direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D4.S3, C30⋊12SD16, C60.149D4, C60.229C23, C6⋊2(C5×SD16), C3⋊3(C10×SD16), (C5×D4).36D6, D4.7(S3×C10), (C6×D4).3C10, C12.16(C5×D4), C6.46(D4×C10), C15⋊24(C2×SD16), (C2×Dic6)⋊9C10, Dic6⋊6(C2×C10), (D4×C30).13C2, (D4×C10).10S3, (C2×C20).359D6, C30.429(C2×D4), (C2×C30).183D4, (C10×Dic6)⋊25C2, C20.71(C3⋊D4), C20.202(C22×S3), (C2×C60).359C22, C12.13(C22×C10), (C5×Dic6)⋊33C22, (D4×C15).46C22, (C2×C3⋊C8)⋊5C10, C3⋊C8⋊8(C2×C10), (C10×C3⋊C8)⋊19C2, C4.13(S3×C2×C10), C4.6(C5×C3⋊D4), (C5×C3⋊C8)⋊41C22, (C2×D4).4(C5×S3), (C2×C6).40(C5×D4), (C2×C4).48(S3×C10), (C3×D4).7(C2×C10), C2.10(C10×C3⋊D4), (C2×C12).32(C2×C10), C10.131(C2×C3⋊D4), C22.22(C5×C3⋊D4), (C2×C10).94(C3⋊D4), SmallGroup(480,812)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D4.S3
G = < a,b,c,d,e | a10=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 292 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, C30, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, C5×Dic3, C60, C2×C30, C2×C30, C2×C40, C5×SD16, D4×C10, Q8×C10, C2×D4.S3, C5×C3⋊C8, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, D4×C15, D4×C15, C22×C30, C10×SD16, C10×C3⋊C8, C5×D4.S3, C10×Dic6, D4×C30, C10×D4.S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, SD16, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C2×SD16, C5×D4, C22×C10, D4.S3, C2×C3⋊D4, S3×C10, C5×SD16, D4×C10, C2×D4.S3, C5×C3⋊D4, S3×C2×C10, C10×SD16, C5×D4.S3, C10×C3⋊D4, C10×D4.S3
►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)
(1 183 154 124)(2 184 155 125)(3 185 156 126)(4 186 157 127)(5 187 158 128)(6 188 159 129)(7 189 160 130)(8 190 151 121)(9 181 152 122)(10 182 153 123)(11 239 76 99)(12 240 77 100)(13 231 78 91)(14 232 79 92)(15 233 80 93)(16 234 71 94)(17 235 72 95)(18 236 73 96)(19 237 74 97)(20 238 75 98)(21 196 86 46)(22 197 87 47)(23 198 88 48)(24 199 89 49)(25 200 90 50)(26 191 81 41)(27 192 82 42)(28 193 83 43)(29 194 84 44)(30 195 85 45)(31 103 143 69)(32 104 144 70)(33 105 145 61)(34 106 146 62)(35 107 147 63)(36 108 148 64)(37 109 149 65)(38 110 150 66)(39 101 141 67)(40 102 142 68)(51 164 177 220)(52 165 178 211)(53 166 179 212)(54 167 180 213)(55 168 171 214)(56 169 172 215)(57 170 173 216)(58 161 174 217)(59 162 175 218)(60 163 176 219)(111 228 208 139)(112 229 209 140)(113 230 210 131)(114 221 201 132)(115 222 202 133)(116 223 203 134)(117 224 204 135)(118 225 205 136)(119 226 206 137)(120 227 207 138)
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 121)(9 122)(10 123)(11 76)(12 77)(13 78)(14 79)(15 80)(16 71)(17 72)(18 73)(19 74)(20 75)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 141)(40 142)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 220)(52 211)(53 212)(54 213)(55 214)(56 215)(57 216)(58 217)(59 218)(60 219)(111 228)(112 229)(113 230)(114 221)(115 222)(116 223)(117 224)(118 225)(119 226)(120 227)(131 210)(132 201)(133 202)(134 203)(135 204)(136 205)(137 206)(138 207)(139 208)(140 209)(151 190)(152 181)(153 182)(154 183)(155 184)(156 185)(157 186)(158 187)(159 188)(160 189)(161 174)(162 175)(163 176)(164 177)(165 178)(166 179)(167 180)(168 171)(169 172)(170 173)
(1 60 221)(2 51 222)(3 52 223)(4 53 224)(5 54 225)(6 55 226)(7 56 227)(8 57 228)(9 58 229)(10 59 230)(11 193 145)(12 194 146)(13 195 147)(14 196 148)(15 197 149)(16 198 150)(17 199 141)(18 200 142)(19 191 143)(20 192 144)(21 108 92)(22 109 93)(23 110 94)(24 101 95)(25 102 96)(26 103 97)(27 104 98)(28 105 99)(29 106 100)(30 107 91)(31 74 41)(32 75 42)(33 76 43)(34 77 44)(35 78 45)(36 79 46)(37 80 47)(38 71 48)(39 72 49)(40 73 50)(61 239 83)(62 240 84)(63 231 85)(64 232 86)(65 233 87)(66 234 88)(67 235 89)(68 236 90)(69 237 81)(70 238 82)(111 121 216)(112 122 217)(113 123 218)(114 124 219)(115 125 220)(116 126 211)(117 127 212)(118 128 213)(119 129 214)(120 130 215)(131 153 175)(132 154 176)(133 155 177)(134 156 178)(135 157 179)(136 158 180)(137 159 171)(138 160 172)(139 151 173)(140 152 174)(161 209 181)(162 210 182)(163 201 183)(164 202 184)(165 203 185)(166 204 186)(167 205 187)(168 206 188)(169 207 189)(170 208 190)
(1 33 154 145)(2 34 155 146)(3 35 156 147)(4 36 157 148)(5 37 158 149)(6 38 159 150)(7 39 160 141)(8 40 151 142)(9 31 152 143)(10 32 153 144)(11 221 76 132)(12 222 77 133)(13 223 78 134)(14 224 79 135)(15 225 80 136)(16 226 71 137)(17 227 72 138)(18 228 73 139)(19 229 74 140)(20 230 75 131)(21 166 86 212)(22 167 87 213)(23 168 88 214)(24 169 89 215)(25 170 90 216)(26 161 81 217)(27 162 82 218)(28 163 83 219)(29 164 84 220)(30 165 85 211)(41 174 191 58)(42 175 192 59)(43 176 193 60)(44 177 194 51)(45 178 195 52)(46 179 196 53)(47 180 197 54)(48 171 198 55)(49 172 199 56)(50 173 200 57)(61 124 105 183)(62 125 106 184)(63 126 107 185)(64 127 108 186)(65 128 109 187)(66 129 110 188)(67 130 101 189)(68 121 102 190)(69 122 103 181)(70 123 104 182)(91 203 231 116)(92 204 232 117)(93 205 233 118)(94 206 234 119)(95 207 235 120)(96 208 236 111)(97 209 237 112)(98 210 238 113)(99 201 239 114)(100 202 240 115)
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), (1,183,154,124)(2,184,155,125)(3,185,156,126)(4,186,157,127)(5,187,158,128)(6,188,159,129)(7,189,160,130)(8,190,151,121)(9,181,152,122)(10,182,153,123)(11,239,76,99)(12,240,77,100)(13,231,78,91)(14,232,79,92)(15,233,80,93)(16,234,71,94)(17,235,72,95)(18,236,73,96)(19,237,74,97)(20,238,75,98)(21,196,86,46)(22,197,87,47)(23,198,88,48)(24,199,89,49)(25,200,90,50)(26,191,81,41)(27,192,82,42)(28,193,83,43)(29,194,84,44)(30,195,85,45)(31,103,143,69)(32,104,144,70)(33,105,145,61)(34,106,146,62)(35,107,147,63)(36,108,148,64)(37,109,149,65)(38,110,150,66)(39,101,141,67)(40,102,142,68)(51,164,177,220)(52,165,178,211)(53,166,179,212)(54,167,180,213)(55,168,171,214)(56,169,172,215)(57,170,173,216)(58,161,174,217)(59,162,175,218)(60,163,176,219)(111,228,208,139)(112,229,209,140)(113,230,210,131)(114,221,201,132)(115,222,202,133)(116,223,203,134)(117,224,204,135)(118,225,205,136)(119,226,206,137)(120,227,207,138), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,121)(9,122)(10,123)(11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,141)(40,142)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,220)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(111,228)(112,229)(113,230)(114,221)(115,222)(116,223)(117,224)(118,225)(119,226)(120,227)(131,210)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(151,190)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,171)(169,172)(170,173), (1,60,221)(2,51,222)(3,52,223)(4,53,224)(5,54,225)(6,55,226)(7,56,227)(8,57,228)(9,58,229)(10,59,230)(11,193,145)(12,194,146)(13,195,147)(14,196,148)(15,197,149)(16,198,150)(17,199,141)(18,200,142)(19,191,143)(20,192,144)(21,108,92)(22,109,93)(23,110,94)(24,101,95)(25,102,96)(26,103,97)(27,104,98)(28,105,99)(29,106,100)(30,107,91)(31,74,41)(32,75,42)(33,76,43)(34,77,44)(35,78,45)(36,79,46)(37,80,47)(38,71,48)(39,72,49)(40,73,50)(61,239,83)(62,240,84)(63,231,85)(64,232,86)(65,233,87)(66,234,88)(67,235,89)(68,236,90)(69,237,81)(70,238,82)(111,121,216)(112,122,217)(113,123,218)(114,124,219)(115,125,220)(116,126,211)(117,127,212)(118,128,213)(119,129,214)(120,130,215)(131,153,175)(132,154,176)(133,155,177)(134,156,178)(135,157,179)(136,158,180)(137,159,171)(138,160,172)(139,151,173)(140,152,174)(161,209,181)(162,210,182)(163,201,183)(164,202,184)(165,203,185)(166,204,186)(167,205,187)(168,206,188)(169,207,189)(170,208,190), (1,33,154,145)(2,34,155,146)(3,35,156,147)(4,36,157,148)(5,37,158,149)(6,38,159,150)(7,39,160,141)(8,40,151,142)(9,31,152,143)(10,32,153,144)(11,221,76,132)(12,222,77,133)(13,223,78,134)(14,224,79,135)(15,225,80,136)(16,226,71,137)(17,227,72,138)(18,228,73,139)(19,229,74,140)(20,230,75,131)(21,166,86,212)(22,167,87,213)(23,168,88,214)(24,169,89,215)(25,170,90,216)(26,161,81,217)(27,162,82,218)(28,163,83,219)(29,164,84,220)(30,165,85,211)(41,174,191,58)(42,175,192,59)(43,176,193,60)(44,177,194,51)(45,178,195,52)(46,179,196,53)(47,180,197,54)(48,171,198,55)(49,172,199,56)(50,173,200,57)(61,124,105,183)(62,125,106,184)(63,126,107,185)(64,127,108,186)(65,128,109,187)(66,129,110,188)(67,130,101,189)(68,121,102,190)(69,122,103,181)(70,123,104,182)(91,203,231,116)(92,204,232,117)(93,205,233,118)(94,206,234,119)(95,207,235,120)(96,208,236,111)(97,209,237,112)(98,210,238,113)(99,201,239,114)(100,202,240,115)>;
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), (1,183,154,124)(2,184,155,125)(3,185,156,126)(4,186,157,127)(5,187,158,128)(6,188,159,129)(7,189,160,130)(8,190,151,121)(9,181,152,122)(10,182,153,123)(11,239,76,99)(12,240,77,100)(13,231,78,91)(14,232,79,92)(15,233,80,93)(16,234,71,94)(17,235,72,95)(18,236,73,96)(19,237,74,97)(20,238,75,98)(21,196,86,46)(22,197,87,47)(23,198,88,48)(24,199,89,49)(25,200,90,50)(26,191,81,41)(27,192,82,42)(28,193,83,43)(29,194,84,44)(30,195,85,45)(31,103,143,69)(32,104,144,70)(33,105,145,61)(34,106,146,62)(35,107,147,63)(36,108,148,64)(37,109,149,65)(38,110,150,66)(39,101,141,67)(40,102,142,68)(51,164,177,220)(52,165,178,211)(53,166,179,212)(54,167,180,213)(55,168,171,214)(56,169,172,215)(57,170,173,216)(58,161,174,217)(59,162,175,218)(60,163,176,219)(111,228,208,139)(112,229,209,140)(113,230,210,131)(114,221,201,132)(115,222,202,133)(116,223,203,134)(117,224,204,135)(118,225,205,136)(119,226,206,137)(120,227,207,138), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,121)(9,122)(10,123)(11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,141)(40,142)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,220)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(111,228)(112,229)(113,230)(114,221)(115,222)(116,223)(117,224)(118,225)(119,226)(120,227)(131,210)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207)(139,208)(140,209)(151,190)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,171)(169,172)(170,173), (1,60,221)(2,51,222)(3,52,223)(4,53,224)(5,54,225)(6,55,226)(7,56,227)(8,57,228)(9,58,229)(10,59,230)(11,193,145)(12,194,146)(13,195,147)(14,196,148)(15,197,149)(16,198,150)(17,199,141)(18,200,142)(19,191,143)(20,192,144)(21,108,92)(22,109,93)(23,110,94)(24,101,95)(25,102,96)(26,103,97)(27,104,98)(28,105,99)(29,106,100)(30,107,91)(31,74,41)(32,75,42)(33,76,43)(34,77,44)(35,78,45)(36,79,46)(37,80,47)(38,71,48)(39,72,49)(40,73,50)(61,239,83)(62,240,84)(63,231,85)(64,232,86)(65,233,87)(66,234,88)(67,235,89)(68,236,90)(69,237,81)(70,238,82)(111,121,216)(112,122,217)(113,123,218)(114,124,219)(115,125,220)(116,126,211)(117,127,212)(118,128,213)(119,129,214)(120,130,215)(131,153,175)(132,154,176)(133,155,177)(134,156,178)(135,157,179)(136,158,180)(137,159,171)(138,160,172)(139,151,173)(140,152,174)(161,209,181)(162,210,182)(163,201,183)(164,202,184)(165,203,185)(166,204,186)(167,205,187)(168,206,188)(169,207,189)(170,208,190), (1,33,154,145)(2,34,155,146)(3,35,156,147)(4,36,157,148)(5,37,158,149)(6,38,159,150)(7,39,160,141)(8,40,151,142)(9,31,152,143)(10,32,153,144)(11,221,76,132)(12,222,77,133)(13,223,78,134)(14,224,79,135)(15,225,80,136)(16,226,71,137)(17,227,72,138)(18,228,73,139)(19,229,74,140)(20,230,75,131)(21,166,86,212)(22,167,87,213)(23,168,88,214)(24,169,89,215)(25,170,90,216)(26,161,81,217)(27,162,82,218)(28,163,83,219)(29,164,84,220)(30,165,85,211)(41,174,191,58)(42,175,192,59)(43,176,193,60)(44,177,194,51)(45,178,195,52)(46,179,196,53)(47,180,197,54)(48,171,198,55)(49,172,199,56)(50,173,200,57)(61,124,105,183)(62,125,106,184)(63,126,107,185)(64,127,108,186)(65,128,109,187)(66,129,110,188)(67,130,101,189)(68,121,102,190)(69,122,103,181)(70,123,104,182)(91,203,231,116)(92,204,232,117)(93,205,233,118)(94,206,234,119)(95,207,235,120)(96,208,236,111)(97,209,237,112)(98,210,238,113)(99,201,239,114)(100,202,240,115) );
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)], [(1,183,154,124),(2,184,155,125),(3,185,156,126),(4,186,157,127),(5,187,158,128),(6,188,159,129),(7,189,160,130),(8,190,151,121),(9,181,152,122),(10,182,153,123),(11,239,76,99),(12,240,77,100),(13,231,78,91),(14,232,79,92),(15,233,80,93),(16,234,71,94),(17,235,72,95),(18,236,73,96),(19,237,74,97),(20,238,75,98),(21,196,86,46),(22,197,87,47),(23,198,88,48),(24,199,89,49),(25,200,90,50),(26,191,81,41),(27,192,82,42),(28,193,83,43),(29,194,84,44),(30,195,85,45),(31,103,143,69),(32,104,144,70),(33,105,145,61),(34,106,146,62),(35,107,147,63),(36,108,148,64),(37,109,149,65),(38,110,150,66),(39,101,141,67),(40,102,142,68),(51,164,177,220),(52,165,178,211),(53,166,179,212),(54,167,180,213),(55,168,171,214),(56,169,172,215),(57,170,173,216),(58,161,174,217),(59,162,175,218),(60,163,176,219),(111,228,208,139),(112,229,209,140),(113,230,210,131),(114,221,201,132),(115,222,202,133),(116,223,203,134),(117,224,204,135),(118,225,205,136),(119,226,206,137),(120,227,207,138)], [(1,124),(2,125),(3,126),(4,127),(5,128),(6,129),(7,130),(8,121),(9,122),(10,123),(11,76),(12,77),(13,78),(14,79),(15,80),(16,71),(17,72),(18,73),(19,74),(20,75),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,141),(40,142),(41,191),(42,192),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,199),(50,200),(51,220),(52,211),(53,212),(54,213),(55,214),(56,215),(57,216),(58,217),(59,218),(60,219),(111,228),(112,229),(113,230),(114,221),(115,222),(116,223),(117,224),(118,225),(119,226),(120,227),(131,210),(132,201),(133,202),(134,203),(135,204),(136,205),(137,206),(138,207),(139,208),(140,209),(151,190),(152,181),(153,182),(154,183),(155,184),(156,185),(157,186),(158,187),(159,188),(160,189),(161,174),(162,175),(163,176),(164,177),(165,178),(166,179),(167,180),(168,171),(169,172),(170,173)], [(1,60,221),(2,51,222),(3,52,223),(4,53,224),(5,54,225),(6,55,226),(7,56,227),(8,57,228),(9,58,229),(10,59,230),(11,193,145),(12,194,146),(13,195,147),(14,196,148),(15,197,149),(16,198,150),(17,199,141),(18,200,142),(19,191,143),(20,192,144),(21,108,92),(22,109,93),(23,110,94),(24,101,95),(25,102,96),(26,103,97),(27,104,98),(28,105,99),(29,106,100),(30,107,91),(31,74,41),(32,75,42),(33,76,43),(34,77,44),(35,78,45),(36,79,46),(37,80,47),(38,71,48),(39,72,49),(40,73,50),(61,239,83),(62,240,84),(63,231,85),(64,232,86),(65,233,87),(66,234,88),(67,235,89),(68,236,90),(69,237,81),(70,238,82),(111,121,216),(112,122,217),(113,123,218),(114,124,219),(115,125,220),(116,126,211),(117,127,212),(118,128,213),(119,129,214),(120,130,215),(131,153,175),(132,154,176),(133,155,177),(134,156,178),(135,157,179),(136,158,180),(137,159,171),(138,160,172),(139,151,173),(140,152,174),(161,209,181),(162,210,182),(163,201,183),(164,202,184),(165,203,185),(166,204,186),(167,205,187),(168,206,188),(169,207,189),(170,208,190)], [(1,33,154,145),(2,34,155,146),(3,35,156,147),(4,36,157,148),(5,37,158,149),(6,38,159,150),(7,39,160,141),(8,40,151,142),(9,31,152,143),(10,32,153,144),(11,221,76,132),(12,222,77,133),(13,223,78,134),(14,224,79,135),(15,225,80,136),(16,226,71,137),(17,227,72,138),(18,228,73,139),(19,229,74,140),(20,230,75,131),(21,166,86,212),(22,167,87,213),(23,168,88,214),(24,169,89,215),(25,170,90,216),(26,161,81,217),(27,162,82,218),(28,163,83,219),(29,164,84,220),(30,165,85,211),(41,174,191,58),(42,175,192,59),(43,176,193,60),(44,177,194,51),(45,178,195,52),(46,179,196,53),(47,180,197,54),(48,171,198,55),(49,172,199,56),(50,173,200,57),(61,124,105,183),(62,125,106,184),(63,126,107,185),(64,127,108,186),(65,128,109,187),(66,129,110,188),(67,130,101,189),(68,121,102,190),(69,122,103,181),(70,123,104,182),(91,203,231,116),(92,204,232,117),(93,205,233,118),(94,206,234,119),(95,207,235,120),(96,208,236,111),(97,209,237,112),(98,210,238,113),(99,201,239,114),(100,202,240,115)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 30M | ··· | 30AB | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×SD16 | C5×C3⋊D4 | C5×C3⋊D4 | D4.S3 | C5×D4.S3 |
| kernel | C10×D4.S3 | C10×C3⋊C8 | C5×D4.S3 | C10×Dic6 | D4×C30 | C2×D4.S3 | C2×C3⋊C8 | D4.S3 | C2×Dic6 | C6×D4 | D4×C10 | C60 | C2×C30 | C2×C20 | C5×D4 | C30 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C6 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 16 | 8 | 8 | 2 | 8 |
Matrix representation of C10×D4.S3 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 29 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 153 | 225 |
| 222 | 19 | 0 | 0 |
| 19 | 19 | 0 | 0 |
| 0 | 0 | 38 | 227 |
| 0 | 0 | 17 | 203 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,36,0,0,0,0,36],[0,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,240,29,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,15,153,0,0,0,225],[222,19,0,0,19,19,0,0,0,0,38,17,0,0,227,203] >;
C10×D4.S3 in GAP, Magma, Sage, TeX
C_{10}\times D_4.S_3 % in TeX
G:=Group("C10xD4.S3"); // GroupNames label
G:=SmallGroup(480,812);
// by ID
G=gap.SmallGroup(480,812);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations