direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C4○D12, C30.87C24, C60.291C23, (C2×C20)⋊37D6, (C2×D12)⋊14C10, (C10×D12)⋊30C2, D12⋊12(C2×C10), C30⋊13(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, Dic6⋊11(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, C6⋊1(C5×C4○D4), C3⋊1(C10×C4○D4), (S3×C2×C4)⋊15C10, (S3×C2×C20)⋊31C2, C15⋊22(C2×C4○D4), C4.43(S3×C2×C10), (C4×S3)⋊6(C2×C10), (C2×C4)⋊10(S3×C10), C3⋊D4⋊6(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
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, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C30, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C22×C20, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4○D12, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C2×C60, S3×C2×C10, C22×C30, C10×C4○D4, C10×Dic6, S3×C2×C20, C10×D12, C5×C4○D12, C10×C3⋊D4, C22×C60, C10×C4○D12
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C24, C2×C10, C22×S3, C5×S3, C2×C4○D4, C22×C10, C4○D12, S3×C23, S3×C10, C5×C4○D4, C23×C10, C2×C4○D12, S3×C2×C10, C10×C4○D4, C5×C4○D12, S3×C22×C10, C10×C4○D12
►On 240 points
Generators in S240
(1 239 160 182 43 148 119 95 69 175)(2 240 161 183 44 149 120 96 70 176)(3 229 162 184 45 150 109 85 71 177)(4 230 163 185 46 151 110 86 72 178)(5 231 164 186 47 152 111 87 61 179)(6 232 165 187 48 153 112 88 62 180)(7 233 166 188 37 154 113 89 63 169)(8 234 167 189 38 155 114 90 64 170)(9 235 168 190 39 156 115 91 65 171)(10 236 157 191 40 145 116 92 66 172)(11 237 158 192 41 146 117 93 67 173)(12 238 159 181 42 147 118 94 68 174)(13 224 82 124 31 133 104 203 60 215)(14 225 83 125 32 134 105 204 49 216)(15 226 84 126 33 135 106 193 50 205)(16 227 73 127 34 136 107 194 51 206)(17 228 74 128 35 137 108 195 52 207)(18 217 75 129 36 138 97 196 53 208)(19 218 76 130 25 139 98 197 54 209)(20 219 77 131 26 140 99 198 55 210)(21 220 78 132 27 141 100 199 56 211)(22 221 79 121 28 142 101 200 57 212)(23 222 80 122 29 143 102 201 58 213)(24 223 81 123 30 144 103 202 59 214)
(1 82 7 76)(2 83 8 77)(3 84 9 78)(4 73 10 79)(5 74 11 80)(6 75 12 81)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 160 31 166)(26 161 32 167)(27 162 33 168)(28 163 34 157)(29 164 35 158)(30 165 36 159)(37 98 43 104)(38 99 44 105)(39 100 45 106)(40 101 46 107)(41 102 47 108)(42 103 48 97)(49 114 55 120)(50 115 56 109)(51 116 57 110)(52 117 58 111)(53 118 59 112)(54 119 60 113)(85 205 91 211)(86 206 92 212)(87 207 93 213)(88 208 94 214)(89 209 95 215)(90 210 96 216)(121 230 127 236)(122 231 128 237)(123 232 129 238)(124 233 130 239)(125 234 131 240)(126 235 132 229)(133 188 139 182)(134 189 140 183)(135 190 141 184)(136 191 142 185)(137 192 143 186)(138 181 144 187)(145 200 151 194)(146 201 152 195)(147 202 153 196)(148 203 154 197)(149 204 155 198)(150 193 156 199)(169 218 175 224)(170 219 176 225)(171 220 177 226)(172 221 178 227)(173 222 179 228)(174 223 180 217)
(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 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)(121 126)(122 125)(123 124)(127 132)(128 131)(129 130)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(145 150)(146 149)(147 148)(151 156)(152 155)(153 154)(157 162)(158 161)(159 160)(163 168)(164 167)(165 166)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 182)(183 192)(184 191)(185 190)(186 189)(187 188)(193 200)(194 199)(195 198)(196 197)(201 204)(202 203)(205 212)(206 211)(207 210)(208 209)(213 216)(214 215)(217 218)(219 228)(220 227)(221 226)(222 225)(223 224)(229 236)(230 235)(231 234)(232 233)(237 240)(238 239)
G:=sub<Sym(240)| (1,239,160,182,43,148,119,95,69,175)(2,240,161,183,44,149,120,96,70,176)(3,229,162,184,45,150,109,85,71,177)(4,230,163,185,46,151,110,86,72,178)(5,231,164,186,47,152,111,87,61,179)(6,232,165,187,48,153,112,88,62,180)(7,233,166,188,37,154,113,89,63,169)(8,234,167,189,38,155,114,90,64,170)(9,235,168,190,39,156,115,91,65,171)(10,236,157,191,40,145,116,92,66,172)(11,237,158,192,41,146,117,93,67,173)(12,238,159,181,42,147,118,94,68,174)(13,224,82,124,31,133,104,203,60,215)(14,225,83,125,32,134,105,204,49,216)(15,226,84,126,33,135,106,193,50,205)(16,227,73,127,34,136,107,194,51,206)(17,228,74,128,35,137,108,195,52,207)(18,217,75,129,36,138,97,196,53,208)(19,218,76,130,25,139,98,197,54,209)(20,219,77,131,26,140,99,198,55,210)(21,220,78,132,27,141,100,199,56,211)(22,221,79,121,28,142,101,200,57,212)(23,222,80,122,29,143,102,201,58,213)(24,223,81,123,30,144,103,202,59,214), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,73,10,79)(5,74,11,80)(6,75,12,81)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,160,31,166)(26,161,32,167)(27,162,33,168)(28,163,34,157)(29,164,35,158)(30,165,36,159)(37,98,43,104)(38,99,44,105)(39,100,45,106)(40,101,46,107)(41,102,47,108)(42,103,48,97)(49,114,55,120)(50,115,56,109)(51,116,57,110)(52,117,58,111)(53,118,59,112)(54,119,60,113)(85,205,91,211)(86,206,92,212)(87,207,93,213)(88,208,94,214)(89,209,95,215)(90,210,96,216)(121,230,127,236)(122,231,128,237)(123,232,129,238)(124,233,130,239)(125,234,131,240)(126,235,132,229)(133,188,139,182)(134,189,140,183)(135,190,141,184)(136,191,142,185)(137,192,143,186)(138,181,144,187)(145,200,151,194)(146,201,152,195)(147,202,153,196)(148,203,154,197)(149,204,155,198)(150,193,156,199)(169,218,175,224)(170,219,176,225)(171,220,177,226)(172,221,178,227)(173,222,179,228)(174,223,180,217), (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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,116)(110,115)(111,114)(112,113)(117,120)(118,119)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,162)(158,161)(159,160)(163,168)(164,167)(165,166)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,182)(183,192)(184,191)(185,190)(186,189)(187,188)(193,200)(194,199)(195,198)(196,197)(201,204)(202,203)(205,212)(206,211)(207,210)(208,209)(213,216)(214,215)(217,218)(219,228)(220,227)(221,226)(222,225)(223,224)(229,236)(230,235)(231,234)(232,233)(237,240)(238,239)>;
G:=Group( (1,239,160,182,43,148,119,95,69,175)(2,240,161,183,44,149,120,96,70,176)(3,229,162,184,45,150,109,85,71,177)(4,230,163,185,46,151,110,86,72,178)(5,231,164,186,47,152,111,87,61,179)(6,232,165,187,48,153,112,88,62,180)(7,233,166,188,37,154,113,89,63,169)(8,234,167,189,38,155,114,90,64,170)(9,235,168,190,39,156,115,91,65,171)(10,236,157,191,40,145,116,92,66,172)(11,237,158,192,41,146,117,93,67,173)(12,238,159,181,42,147,118,94,68,174)(13,224,82,124,31,133,104,203,60,215)(14,225,83,125,32,134,105,204,49,216)(15,226,84,126,33,135,106,193,50,205)(16,227,73,127,34,136,107,194,51,206)(17,228,74,128,35,137,108,195,52,207)(18,217,75,129,36,138,97,196,53,208)(19,218,76,130,25,139,98,197,54,209)(20,219,77,131,26,140,99,198,55,210)(21,220,78,132,27,141,100,199,56,211)(22,221,79,121,28,142,101,200,57,212)(23,222,80,122,29,143,102,201,58,213)(24,223,81,123,30,144,103,202,59,214), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,73,10,79)(5,74,11,80)(6,75,12,81)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,160,31,166)(26,161,32,167)(27,162,33,168)(28,163,34,157)(29,164,35,158)(30,165,36,159)(37,98,43,104)(38,99,44,105)(39,100,45,106)(40,101,46,107)(41,102,47,108)(42,103,48,97)(49,114,55,120)(50,115,56,109)(51,116,57,110)(52,117,58,111)(53,118,59,112)(54,119,60,113)(85,205,91,211)(86,206,92,212)(87,207,93,213)(88,208,94,214)(89,209,95,215)(90,210,96,216)(121,230,127,236)(122,231,128,237)(123,232,129,238)(124,233,130,239)(125,234,131,240)(126,235,132,229)(133,188,139,182)(134,189,140,183)(135,190,141,184)(136,191,142,185)(137,192,143,186)(138,181,144,187)(145,200,151,194)(146,201,152,195)(147,202,153,196)(148,203,154,197)(149,204,155,198)(150,193,156,199)(169,218,175,224)(170,219,176,225)(171,220,177,226)(172,221,178,227)(173,222,179,228)(174,223,180,217), (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)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,116)(110,115)(111,114)(112,113)(117,120)(118,119)(121,126)(122,125)(123,124)(127,132)(128,131)(129,130)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,162)(158,161)(159,160)(163,168)(164,167)(165,166)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,182)(183,192)(184,191)(185,190)(186,189)(187,188)(193,200)(194,199)(195,198)(196,197)(201,204)(202,203)(205,212)(206,211)(207,210)(208,209)(213,216)(214,215)(217,218)(219,228)(220,227)(221,226)(222,225)(223,224)(229,236)(230,235)(231,234)(232,233)(237,240)(238,239) );
G=PermutationGroup([[(1,239,160,182,43,148,119,95,69,175),(2,240,161,183,44,149,120,96,70,176),(3,229,162,184,45,150,109,85,71,177),(4,230,163,185,46,151,110,86,72,178),(5,231,164,186,47,152,111,87,61,179),(6,232,165,187,48,153,112,88,62,180),(7,233,166,188,37,154,113,89,63,169),(8,234,167,189,38,155,114,90,64,170),(9,235,168,190,39,156,115,91,65,171),(10,236,157,191,40,145,116,92,66,172),(11,237,158,192,41,146,117,93,67,173),(12,238,159,181,42,147,118,94,68,174),(13,224,82,124,31,133,104,203,60,215),(14,225,83,125,32,134,105,204,49,216),(15,226,84,126,33,135,106,193,50,205),(16,227,73,127,34,136,107,194,51,206),(17,228,74,128,35,137,108,195,52,207),(18,217,75,129,36,138,97,196,53,208),(19,218,76,130,25,139,98,197,54,209),(20,219,77,131,26,140,99,198,55,210),(21,220,78,132,27,141,100,199,56,211),(22,221,79,121,28,142,101,200,57,212),(23,222,80,122,29,143,102,201,58,213),(24,223,81,123,30,144,103,202,59,214)], [(1,82,7,76),(2,83,8,77),(3,84,9,78),(4,73,10,79),(5,74,11,80),(6,75,12,81),(13,63,19,69),(14,64,20,70),(15,65,21,71),(16,66,22,72),(17,67,23,61),(18,68,24,62),(25,160,31,166),(26,161,32,167),(27,162,33,168),(28,163,34,157),(29,164,35,158),(30,165,36,159),(37,98,43,104),(38,99,44,105),(39,100,45,106),(40,101,46,107),(41,102,47,108),(42,103,48,97),(49,114,55,120),(50,115,56,109),(51,116,57,110),(52,117,58,111),(53,118,59,112),(54,119,60,113),(85,205,91,211),(86,206,92,212),(87,207,93,213),(88,208,94,214),(89,209,95,215),(90,210,96,216),(121,230,127,236),(122,231,128,237),(123,232,129,238),(124,233,130,239),(125,234,131,240),(126,235,132,229),(133,188,139,182),(134,189,140,183),(135,190,141,184),(136,191,142,185),(137,192,143,186),(138,181,144,187),(145,200,151,194),(146,201,152,195),(147,202,153,196),(148,203,154,197),(149,204,155,198),(150,193,156,199),(169,218,175,224),(170,219,176,225),(171,220,177,226),(172,221,178,227),(173,222,179,228),(174,223,180,217)], [(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),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(85,92),(86,91),(87,90),(88,89),(93,96),(94,95),(97,98),(99,108),(100,107),(101,106),(102,105),(103,104),(109,116),(110,115),(111,114),(112,113),(117,120),(118,119),(121,126),(122,125),(123,124),(127,132),(128,131),(129,130),(133,144),(134,143),(135,142),(136,141),(137,140),(138,139),(145,150),(146,149),(147,148),(151,156),(152,155),(153,154),(157,162),(158,161),(159,160),(163,168),(164,167),(165,166),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,182),(183,192),(184,191),(185,190),(186,189),(187,188),(193,200),(194,199),(195,198),(196,197),(201,204),(202,203),(205,212),(206,211),(207,210),(208,209),(213,216),(214,215),(217,218),(219,228),(220,227),(221,226),(222,225),(223,224),(229,236),(230,235),(231,234),(232,233),(237,240),(238,239)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C4○D4 | C5×S3 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 |
| kernel | C10×C4○D12 | C10×Dic6 | S3×C2×C20 | C10×D12 | C5×C4○D12 | C10×C3⋊D4 | C22×C60 | C2×C4○D12 | C2×Dic6 | S3×C2×C4 | C2×D12 | C4○D12 | C2×C3⋊D4 | C22×C12 | C22×C20 | C2×C20 | C22×C10 | C30 | C22×C4 | C10 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 4 | 4 | 8 | 4 | 32 | 8 | 4 | 1 | 6 | 1 | 4 | 4 | 8 | 24 | 4 | 16 | 32 |
Matrix representation of C10×C4○D12 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 60 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 46 | 23 |
| 0 | 38 | 23 |
| 60 | 0 | 0 |
| 0 | 46 | 23 |
| 0 | 38 | 15 |
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