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G = C10×C4○D12order 480 = 25·3·5

Direct product of C10 and C4○D12

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

Aliases: C10×C4○D12, C30.87C24, C60.291C23, (C2×C20)⋊37D6, (C2×D12)⋊14C10, (C10×D12)⋊30C2, D1212(C2×C10), C3013(C4○D4), (C2×C60)⋊50C22, (C22×C12)⋊8C10, (C22×C20)⋊17S3, (C22×C60)⋊20C2, C6.4(C23×C10), (S3×C20)⋊25C22, (C10×Dic6)⋊31C2, (C2×Dic6)⋊15C10, Dic611(C2×C10), (C5×D12)⋊42C22, C10.72(S3×C23), C23.31(S3×C10), D6.1(C22×C10), (S3×C10).36C23, (C2×C30).444C23, C12.43(C22×C10), C20.238(C22×S3), (C5×Dic6)⋊38C22, (C22×C10).129D6, (C5×Dic3).38C23, Dic3.2(C22×C10), (C22×C30).184C22, (C10×Dic3).235C22, C61(C5×C4○D4), C31(C10×C4○D4), (S3×C2×C4)⋊15C10, (S3×C2×C20)⋊31C2, C1522(C2×C4○D4), C4.43(S3×C2×C10), (C4×S3)⋊6(C2×C10), (C2×C4)⋊10(S3×C10), C3⋊D46(C2×C10), (C22×C4)⋊8(C5×S3), (C2×C12)⋊13(C2×C10), C22.6(S3×C2×C10), C2.5(S3×C22×C10), (C2×C3⋊D4)⋊12C10, (C10×C3⋊D4)⋊27C2, (C5×C3⋊D4)⋊22C22, (S3×C2×C10).120C22, (C22×C6).46(C2×C10), (C2×C6).65(C22×C10), (C22×S3).29(C2×C10), (C2×C10).257(C22×S3), (C2×Dic3).44(C2×C10), SmallGroup(480,1153)

Series: Derived Chief Lower central Upper central

C1C6 — C10×C4○D12
C1C3C6C30S3×C10S3×C2×C10S3×C2×C20 — C10×C4○D12
C3C6 — C10×C4○D12
C1C2×C20C22×C20

Generators and relations for C10×C4○D12
 G = < a,b,c,d | a10=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 644 in 328 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C10, C10 [×2], C10 [×6], Dic3 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×10], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C5×S3 [×4], C30, C30 [×2], C30 [×2], C2×C4○D4, C2×C20 [×2], C2×C20 [×4], C2×C20 [×10], C5×D4 [×12], C5×Q8 [×4], C22×C10, C22×C10 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×4], C60 [×4], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C22×C20, C22×C20 [×2], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4○D12, C5×Dic6 [×4], S3×C20 [×8], C5×D12 [×4], C10×Dic3 [×2], C5×C3⋊D4 [×8], C2×C60 [×2], C2×C60 [×4], S3×C2×C10 [×2], C22×C30, C10×C4○D4, C10×Dic6, S3×C2×C20 [×2], C10×D12, C5×C4○D12 [×8], C10×C3⋊D4 [×2], C22×C60, C10×C4○D12
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C4○D4 [×2], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C2×C4○D4, C22×C10 [×15], C4○D12 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×C4○D12, S3×C2×C10 [×7], C10×C4○D4, C5×C4○D12 [×2], S3×C22×C10, C10×C4○D12

Smallest permutation representation of C10×C4○D12
On 240 points
Generators in S240
(1 56 198 125 97 238 119 146 80 136)(2 57 199 126 98 239 120 147 81 137)(3 58 200 127 99 240 109 148 82 138)(4 59 201 128 100 229 110 149 83 139)(5 60 202 129 101 230 111 150 84 140)(6 49 203 130 102 231 112 151 73 141)(7 50 204 131 103 232 113 152 74 142)(8 51 193 132 104 233 114 153 75 143)(9 52 194 121 105 234 115 154 76 144)(10 53 195 122 106 235 116 155 77 133)(11 54 196 123 107 236 117 156 78 134)(12 55 197 124 108 237 118 145 79 135)(13 42 218 205 178 31 188 68 164 90)(14 43 219 206 179 32 189 69 165 91)(15 44 220 207 180 33 190 70 166 92)(16 45 221 208 169 34 191 71 167 93)(17 46 222 209 170 35 192 72 168 94)(18 47 223 210 171 36 181 61 157 95)(19 48 224 211 172 25 182 62 158 96)(20 37 225 212 173 26 183 63 159 85)(21 38 226 213 174 27 184 64 160 86)(22 39 227 214 175 28 185 65 161 87)(23 40 228 215 176 29 186 66 162 88)(24 41 217 216 177 30 187 67 163 89)
(1 210 7 216)(2 211 8 205)(3 212 9 206)(4 213 10 207)(5 214 11 208)(6 215 12 209)(13 147 19 153)(14 148 20 154)(15 149 21 155)(16 150 22 156)(17 151 23 145)(18 152 24 146)(25 193 31 199)(26 194 32 200)(27 195 33 201)(28 196 34 202)(29 197 35 203)(30 198 36 204)(37 76 43 82)(38 77 44 83)(39 78 45 84)(40 79 46 73)(41 80 47 74)(42 81 48 75)(49 176 55 170)(50 177 56 171)(51 178 57 172)(52 179 58 173)(53 180 59 174)(54 169 60 175)(61 103 67 97)(62 104 68 98)(63 105 69 99)(64 106 70 100)(65 107 71 101)(66 108 72 102)(85 115 91 109)(86 116 92 110)(87 117 93 111)(88 118 94 112)(89 119 95 113)(90 120 96 114)(121 189 127 183)(122 190 128 184)(123 191 129 185)(124 192 130 186)(125 181 131 187)(126 182 132 188)(133 220 139 226)(134 221 140 227)(135 222 141 228)(136 223 142 217)(137 224 143 218)(138 225 144 219)(157 232 163 238)(158 233 164 239)(159 234 165 240)(160 235 166 229)(161 236 167 230)(162 237 168 231)
(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 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)(121 128)(122 127)(123 126)(124 125)(129 132)(130 131)(133 138)(134 137)(135 136)(139 144)(140 143)(141 142)(145 146)(147 156)(148 155)(149 154)(150 153)(151 152)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)(169 172)(170 171)(173 180)(174 179)(175 178)(176 177)(181 192)(182 191)(183 190)(184 189)(185 188)(186 187)(193 202)(194 201)(195 200)(196 199)(197 198)(203 204)(205 214)(206 213)(207 212)(208 211)(209 210)(215 216)(217 228)(218 227)(219 226)(220 225)(221 224)(222 223)(229 234)(230 233)(231 232)(235 240)(236 239)(237 238)

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

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

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

180 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B5C5D6A···6G10A···10L10M···10T10U···10AJ12A···12H15A15B15C15D20A···20P20Q···20X20Y···20AN30A···30AB60A···60AF
order12222222223444444444455556···610···1010···1010···1012···121515151520···2020···2020···2030···3060···60
size11112266662111122666611112···21···12···26···62···222221···12···26···62···22···2

180 irreducible representations

dim111111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10S3D6D6C4○D4C5×S3C4○D12S3×C10S3×C10C5×C4○D4C5×C4○D12
kernelC10×C4○D12C10×Dic6S3×C2×C20C10×D12C5×C4○D12C10×C3⋊D4C22×C60C2×C4○D12C2×Dic6S3×C2×C4C2×D12C4○D12C2×C3⋊D4C22×C12C22×C20C2×C20C22×C10C30C22×C4C10C2×C4C23C6C2
# reps1121821448432841614482441632

Matrix representation of C10×C4○D12 in GL3(𝔽61) generated by

6000
030
003
,
6000
0110
0011
,
100
04623
03823
,
6000
04623
03815
G:=sub<GL(3,GF(61))| [60,0,0,0,3,0,0,0,3],[60,0,0,0,11,0,0,0,11],[1,0,0,0,46,38,0,23,23],[60,0,0,0,46,38,0,23,15] >;

C10×C4○D12 in GAP, Magma, Sage, TeX

C_{10}\times C_4\circ D_{12}
% in TeX

G:=Group("C10xC4oD12");
// GroupNames label

G:=SmallGroup(480,1153);
// by ID

G=gap.SmallGroup(480,1153);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,2467,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations

׿
×
𝔽