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G = C5×C8○D12order 480 = 25·3·5

Direct product of C5 and C8○D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C8○D12, C40.84D6, D12.2C20, Dic6.2C20, C60.285C23, C120.111C22, (S3×C8)⋊6C10, (C2×C40)⋊15S3, (S3×C40)⋊15C2, C8⋊S37C10, (C2×C24)⋊12C10, (C2×C120)⋊28C2, C1523(C8○D4), C4.10(S3×C20), C8.22(S3×C10), C20.99(C4×S3), D6.1(C2×C20), (C5×D12).6C4, C3⋊D4.2C20, C12.20(C2×C20), C60.207(C2×C4), C24.27(C2×C10), C4○D12.6C10, (C2×C20).431D6, C22.2(S3×C20), (C5×Dic6).6C4, C4.Dic311C10, C6.14(C22×C20), Dic3.3(C2×C20), (S3×C20).64C22, C30.205(C22×C4), C20.243(C22×S3), (C2×C60).536C22, C12.37(C22×C10), C31(C5×C8○D4), (C2×C8)⋊7(C5×S3), C2.15(S3×C2×C20), C4.37(S3×C2×C10), C3⋊C8.11(C2×C10), C10.141(S3×C2×C4), (C5×C3⋊D4).6C4, (C5×C8⋊S3)⋊15C2, (C2×C6).16(C2×C20), (C2×C10).50(C4×S3), (C2×C4).79(S3×C10), (C5×C3⋊C8).47C22, (S3×C10).31(C2×C4), (C4×S3).15(C2×C10), (C2×C30).161(C2×C4), (C5×C4○D12).12C2, (C5×C4.Dic3)⋊23C2, (C2×C12).103(C2×C10), (C5×Dic3).39(C2×C4), SmallGroup(480,780)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C8○D12
C1C3C6C12C60S3×C20C5×C4○D12 — C5×C8○D12
C3C6 — C5×C8○D12
C1C40C2×C40

Generators and relations for C5×C8○D12
 G = < a,b,c,d | a5=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >

Subgroups: 228 in 124 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C10, C10 [×3], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C5×S3 [×2], C30, C30, C8○D4, C40 [×2], C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], C2×C30, C2×C40, C2×C40 [×2], C5×M4(2) [×3], C5×C4○D4, C8○D12, C5×C3⋊C8 [×2], C120 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C2×C60, C5×C8○D4, S3×C40 [×2], C5×C8⋊S3 [×2], C5×C4.Dic3, C2×C120, C5×C4○D12, C5×C8○D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C8○D4, C2×C20 [×6], C22×C10, S3×C2×C4, S3×C10 [×3], C22×C20, C8○D12, S3×C20 [×2], S3×C2×C10, C5×C8○D4, S3×C2×C20, C5×C8○D12

Smallest permutation representation of C5×C8○D12
On 240 points
Generators in S240
(1 188 57 200 117)(2 189 58 201 118)(3 190 59 202 119)(4 191 60 203 120)(5 192 49 204 109)(6 181 50 193 110)(7 182 51 194 111)(8 183 52 195 112)(9 184 53 196 113)(10 185 54 197 114)(11 186 55 198 115)(12 187 56 199 116)(13 215 144 65 155)(14 216 133 66 156)(15 205 134 67 145)(16 206 135 68 146)(17 207 136 69 147)(18 208 137 70 148)(19 209 138 71 149)(20 210 139 72 150)(21 211 140 61 151)(22 212 141 62 152)(23 213 142 63 153)(24 214 143 64 154)(25 169 89 103 76)(26 170 90 104 77)(27 171 91 105 78)(28 172 92 106 79)(29 173 93 107 80)(30 174 94 108 81)(31 175 95 97 82)(32 176 96 98 83)(33 177 85 99 84)(34 178 86 100 73)(35 179 87 101 74)(36 180 88 102 75)(37 122 226 168 238)(38 123 227 157 239)(39 124 228 158 240)(40 125 217 159 229)(41 126 218 160 230)(42 127 219 161 231)(43 128 220 162 232)(44 129 221 163 233)(45 130 222 164 234)(46 131 223 165 235)(47 132 224 166 236)(48 121 225 167 237)
(1 35 20 167 7 29 14 161)(2 36 21 168 8 30 15 162)(3 25 22 157 9 31 16 163)(4 26 23 158 10 32 17 164)(5 27 24 159 11 33 18 165)(6 28 13 160 12 34 19 166)(37 52 94 134 43 58 88 140)(38 53 95 135 44 59 89 141)(39 54 96 136 45 60 90 142)(40 55 85 137 46 49 91 143)(41 56 86 138 47 50 92 144)(42 57 87 139 48 51 93 133)(61 122 195 108 67 128 201 102)(62 123 196 97 68 129 202 103)(63 124 197 98 69 130 203 104)(64 125 198 99 70 131 204 105)(65 126 199 100 71 132 193 106)(66 127 200 101 72 121 194 107)(73 149 224 110 79 155 218 116)(74 150 225 111 80 156 219 117)(75 151 226 112 81 145 220 118)(76 152 227 113 82 146 221 119)(77 153 228 114 83 147 222 120)(78 154 217 115 84 148 223 109)(169 212 239 184 175 206 233 190)(170 213 240 185 176 207 234 191)(171 214 229 186 177 208 235 192)(172 215 230 187 178 209 236 181)(173 216 231 188 179 210 237 182)(174 205 232 189 180 211 238 183)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 52)(50 51)(53 60)(54 59)(55 58)(56 57)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)(85 88)(86 87)(89 96)(90 95)(91 94)(92 93)(97 104)(98 103)(99 102)(100 101)(105 108)(106 107)(109 112)(110 111)(113 120)(114 119)(115 118)(116 117)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(145 154)(146 153)(147 152)(148 151)(149 150)(155 156)(157 164)(158 163)(159 162)(160 161)(165 168)(166 167)(169 176)(170 175)(171 174)(172 173)(177 180)(178 179)(181 182)(183 192)(184 191)(185 190)(186 189)(187 188)(193 194)(195 204)(196 203)(197 202)(198 201)(199 200)(205 214)(206 213)(207 212)(208 211)(209 210)(215 216)(217 220)(218 219)(221 228)(222 227)(223 226)(224 225)(229 232)(230 231)(233 240)(234 239)(235 238)(236 237)

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

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

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

180 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D6A6B6C8A8B8C8D8E8F8G8H8I8J10A10B10C10D10E10F10G10H10I···10P12A12B12C12D15A15B15C15D20A···20H20I20J20K20L20M···20T24A···24H30A···30L40A···40P40Q···40X40Y···40AN60A···60P120A···120AF
order1222234444455556668888888888101010101010101010···10121212121515151520···202020202020···2024···2430···3040···4040···4040···4060···60120···120
size1126621126611112221111226666111122226···6222222221···122226···62···22···21···12···26···62···22···2

180 irreducible representations

dim11111111111111111122222222222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4C5C10C10C10C10C10C20C20C20S3D6D6C4×S3C4×S3C5×S3C8○D4S3×C10S3×C10C8○D12S3×C20S3×C20C5×C8○D4C5×C8○D12
kernelC5×C8○D12S3×C40C5×C8⋊S3C5×C4.Dic3C2×C120C5×C4○D12C5×Dic6C5×D12C5×C3⋊D4C8○D12S3×C8C8⋊S3C4.Dic3C2×C24C4○D12Dic6D12C3⋊D4C2×C40C40C2×C20C20C2×C10C2×C8C15C8C2×C4C5C4C22C3C1
# reps12211122448844488161212244848881632

Matrix representation of C5×C8○D12 in GL2(𝔽241) generated by

870
087
,
80
08
,
142142
9943
,
9943
142142
G:=sub<GL(2,GF(241))| [87,0,0,87],[8,0,0,8],[142,99,142,43],[99,142,43,142] >;

C5×C8○D12 in GAP, Magma, Sage, TeX

C_5\times C_8\circ D_{12}
% in TeX

G:=Group("C5xC8oD12");
// GroupNames label

G:=SmallGroup(480,780);
// by ID

G=gap.SmallGroup(480,780);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,226,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^6=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^5>;
// generators/relations

׿
×
𝔽