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G = C5×C4.D12order 480 = 25·3·5

Direct product of C5 and C4.D12

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

Aliases: C5×C4.D12, C20.65D12, C60.144D4, D62(C5×Q8), C6.8(D4×C10), D6⋊C4.2C10, C4⋊Dic36C10, (S3×C10)⋊10Q8, C12.11(C5×D4), C4.13(C5×D12), C6.14(Q8×C10), C10.52(S3×Q8), (C2×Dic6)⋊7C10, C30.295(C2×D4), C10.79(C2×D12), (C2×C20).355D6, C2.10(C10×D12), C1533(C22⋊Q8), C30.112(C2×Q8), (C10×Dic6)⋊23C2, C30.251(C4○D4), (C2×C30).417C23, (C2×C60).336C22, C10.117(D42S3), (C10×Dic3).146C22, C4⋊C45(C5×S3), C2.7(C5×S3×Q8), (C3×C4⋊C4)⋊8C10, (C5×C4⋊C4)⋊14S3, C33(C5×C22⋊Q8), (S3×C2×C4).3C10, (C15×C4⋊C4)⋊26C2, (S3×C2×C20).14C2, C6.26(C5×C4○D4), (C2×C4).12(S3×C10), (C5×D6⋊C4).10C2, (C5×C4⋊Dic3)⋊24C2, C22.52(S3×C2×C10), (C2×C12).25(C2×C10), C2.13(C5×D42S3), (S3×C2×C10).114C22, (C2×C6).38(C22×C10), (C22×S3).23(C2×C10), (C2×C10).351(C22×S3), (C2×Dic3).10(C2×C10), SmallGroup(480,776)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C4.D12
C1C3C6C2×C6C2×C30S3×C2×C10S3×C2×C20 — C5×C4.D12
C3C2×C6 — C5×C4.D12
C1C2×C10C5×C4⋊C4

Generators and relations for C5×C4.D12
 G = < a,b,c,d | a5=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 340 in 148 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, C2×C20, C2×C20 [×2], C2×C20 [×5], C5×Q8 [×2], C22×C10, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×3], C60 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C22×C20, Q8×C10, C4.D12, C5×Dic6 [×2], S3×C20 [×2], C10×Dic3, C10×Dic3 [×2], C2×C60, C2×C60 [×2], S3×C2×C10, C5×C22⋊Q8, C5×C4⋊Dic3 [×2], C5×D6⋊C4 [×2], C15×C4⋊C4, C10×Dic6, S3×C2×C20, C5×C4.D12
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, C2×D12, D42S3, S3×Q8, S3×C10 [×3], D4×C10, Q8×C10, C5×C4○D4, C4.D12, C5×D12 [×2], S3×C2×C10, C5×C22⋊Q8, C10×D12, C5×D42S3, C5×S3×Q8, C5×C4.D12

Smallest permutation representation of C5×C4.D12
On 240 points
Generators in S240
(1 26 177 152 193)(2 27 178 153 194)(3 28 179 154 195)(4 29 180 155 196)(5 30 169 156 197)(6 31 170 145 198)(7 32 171 146 199)(8 33 172 147 200)(9 34 173 148 201)(10 35 174 149 202)(11 36 175 150 203)(12 25 176 151 204)(13 127 186 111 40)(14 128 187 112 41)(15 129 188 113 42)(16 130 189 114 43)(17 131 190 115 44)(18 132 191 116 45)(19 121 192 117 46)(20 122 181 118 47)(21 123 182 119 48)(22 124 183 120 37)(23 125 184 109 38)(24 126 185 110 39)(49 107 143 165 82)(50 108 144 166 83)(51 97 133 167 84)(52 98 134 168 73)(53 99 135 157 74)(54 100 136 158 75)(55 101 137 159 76)(56 102 138 160 77)(57 103 139 161 78)(58 104 140 162 79)(59 105 141 163 80)(60 106 142 164 81)(61 206 225 90 234)(62 207 226 91 235)(63 208 227 92 236)(64 209 228 93 237)(65 210 217 94 238)(66 211 218 95 239)(67 212 219 96 240)(68 213 220 85 229)(69 214 221 86 230)(70 215 222 87 231)(71 216 223 88 232)(72 205 224 89 233)
(1 216 100 190)(2 191 101 205)(3 206 102 192)(4 181 103 207)(5 208 104 182)(6 183 105 209)(7 210 106 184)(8 185 107 211)(9 212 108 186)(10 187 97 213)(11 214 98 188)(12 189 99 215)(13 148 240 83)(14 84 229 149)(15 150 230 73)(16 74 231 151)(17 152 232 75)(18 76 233 153)(19 154 234 77)(20 78 235 155)(21 156 236 79)(22 80 237 145)(23 146 238 81)(24 82 239 147)(25 114 135 222)(26 223 136 115)(27 116 137 224)(28 225 138 117)(29 118 139 226)(30 227 140 119)(31 120 141 228)(32 217 142 109)(33 110 143 218)(34 219 144 111)(35 112 133 220)(36 221 134 113)(37 163 93 170)(38 171 94 164)(39 165 95 172)(40 173 96 166)(41 167 85 174)(42 175 86 168)(43 157 87 176)(44 177 88 158)(45 159 89 178)(46 179 90 160)(47 161 91 180)(48 169 92 162)(49 66 200 126)(50 127 201 67)(51 68 202 128)(52 129 203 69)(53 70 204 130)(54 131 193 71)(55 72 194 132)(56 121 195 61)(57 62 196 122)(58 123 197 63)(59 64 198 124)(60 125 199 65)
(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 99 100 12)(2 11 101 98)(3 97 102 10)(4 9 103 108)(5 107 104 8)(6 7 105 106)(13 235 240 20)(14 19 229 234)(15 233 230 18)(16 17 231 232)(21 239 236 24)(22 23 237 238)(25 26 135 136)(27 36 137 134)(28 133 138 35)(29 34 139 144)(30 143 140 33)(31 32 141 142)(37 38 93 94)(39 48 95 92)(40 91 96 47)(41 46 85 90)(42 89 86 45)(43 44 87 88)(49 58 200 197)(50 196 201 57)(51 56 202 195)(52 194 203 55)(53 54 204 193)(59 60 198 199)(61 128 121 68)(62 67 122 127)(63 126 123 66)(64 65 124 125)(69 132 129 72)(70 71 130 131)(73 153 150 76)(74 75 151 152)(77 149 154 84)(78 83 155 148)(79 147 156 82)(80 81 145 146)(109 228 217 120)(110 119 218 227)(111 226 219 118)(112 117 220 225)(113 224 221 116)(114 115 222 223)(157 158 176 177)(159 168 178 175)(160 174 179 167)(161 166 180 173)(162 172 169 165)(163 164 170 171)(181 186 207 212)(182 211 208 185)(183 184 209 210)(187 192 213 206)(188 205 214 191)(189 190 215 216)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C10A···10L10M···10T12A···12F15A15B15C15D20A···20H20I···20P20Q···20X20Y···20AF30A···30L60A···60X
order122222344444444555566610···1010···1012···121515151520···2020···2020···2020···2030···3060···60
size1111662224466121211112221···16···64···422222···24···46···612···122···24···4

120 irreducible representations

dim1111111111112222222222224444
type++++++++-++--
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4Q8D6C4○D4D12C5×S3C5×D4C5×Q8S3×C10C5×C4○D4C5×D12D42S3S3×Q8C5×D42S3C5×S3×Q8
kernelC5×C4.D12C5×C4⋊Dic3C5×D6⋊C4C15×C4⋊C4C10×Dic6S3×C2×C20C4.D12C4⋊Dic3D6⋊C4C3×C4⋊C4C2×Dic6S3×C2×C4C5×C4⋊C4C60S3×C10C2×C20C30C20C4⋊C4C12D6C2×C4C6C4C10C10C2C2
# reps122111488444122324488128161144

Matrix representation of C5×C4.D12 in GL4(𝔽61) generated by

34000
03400
0010
0001
,
50000
281100
0010
0001
,
14000
06000
002323
003846
,
14000
36000
003846
002323
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[50,28,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,40,60,0,0,0,0,23,38,0,0,23,46],[1,3,0,0,40,60,0,0,0,0,38,23,0,0,46,23] >;

C5×C4.D12 in GAP, Magma, Sage, TeX

C_5\times C_4.D_{12}
% in TeX

G:=Group("C5xC4.D12");
// GroupNames label

G:=SmallGroup(480,776);
// by ID

G=gap.SmallGroup(480,776);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,646,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

׿
×
𝔽