C5⋊D4 [×2], C22×D5, C2×D12, D42S3 [×2], S3×D5, C4○D20 [×2], C2×C5⋊D4, C23.21D6, C5⋊D12 [×2], C2×S3×D5, C23.23D10, Dic5.D6 [×2], C2×C5⋊D12, C10.(C2×D12)

Smallest permutation representation of C10.(C2×D12)
On 240 points
Generators in S240
(1 16 187 219 30 126 195 142 88 233)(2 234 89 143 196 127 31 220 188 17)(3 18 189 221 32 128 197 144 90 235)(4 236 91 133 198 129 33 222 190 19)(5 20 191 223 34 130 199 134 92 237)(6 238 93 135 200 131 35 224 192 21)(7 22 181 225 36 132 201 136 94 239)(8 240 95 137 202 121 25 226 182 23)(9 24 183 227 26 122 203 138 96 229)(10 230 85 139 204 123 27 228 184 13)(11 14 185 217 28 124 193 140 86 231)(12 232 87 141 194 125 29 218 186 15)(37 55 98 174 158 153 208 109 81 72)(38 61 82 110 209 154 159 175 99 56)(39 57 100 176 160 155 210 111 83 62)(40 63 84 112 211 156 161 177 101 58)(41 59 102 178 162 145 212 113 73 64)(42 65 74 114 213 146 163 179 103 60)(43 49 104 180 164 147 214 115 75 66)(44 67 76 116 215 148 165 169 105 50)(45 51 106 170 166 149 216 117 77 68)(46 69 78 118 205 150 167 171 107 52)(47 53 108 172 168 151 206 119 79 70)(48 71 80 120 207 152 157 173 97 54)
(1 7)(2 121)(3 9)(4 123)(5 11)(6 125)(8 127)(10 129)(12 131)(13 198)(14 20)(15 200)(16 22)(17 202)(18 24)(19 204)(21 194)(23 196)(25 234)(26 32)(27 236)(28 34)(29 238)(30 36)(31 240)(33 230)(35 232)(37 147)(38 44)(39 149)(40 46)(41 151)(42 48)(43 153)(45 155)(47 145)(49 208)(50 56)(51 210)(52 58)(53 212)(54 60)(55 214)(57 216)(59 206)(61 67)(62 166)(63 69)(64 168)(65 71)(66 158)(68 160)(70 162)(72 164)(73 172)(74 80)(75 174)(76 82)(77 176)(78 84)(79 178)(81 180)(83 170)(85 222)(86 92)(87 224)(88 94)(89 226)(90 96)(91 228)(93 218)(95 220)(97 103)(98 115)(99 105)(100 117)(101 107)(102 119)(104 109)(106 111)(108 113)(110 116)(112 118)(114 120)(122 128)(124 130)(126 132)(133 184)(134 140)(135 186)(136 142)(137 188)(138 144)(139 190)(141 192)(143 182)(146 152)(148 154)(150 156)(157 163)(159 165)(161 167)(169 175)(171 177)(173 179)(181 187)(183 189)(185 191)(193 199)(195 201)(197 203)(205 211)(207 213)(209 215)(217 223)(219 225)(221 227)(229 235)(231 237)(233 239)
(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 43 126 147)(2 42 127 146)(3 41 128 145)(4 40 129 156)(5 39 130 155)(6 38 131 154)(7 37 132 153)(8 48 121 152)(9 47 122 151)(10 46 123 150)(11 45 124 149)(12 44 125 148)(13 69 204 167)(14 68 193 166)(15 67 194 165)(16 66 195 164)(17 65 196 163)(18 64 197 162)(19 63 198 161)(20 62 199 160)(21 61 200 159)(22 72 201 158)(23 71 202 157)(24 70 203 168)(25 207 240 54)(26 206 229 53)(27 205 230 52)(28 216 231 51)(29 215 232 50)(30 214 233 49)(31 213 234 60)(32 212 235 59)(33 211 236 58)(34 210 237 57)(35 209 238 56)(36 208 239 55)(73 144 178 189)(74 143 179 188)(75 142 180 187)(76 141 169 186)(77 140 170 185)(78 139 171 184)(79 138 172 183)(80 137 173 182)(81 136 174 181)(82 135 175 192)(83 134 176 191)(84 133 177 190)(85 107 228 118)(86 106 217 117)(87 105 218 116)(88 104 219 115)(89 103 220 114)(90 102 221 113)(91 101 222 112)(92 100 223 111)(93 99 224 110)(94 98 225 109)(95 97 226 120)(96 108 227 119)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10N12A12B12C12D15A15B20A···20P30A···30N
order122222234444444556666610···1012121212151520···2030···30
size111122602666620206022222442···220202020446···64···4

72 irreducible representations

dim1111112222222222244444
type++++++++++++++-+++
imageC1C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D12C5⋊D4C4○D20D42S3S3×D5C5⋊D12C2×S3×D5Dic5.D6
kernelC10.(C2×D12)D304C4C30.Q8C3×C23.D5Dic3×C2×C10C2×C157D4C23.D5C2×C30C22×Dic3C2×Dic5C22×C10C30C2×Dic3C22×C6C2×C10C2×C6C6C10C23C22C22C2
# reps12211112221442481622428

Matrix representation of C10.(C2×D12) in GL6(𝔽61)

3400000
090000
001000
000100
000030
00004141
,
6000000
0600000
001000
000100
0000600
0000151
,
010000
6000000
0006000
001100
0000608
0000151
,
0600000
6000000
00512400
00341000
00005027
0000011

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,41,0,0,0,0,0,41],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,15,0,0,0,0,0,1],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,51,34,0,0,0,0,24,10,0,0,0,0,0,0,50,0,0,0,0,0,27,11] >;

C10.(C2×D12) in GAP, Magma, Sage, TeX

C_{10}.(C_2\times D_{12})
% in TeX

G:=Group("C10.(C2xD12)");
// GroupNames label

G:=SmallGroup(480,618);
// by ID

G=gap.SmallGroup(480,618);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^5,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽