metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (S3×C10)⋊7D4, C5⋊4(D6⋊3D4), D6⋊4(C5⋊D4), (C3×Dic5)⋊4D4, C6.163(D4×D5), C15⋊21(C4⋊D4), C30.245(C2×D4), C10.162(S3×D4), C23.24(S3×D5), Dic5⋊3(C3⋊D4), C3⋊5(Dic5⋊D4), C30.Q8⋊37C2, (C22×D5).30D6, (C22×C6).42D10, (C22×C10).59D6, C30.156(C4○D4), C30.38D4⋊28C2, C6.85(D4⋊2D5), D10⋊Dic3⋊35C2, (C2×C30).207C23, (C2×Dic5).133D6, (C2×Dic3).63D10, (C22×S3).56D10, C10.85(D4⋊2S3), (C22×C30).69C22, C2.29(C30.C23), (C6×Dic5).120C22, (C2×Dic15).141C22, (C10×Dic3).120C22, (C6×C5⋊D4)⋊4C2, (C2×C5⋊D4)⋊4S3, (C2×C3⋊D4)⋊5D5, (C10×C3⋊D4)⋊5C2, (C2×S3×Dic5)⋊18C2, C2.43(D5×C3⋊D4), C6.66(C2×C5⋊D4), C2.41(S3×C5⋊D4), (C2×C15⋊D4)⋊13C2, C10.67(C2×C3⋊D4), (D5×C2×C6).53C22, C22.236(C2×S3×D5), (S3×C2×C10).53C22, (C2×C6).219(C22×D5), (C2×C10).219(C22×S3), SmallGroup(480,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S3×C10)⋊D4
G = < a,b,c,d,e | a10=b3=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=a5c, ede=d-1 >
Subgroups: 924 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, S3×C10, S3×C10, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D6⋊3D4, S3×Dic5, C15⋊D4, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, D5×C2×C6, S3×C2×C10, C22×C30, Dic5⋊D4, D10⋊Dic3, C30.Q8, C30.38D4, C2×S3×Dic5, C2×C15⋊D4, C6×C5⋊D4, C10×C3⋊D4, (S3×C10)⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, D6⋊3D4, C2×S3×D5, Dic5⋊D4, C30.C23, D5×C3⋊D4, S3×C5⋊D4, (S3×C10)⋊D4
►On 240 points
Generators in S240
(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 40 27)(2 31 28)(3 32 29)(4 33 30)(5 34 21)(6 35 22)(7 36 23)(8 37 24)(9 38 25)(10 39 26)(11 229 232)(12 230 233)(13 221 234)(14 222 235)(15 223 236)(16 224 237)(17 225 238)(18 226 239)(19 227 240)(20 228 231)(41 53 66)(42 54 67)(43 55 68)(44 56 69)(45 57 70)(46 58 61)(47 59 62)(48 60 63)(49 51 64)(50 52 65)(71 83 96)(72 84 97)(73 85 98)(74 86 99)(75 87 100)(76 88 91)(77 89 92)(78 90 93)(79 81 94)(80 82 95)(101 113 125)(102 114 126)(103 115 127)(104 116 128)(105 117 129)(106 118 130)(107 119 121)(108 120 122)(109 111 123)(110 112 124)(131 144 156)(132 145 157)(133 146 158)(134 147 159)(135 148 160)(136 149 151)(137 150 152)(138 141 153)(139 142 154)(140 143 155)(161 174 186)(162 175 187)(163 176 188)(164 177 189)(165 178 190)(166 179 181)(167 180 182)(168 171 183)(169 172 184)(170 173 185)(191 204 216)(192 205 217)(193 206 218)(194 207 219)(195 208 220)(196 209 211)(197 210 212)(198 201 213)(199 202 214)(200 203 215)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 224)(12 225)(13 226)(14 227)(15 228)(16 229)(17 230)(18 221)(19 222)(20 223)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 56)(52 57)(53 58)(54 59)(55 60)(71 96)(72 97)(73 98)(74 99)(75 100)(76 91)(77 92)(78 93)(79 94)(80 95)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 121)(108 122)(109 123)(110 124)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)(161 186)(162 187)(163 188)(164 189)(165 190)(166 181)(167 182)(168 183)(169 184)(170 185)(191 211)(192 212)(193 213)(194 214)(195 215)(196 216)(197 217)(198 218)(199 219)(200 220)(201 206)(202 207)(203 208)(204 209)(205 210)(231 236)(232 237)(233 238)(234 239)(235 240)
(1 233 57 203)(2 232 58 202)(3 231 59 201)(4 240 60 210)(5 239 51 209)(6 238 52 208)(7 237 53 207)(8 236 54 206)(9 235 55 205)(10 234 56 204)(11 61 214 31)(12 70 215 40)(13 69 216 39)(14 68 217 38)(15 67 218 37)(16 66 219 36)(17 65 220 35)(18 64 211 34)(19 63 212 33)(20 62 213 32)(21 226 49 196)(22 225 50 195)(23 224 41 194)(24 223 42 193)(25 222 43 192)(26 221 44 191)(27 230 45 200)(28 229 46 199)(29 228 47 198)(30 227 48 197)(71 164 101 134)(72 163 102 133)(73 162 103 132)(74 161 104 131)(75 170 105 140)(76 169 106 139)(77 168 107 138)(78 167 108 137)(79 166 109 136)(80 165 110 135)(81 179 111 149)(82 178 112 148)(83 177 113 147)(84 176 114 146)(85 175 115 145)(86 174 116 144)(87 173 117 143)(88 172 118 142)(89 171 119 141)(90 180 120 150)(91 184 130 154)(92 183 121 153)(93 182 122 152)(94 181 123 151)(95 190 124 160)(96 189 125 159)(97 188 126 158)(98 187 127 157)(99 186 128 156)(100 185 129 155)
(1 143)(2 142)(3 141)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 130)(12 129)(13 128)(14 127)(15 126)(16 125)(17 124)(18 123)(19 122)(20 121)(21 136)(22 135)(23 134)(24 133)(25 132)(26 131)(27 140)(28 139)(29 138)(30 137)(31 154)(32 153)(33 152)(34 151)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 164)(42 163)(43 162)(44 161)(45 170)(46 169)(47 168)(48 167)(49 166)(50 165)(51 179)(52 178)(53 177)(54 176)(55 175)(56 174)(57 173)(58 172)(59 171)(60 180)(61 184)(62 183)(63 182)(64 181)(65 190)(66 189)(67 188)(68 187)(69 186)(70 185)(71 194)(72 193)(73 192)(74 191)(75 200)(76 199)(77 198)(78 197)(79 196)(80 195)(81 209)(82 208)(83 207)(84 206)(85 205)(86 204)(87 203)(88 202)(89 201)(90 210)(91 214)(92 213)(93 212)(94 211)(95 220)(96 219)(97 218)(98 217)(99 216)(100 215)(101 224)(102 223)(103 222)(104 221)(105 230)(106 229)(107 228)(108 227)(109 226)(110 225)(111 239)(112 238)(113 237)(114 236)(115 235)(116 234)(117 233)(118 232)(119 231)(120 240)
G:=sub<Sym(240)| (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), (1,40,27)(2,31,28)(3,32,29)(4,33,30)(5,34,21)(6,35,22)(7,36,23)(8,37,24)(9,38,25)(10,39,26)(11,229,232)(12,230,233)(13,221,234)(14,222,235)(15,223,236)(16,224,237)(17,225,238)(18,226,239)(19,227,240)(20,228,231)(41,53,66)(42,54,67)(43,55,68)(44,56,69)(45,57,70)(46,58,61)(47,59,62)(48,60,63)(49,51,64)(50,52,65)(71,83,96)(72,84,97)(73,85,98)(74,86,99)(75,87,100)(76,88,91)(77,89,92)(78,90,93)(79,81,94)(80,82,95)(101,113,125)(102,114,126)(103,115,127)(104,116,128)(105,117,129)(106,118,130)(107,119,121)(108,120,122)(109,111,123)(110,112,124)(131,144,156)(132,145,157)(133,146,158)(134,147,159)(135,148,160)(136,149,151)(137,150,152)(138,141,153)(139,142,154)(140,143,155)(161,174,186)(162,175,187)(163,176,188)(164,177,189)(165,178,190)(166,179,181)(167,180,182)(168,171,183)(169,172,184)(170,173,185)(191,204,216)(192,205,217)(193,206,218)(194,207,219)(195,208,220)(196,209,211)(197,210,212)(198,201,213)(199,202,214)(200,203,215), (1,6)(2,7)(3,8)(4,9)(5,10)(11,224)(12,225)(13,226)(14,227)(15,228)(16,229)(17,230)(18,221)(19,222)(20,223)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,56)(52,57)(53,58)(54,59)(55,60)(71,96)(72,97)(73,98)(74,99)(75,100)(76,91)(77,92)(78,93)(79,94)(80,95)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,121)(108,122)(109,123)(110,124)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)(137,152)(138,153)(139,154)(140,155)(161,186)(162,187)(163,188)(164,189)(165,190)(166,181)(167,182)(168,183)(169,184)(170,185)(191,211)(192,212)(193,213)(194,214)(195,215)(196,216)(197,217)(198,218)(199,219)(200,220)(201,206)(202,207)(203,208)(204,209)(205,210)(231,236)(232,237)(233,238)(234,239)(235,240), (1,233,57,203)(2,232,58,202)(3,231,59,201)(4,240,60,210)(5,239,51,209)(6,238,52,208)(7,237,53,207)(8,236,54,206)(9,235,55,205)(10,234,56,204)(11,61,214,31)(12,70,215,40)(13,69,216,39)(14,68,217,38)(15,67,218,37)(16,66,219,36)(17,65,220,35)(18,64,211,34)(19,63,212,33)(20,62,213,32)(21,226,49,196)(22,225,50,195)(23,224,41,194)(24,223,42,193)(25,222,43,192)(26,221,44,191)(27,230,45,200)(28,229,46,199)(29,228,47,198)(30,227,48,197)(71,164,101,134)(72,163,102,133)(73,162,103,132)(74,161,104,131)(75,170,105,140)(76,169,106,139)(77,168,107,138)(78,167,108,137)(79,166,109,136)(80,165,110,135)(81,179,111,149)(82,178,112,148)(83,177,113,147)(84,176,114,146)(85,175,115,145)(86,174,116,144)(87,173,117,143)(88,172,118,142)(89,171,119,141)(90,180,120,150)(91,184,130,154)(92,183,121,153)(93,182,122,152)(94,181,123,151)(95,190,124,160)(96,189,125,159)(97,188,126,158)(98,187,127,157)(99,186,128,156)(100,185,129,155), (1,143)(2,142)(3,141)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,136)(22,135)(23,134)(24,133)(25,132)(26,131)(27,140)(28,139)(29,138)(30,137)(31,154)(32,153)(33,152)(34,151)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,164)(42,163)(43,162)(44,161)(45,170)(46,169)(47,168)(48,167)(49,166)(50,165)(51,179)(52,178)(53,177)(54,176)(55,175)(56,174)(57,173)(58,172)(59,171)(60,180)(61,184)(62,183)(63,182)(64,181)(65,190)(66,189)(67,188)(68,187)(69,186)(70,185)(71,194)(72,193)(73,192)(74,191)(75,200)(76,199)(77,198)(78,197)(79,196)(80,195)(81,209)(82,208)(83,207)(84,206)(85,205)(86,204)(87,203)(88,202)(89,201)(90,210)(91,214)(92,213)(93,212)(94,211)(95,220)(96,219)(97,218)(98,217)(99,216)(100,215)(101,224)(102,223)(103,222)(104,221)(105,230)(106,229)(107,228)(108,227)(109,226)(110,225)(111,239)(112,238)(113,237)(114,236)(115,235)(116,234)(117,233)(118,232)(119,231)(120,240)>;
G:=Group( (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), (1,40,27)(2,31,28)(3,32,29)(4,33,30)(5,34,21)(6,35,22)(7,36,23)(8,37,24)(9,38,25)(10,39,26)(11,229,232)(12,230,233)(13,221,234)(14,222,235)(15,223,236)(16,224,237)(17,225,238)(18,226,239)(19,227,240)(20,228,231)(41,53,66)(42,54,67)(43,55,68)(44,56,69)(45,57,70)(46,58,61)(47,59,62)(48,60,63)(49,51,64)(50,52,65)(71,83,96)(72,84,97)(73,85,98)(74,86,99)(75,87,100)(76,88,91)(77,89,92)(78,90,93)(79,81,94)(80,82,95)(101,113,125)(102,114,126)(103,115,127)(104,116,128)(105,117,129)(106,118,130)(107,119,121)(108,120,122)(109,111,123)(110,112,124)(131,144,156)(132,145,157)(133,146,158)(134,147,159)(135,148,160)(136,149,151)(137,150,152)(138,141,153)(139,142,154)(140,143,155)(161,174,186)(162,175,187)(163,176,188)(164,177,189)(165,178,190)(166,179,181)(167,180,182)(168,171,183)(169,172,184)(170,173,185)(191,204,216)(192,205,217)(193,206,218)(194,207,219)(195,208,220)(196,209,211)(197,210,212)(198,201,213)(199,202,214)(200,203,215), (1,6)(2,7)(3,8)(4,9)(5,10)(11,224)(12,225)(13,226)(14,227)(15,228)(16,229)(17,230)(18,221)(19,222)(20,223)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,56)(52,57)(53,58)(54,59)(55,60)(71,96)(72,97)(73,98)(74,99)(75,100)(76,91)(77,92)(78,93)(79,94)(80,95)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,121)(108,122)(109,123)(110,124)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)(137,152)(138,153)(139,154)(140,155)(161,186)(162,187)(163,188)(164,189)(165,190)(166,181)(167,182)(168,183)(169,184)(170,185)(191,211)(192,212)(193,213)(194,214)(195,215)(196,216)(197,217)(198,218)(199,219)(200,220)(201,206)(202,207)(203,208)(204,209)(205,210)(231,236)(232,237)(233,238)(234,239)(235,240), (1,233,57,203)(2,232,58,202)(3,231,59,201)(4,240,60,210)(5,239,51,209)(6,238,52,208)(7,237,53,207)(8,236,54,206)(9,235,55,205)(10,234,56,204)(11,61,214,31)(12,70,215,40)(13,69,216,39)(14,68,217,38)(15,67,218,37)(16,66,219,36)(17,65,220,35)(18,64,211,34)(19,63,212,33)(20,62,213,32)(21,226,49,196)(22,225,50,195)(23,224,41,194)(24,223,42,193)(25,222,43,192)(26,221,44,191)(27,230,45,200)(28,229,46,199)(29,228,47,198)(30,227,48,197)(71,164,101,134)(72,163,102,133)(73,162,103,132)(74,161,104,131)(75,170,105,140)(76,169,106,139)(77,168,107,138)(78,167,108,137)(79,166,109,136)(80,165,110,135)(81,179,111,149)(82,178,112,148)(83,177,113,147)(84,176,114,146)(85,175,115,145)(86,174,116,144)(87,173,117,143)(88,172,118,142)(89,171,119,141)(90,180,120,150)(91,184,130,154)(92,183,121,153)(93,182,122,152)(94,181,123,151)(95,190,124,160)(96,189,125,159)(97,188,126,158)(98,187,127,157)(99,186,128,156)(100,185,129,155), (1,143)(2,142)(3,141)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,136)(22,135)(23,134)(24,133)(25,132)(26,131)(27,140)(28,139)(29,138)(30,137)(31,154)(32,153)(33,152)(34,151)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,164)(42,163)(43,162)(44,161)(45,170)(46,169)(47,168)(48,167)(49,166)(50,165)(51,179)(52,178)(53,177)(54,176)(55,175)(56,174)(57,173)(58,172)(59,171)(60,180)(61,184)(62,183)(63,182)(64,181)(65,190)(66,189)(67,188)(68,187)(69,186)(70,185)(71,194)(72,193)(73,192)(74,191)(75,200)(76,199)(77,198)(78,197)(79,196)(80,195)(81,209)(82,208)(83,207)(84,206)(85,205)(86,204)(87,203)(88,202)(89,201)(90,210)(91,214)(92,213)(93,212)(94,211)(95,220)(96,219)(97,218)(98,217)(99,216)(100,215)(101,224)(102,223)(103,222)(104,221)(105,230)(106,229)(107,228)(108,227)(109,226)(110,225)(111,239)(112,238)(113,237)(114,236)(115,235)(116,234)(117,233)(118,232)(119,231)(120,240) );
G=PermutationGroup([[(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)], [(1,40,27),(2,31,28),(3,32,29),(4,33,30),(5,34,21),(6,35,22),(7,36,23),(8,37,24),(9,38,25),(10,39,26),(11,229,232),(12,230,233),(13,221,234),(14,222,235),(15,223,236),(16,224,237),(17,225,238),(18,226,239),(19,227,240),(20,228,231),(41,53,66),(42,54,67),(43,55,68),(44,56,69),(45,57,70),(46,58,61),(47,59,62),(48,60,63),(49,51,64),(50,52,65),(71,83,96),(72,84,97),(73,85,98),(74,86,99),(75,87,100),(76,88,91),(77,89,92),(78,90,93),(79,81,94),(80,82,95),(101,113,125),(102,114,126),(103,115,127),(104,116,128),(105,117,129),(106,118,130),(107,119,121),(108,120,122),(109,111,123),(110,112,124),(131,144,156),(132,145,157),(133,146,158),(134,147,159),(135,148,160),(136,149,151),(137,150,152),(138,141,153),(139,142,154),(140,143,155),(161,174,186),(162,175,187),(163,176,188),(164,177,189),(165,178,190),(166,179,181),(167,180,182),(168,171,183),(169,172,184),(170,173,185),(191,204,216),(192,205,217),(193,206,218),(194,207,219),(195,208,220),(196,209,211),(197,210,212),(198,201,213),(199,202,214),(200,203,215)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,224),(12,225),(13,226),(14,227),(15,228),(16,229),(17,230),(18,221),(19,222),(20,223),(21,39),(22,40),(23,31),(24,32),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,56),(52,57),(53,58),(54,59),(55,60),(71,96),(72,97),(73,98),(74,99),(75,100),(76,91),(77,92),(78,93),(79,94),(80,95),(101,125),(102,126),(103,127),(104,128),(105,129),(106,130),(107,121),(108,122),(109,123),(110,124),(131,156),(132,157),(133,158),(134,159),(135,160),(136,151),(137,152),(138,153),(139,154),(140,155),(161,186),(162,187),(163,188),(164,189),(165,190),(166,181),(167,182),(168,183),(169,184),(170,185),(191,211),(192,212),(193,213),(194,214),(195,215),(196,216),(197,217),(198,218),(199,219),(200,220),(201,206),(202,207),(203,208),(204,209),(205,210),(231,236),(232,237),(233,238),(234,239),(235,240)], [(1,233,57,203),(2,232,58,202),(3,231,59,201),(4,240,60,210),(5,239,51,209),(6,238,52,208),(7,237,53,207),(8,236,54,206),(9,235,55,205),(10,234,56,204),(11,61,214,31),(12,70,215,40),(13,69,216,39),(14,68,217,38),(15,67,218,37),(16,66,219,36),(17,65,220,35),(18,64,211,34),(19,63,212,33),(20,62,213,32),(21,226,49,196),(22,225,50,195),(23,224,41,194),(24,223,42,193),(25,222,43,192),(26,221,44,191),(27,230,45,200),(28,229,46,199),(29,228,47,198),(30,227,48,197),(71,164,101,134),(72,163,102,133),(73,162,103,132),(74,161,104,131),(75,170,105,140),(76,169,106,139),(77,168,107,138),(78,167,108,137),(79,166,109,136),(80,165,110,135),(81,179,111,149),(82,178,112,148),(83,177,113,147),(84,176,114,146),(85,175,115,145),(86,174,116,144),(87,173,117,143),(88,172,118,142),(89,171,119,141),(90,180,120,150),(91,184,130,154),(92,183,121,153),(93,182,122,152),(94,181,123,151),(95,190,124,160),(96,189,125,159),(97,188,126,158),(98,187,127,157),(99,186,128,156),(100,185,129,155)], [(1,143),(2,142),(3,141),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,130),(12,129),(13,128),(14,127),(15,126),(16,125),(17,124),(18,123),(19,122),(20,121),(21,136),(22,135),(23,134),(24,133),(25,132),(26,131),(27,140),(28,139),(29,138),(30,137),(31,154),(32,153),(33,152),(34,151),(35,160),(36,159),(37,158),(38,157),(39,156),(40,155),(41,164),(42,163),(43,162),(44,161),(45,170),(46,169),(47,168),(48,167),(49,166),(50,165),(51,179),(52,178),(53,177),(54,176),(55,175),(56,174),(57,173),(58,172),(59,171),(60,180),(61,184),(62,183),(63,182),(64,181),(65,190),(66,189),(67,188),(68,187),(69,186),(70,185),(71,194),(72,193),(73,192),(74,191),(75,200),(76,199),(77,198),(78,197),(79,196),(80,195),(81,209),(82,208),(83,207),(84,206),(85,205),(86,204),(87,203),(88,202),(89,201),(90,210),(91,214),(92,213),(93,212),(94,211),(95,220),(96,219),(97,218),(98,217),(99,216),(100,215),(101,224),(102,223),(103,222),(104,221),(105,230),(106,229),(107,228),(108,227),(109,226),(110,225),(111,239),(112,238),(113,237),(114,236),(115,235),(116,234),(117,233),(118,232),(119,231),(120,240)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 20 | 2 | 10 | 10 | 12 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | C30.C23 | D5×C3⋊D4 | S3×C5⋊D4 |
| kernel | (S3×C10)⋊D4 | D10⋊Dic3 | C30.Q8 | C30.38D4 | C2×S3×Dic5 | C2×C15⋊D4 | C6×C5⋊D4 | C10×C3⋊D4 | C2×C5⋊D4 | C3×Dic5 | S3×C10 | C2×C3⋊D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | Dic5 | D6 | C10 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (S3×C10)⋊D4 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 43 | 18 |
| 0 | 0 | 44 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 4 |
| 0 | 0 | 20 | 38 |
| 9 | 18 | 0 | 0 |
| 43 | 52 | 0 | 0 |
| 0 | 0 | 39 | 36 |
| 0 | 0 | 12 | 22 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,43,44,0,0,18,0],[0,60,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,20,0,0,4,38],[9,43,0,0,18,52,0,0,0,0,39,12,0,0,36,22] >;
(S3×C10)⋊D4 in GAP, Magma, Sage, TeX
(S_3\times C_{10})\rtimes D_4 % in TeX
G:=Group("(S3xC10):D4"); // GroupNames label
G:=SmallGroup(480,641);
// by ID
G=gap.SmallGroup(480,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,e*d*e=d^-1>;
// generators/relations