direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10⋊2Q8, C12.65D20, C60.121D4, C4⋊Dic5⋊6C6, (C6×D5)⋊10Q8, D10⋊2(C3×Q8), C10.6(C6×D4), C6.52(Q8×D5), C2.10(C6×D20), C4.13(C3×D20), C6.79(C2×D20), C20.11(C3×D4), C10.14(C6×Q8), (C2×Dic10)⋊7C6, C30.280(C2×D4), C15⋊31(C22⋊Q8), C30.104(C2×Q8), (C6×Dic10)⋊23C2, D10⋊C4.2C6, (C2×C12).236D10, C30.236(C4○D4), (C2×C30).355C23, (C2×C60).269C22, C6.117(D4⋊2D5), (C6×Dic5).160C22, (C5×C4⋊C4)⋊8C6, C4⋊C4⋊5(C3×D5), C2.7(C3×Q8×D5), (C2×C4×D5).2C6, (C3×C4⋊C4)⋊14D5, C5⋊3(C3×C22⋊Q8), (C15×C4⋊C4)⋊17C2, (D5×C2×C12).13C2, (C2×C4).43(C6×D5), C22.52(D5×C2×C6), (C2×C20).21(C2×C6), (C3×C4⋊Dic5)⋊24C2, C10.25(C3×C4○D4), C2.13(C3×D4⋊2D5), (D5×C2×C6).132C22, (C3×D10⋊C4).8C2, (C2×C10).38(C22×C6), (C2×Dic5).12(C2×C6), (C22×D5).27(C2×C6), (C2×C6).351(C22×D5), SmallGroup(480,690)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10⋊2Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >
Subgroups: 480 in 148 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C3×D5, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C3×C22⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, D10⋊2Q8, C3×C4⋊Dic5, C3×D10⋊C4, C15×C4⋊C4, C6×Dic10, D5×C2×C12, C3×D10⋊2Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, D5, C2×C6, C2×D4, C2×Q8, C4○D4, D10, C3×D4, C3×Q8, C22×C6, C3×D5, C22⋊Q8, D20, C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5, C2×D20, D4⋊2D5, Q8×D5, C3×C22⋊Q8, C3×D20, D5×C2×C6, D10⋊2Q8, C6×D20, C3×D4⋊2D5, C3×Q8×D5, C3×D10⋊2Q8
►On 240 points
Generators in S240
(1 58 38)(2 59 39)(3 60 40)(4 51 31)(5 52 32)(6 53 33)(7 54 34)(8 55 35)(9 56 36)(10 57 37)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 232)(12 231)(13 240)(14 239)(15 238)(16 237)(17 236)(18 235)(19 234)(20 233)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 90)(78 89)(79 88)(80 87)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 110)(98 109)(99 108)(100 107)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 130)(118 129)(119 128)(120 127)(131 141)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(151 161)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(171 181)(172 190)(173 189)(174 188)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)(191 206)(192 205)(193 204)(194 203)(195 202)(196 201)(197 210)(198 209)(199 208)(200 207)(211 226)(212 225)(213 224)(214 223)(215 222)(216 221)(217 230)(218 229)(219 228)(220 227)
(1 73 25 90)(2 72 26 89)(3 71 27 88)(4 80 28 87)(5 79 29 86)(6 78 30 85)(7 77 21 84)(8 76 22 83)(9 75 23 82)(10 74 24 81)(11 190 238 173)(12 189 239 172)(13 188 240 171)(14 187 231 180)(15 186 232 179)(16 185 233 178)(17 184 234 177)(18 183 235 176)(19 182 236 175)(20 181 237 174)(31 100 48 107)(32 99 49 106)(33 98 50 105)(34 97 41 104)(35 96 42 103)(36 95 43 102)(37 94 44 101)(38 93 45 110)(39 92 46 109)(40 91 47 108)(51 120 68 127)(52 119 69 126)(53 118 70 125)(54 117 61 124)(55 116 62 123)(56 115 63 122)(57 114 64 121)(58 113 65 130)(59 112 66 129)(60 111 67 128)(131 207 148 200)(132 206 149 199)(133 205 150 198)(134 204 141 197)(135 203 142 196)(136 202 143 195)(137 201 144 194)(138 210 145 193)(139 209 146 192)(140 208 147 191)(151 227 168 220)(152 226 169 219)(153 225 170 218)(154 224 161 217)(155 223 162 216)(156 222 163 215)(157 221 164 214)(158 230 165 213)(159 229 166 212)(160 228 167 211)
(1 145 25 138)(2 146 26 139)(3 147 27 140)(4 148 28 131)(5 149 29 132)(6 150 30 133)(7 141 21 134)(8 142 22 135)(9 143 23 136)(10 144 24 137)(11 125 238 118)(12 126 239 119)(13 127 240 120)(14 128 231 111)(15 129 232 112)(16 130 233 113)(17 121 234 114)(18 122 235 115)(19 123 236 116)(20 124 237 117)(31 168 48 151)(32 169 49 152)(33 170 50 153)(34 161 41 154)(35 162 42 155)(36 163 43 156)(37 164 44 157)(38 165 45 158)(39 166 46 159)(40 167 47 160)(51 188 68 171)(52 189 69 172)(53 190 70 173)(54 181 61 174)(55 182 62 175)(56 183 63 176)(57 184 64 177)(58 185 65 178)(59 186 66 179)(60 187 67 180)(71 208 88 191)(72 209 89 192)(73 210 90 193)(74 201 81 194)(75 202 82 195)(76 203 83 196)(77 204 84 197)(78 205 85 198)(79 206 86 199)(80 207 87 200)(91 228 108 211)(92 229 109 212)(93 230 110 213)(94 221 101 214)(95 222 102 215)(96 223 103 216)(97 224 104 217)(98 225 105 218)(99 226 106 219)(100 227 107 220)
G:=sub<Sym(240)| (1,58,38)(2,59,39)(3,60,40)(4,51,31)(5,52,32)(6,53,33)(7,54,34)(8,55,35)(9,56,36)(10,57,37)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,232)(12,231)(13,240)(14,239)(15,238)(16,237)(17,236)(18,235)(19,234)(20,233)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,90)(78,89)(79,88)(80,87)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,130)(118,129)(119,128)(120,127)(131,141)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(151,161)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(171,181)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,210)(198,209)(199,208)(200,207)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,230)(218,229)(219,228)(220,227), (1,73,25,90)(2,72,26,89)(3,71,27,88)(4,80,28,87)(5,79,29,86)(6,78,30,85)(7,77,21,84)(8,76,22,83)(9,75,23,82)(10,74,24,81)(11,190,238,173)(12,189,239,172)(13,188,240,171)(14,187,231,180)(15,186,232,179)(16,185,233,178)(17,184,234,177)(18,183,235,176)(19,182,236,175)(20,181,237,174)(31,100,48,107)(32,99,49,106)(33,98,50,105)(34,97,41,104)(35,96,42,103)(36,95,43,102)(37,94,44,101)(38,93,45,110)(39,92,46,109)(40,91,47,108)(51,120,68,127)(52,119,69,126)(53,118,70,125)(54,117,61,124)(55,116,62,123)(56,115,63,122)(57,114,64,121)(58,113,65,130)(59,112,66,129)(60,111,67,128)(131,207,148,200)(132,206,149,199)(133,205,150,198)(134,204,141,197)(135,203,142,196)(136,202,143,195)(137,201,144,194)(138,210,145,193)(139,209,146,192)(140,208,147,191)(151,227,168,220)(152,226,169,219)(153,225,170,218)(154,224,161,217)(155,223,162,216)(156,222,163,215)(157,221,164,214)(158,230,165,213)(159,229,166,212)(160,228,167,211), (1,145,25,138)(2,146,26,139)(3,147,27,140)(4,148,28,131)(5,149,29,132)(6,150,30,133)(7,141,21,134)(8,142,22,135)(9,143,23,136)(10,144,24,137)(11,125,238,118)(12,126,239,119)(13,127,240,120)(14,128,231,111)(15,129,232,112)(16,130,233,113)(17,121,234,114)(18,122,235,115)(19,123,236,116)(20,124,237,117)(31,168,48,151)(32,169,49,152)(33,170,50,153)(34,161,41,154)(35,162,42,155)(36,163,43,156)(37,164,44,157)(38,165,45,158)(39,166,46,159)(40,167,47,160)(51,188,68,171)(52,189,69,172)(53,190,70,173)(54,181,61,174)(55,182,62,175)(56,183,63,176)(57,184,64,177)(58,185,65,178)(59,186,66,179)(60,187,67,180)(71,208,88,191)(72,209,89,192)(73,210,90,193)(74,201,81,194)(75,202,82,195)(76,203,83,196)(77,204,84,197)(78,205,85,198)(79,206,86,199)(80,207,87,200)(91,228,108,211)(92,229,109,212)(93,230,110,213)(94,221,101,214)(95,222,102,215)(96,223,103,216)(97,224,104,217)(98,225,105,218)(99,226,106,219)(100,227,107,220)>;
G:=Group( (1,58,38)(2,59,39)(3,60,40)(4,51,31)(5,52,32)(6,53,33)(7,54,34)(8,55,35)(9,56,36)(10,57,37)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,232)(12,231)(13,240)(14,239)(15,238)(16,237)(17,236)(18,235)(19,234)(20,233)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,90)(78,89)(79,88)(80,87)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,130)(118,129)(119,128)(120,127)(131,141)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(151,161)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(171,181)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,210)(198,209)(199,208)(200,207)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,230)(218,229)(219,228)(220,227), (1,73,25,90)(2,72,26,89)(3,71,27,88)(4,80,28,87)(5,79,29,86)(6,78,30,85)(7,77,21,84)(8,76,22,83)(9,75,23,82)(10,74,24,81)(11,190,238,173)(12,189,239,172)(13,188,240,171)(14,187,231,180)(15,186,232,179)(16,185,233,178)(17,184,234,177)(18,183,235,176)(19,182,236,175)(20,181,237,174)(31,100,48,107)(32,99,49,106)(33,98,50,105)(34,97,41,104)(35,96,42,103)(36,95,43,102)(37,94,44,101)(38,93,45,110)(39,92,46,109)(40,91,47,108)(51,120,68,127)(52,119,69,126)(53,118,70,125)(54,117,61,124)(55,116,62,123)(56,115,63,122)(57,114,64,121)(58,113,65,130)(59,112,66,129)(60,111,67,128)(131,207,148,200)(132,206,149,199)(133,205,150,198)(134,204,141,197)(135,203,142,196)(136,202,143,195)(137,201,144,194)(138,210,145,193)(139,209,146,192)(140,208,147,191)(151,227,168,220)(152,226,169,219)(153,225,170,218)(154,224,161,217)(155,223,162,216)(156,222,163,215)(157,221,164,214)(158,230,165,213)(159,229,166,212)(160,228,167,211), (1,145,25,138)(2,146,26,139)(3,147,27,140)(4,148,28,131)(5,149,29,132)(6,150,30,133)(7,141,21,134)(8,142,22,135)(9,143,23,136)(10,144,24,137)(11,125,238,118)(12,126,239,119)(13,127,240,120)(14,128,231,111)(15,129,232,112)(16,130,233,113)(17,121,234,114)(18,122,235,115)(19,123,236,116)(20,124,237,117)(31,168,48,151)(32,169,49,152)(33,170,50,153)(34,161,41,154)(35,162,42,155)(36,163,43,156)(37,164,44,157)(38,165,45,158)(39,166,46,159)(40,167,47,160)(51,188,68,171)(52,189,69,172)(53,190,70,173)(54,181,61,174)(55,182,62,175)(56,183,63,176)(57,184,64,177)(58,185,65,178)(59,186,66,179)(60,187,67,180)(71,208,88,191)(72,209,89,192)(73,210,90,193)(74,201,81,194)(75,202,82,195)(76,203,83,196)(77,204,84,197)(78,205,85,198)(79,206,86,199)(80,207,87,200)(91,228,108,211)(92,229,109,212)(93,230,110,213)(94,221,101,214)(95,222,102,215)(96,223,103,216)(97,224,104,217)(98,225,105,218)(99,226,106,219)(100,227,107,220) );
G=PermutationGroup([[(1,58,38),(2,59,39),(3,60,40),(4,51,31),(5,52,32),(6,53,33),(7,54,34),(8,55,35),(9,56,36),(10,57,37),(11,225,205),(12,226,206),(13,227,207),(14,228,208),(15,229,209),(16,230,210),(17,221,201),(18,222,202),(19,223,203),(20,224,204),(21,61,41),(22,62,42),(23,63,43),(24,64,44),(25,65,45),(26,66,46),(27,67,47),(28,68,48),(29,69,49),(30,70,50),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100),(81,121,101),(82,122,102),(83,123,103),(84,124,104),(85,125,105),(86,126,106),(87,127,107),(88,128,108),(89,129,109),(90,130,110),(131,171,151),(132,172,152),(133,173,153),(134,174,154),(135,175,155),(136,176,156),(137,177,157),(138,178,158),(139,179,159),(140,180,160),(141,181,161),(142,182,162),(143,183,163),(144,184,164),(145,185,165),(146,186,166),(147,187,167),(148,188,168),(149,189,169),(150,190,170),(191,231,211),(192,232,212),(193,233,213),(194,234,214),(195,235,215),(196,236,216),(197,237,217),(198,238,218),(199,239,219),(200,240,220)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,232),(12,231),(13,240),(14,239),(15,238),(16,237),(17,236),(18,235),(19,234),(20,233),(31,41),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(51,61),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,90),(78,89),(79,88),(80,87),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,110),(98,109),(99,108),(100,107),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,130),(118,129),(119,128),(120,127),(131,141),(132,150),(133,149),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142),(151,161),(152,170),(153,169),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(171,181),(172,190),(173,189),(174,188),(175,187),(176,186),(177,185),(178,184),(179,183),(180,182),(191,206),(192,205),(193,204),(194,203),(195,202),(196,201),(197,210),(198,209),(199,208),(200,207),(211,226),(212,225),(213,224),(214,223),(215,222),(216,221),(217,230),(218,229),(219,228),(220,227)], [(1,73,25,90),(2,72,26,89),(3,71,27,88),(4,80,28,87),(5,79,29,86),(6,78,30,85),(7,77,21,84),(8,76,22,83),(9,75,23,82),(10,74,24,81),(11,190,238,173),(12,189,239,172),(13,188,240,171),(14,187,231,180),(15,186,232,179),(16,185,233,178),(17,184,234,177),(18,183,235,176),(19,182,236,175),(20,181,237,174),(31,100,48,107),(32,99,49,106),(33,98,50,105),(34,97,41,104),(35,96,42,103),(36,95,43,102),(37,94,44,101),(38,93,45,110),(39,92,46,109),(40,91,47,108),(51,120,68,127),(52,119,69,126),(53,118,70,125),(54,117,61,124),(55,116,62,123),(56,115,63,122),(57,114,64,121),(58,113,65,130),(59,112,66,129),(60,111,67,128),(131,207,148,200),(132,206,149,199),(133,205,150,198),(134,204,141,197),(135,203,142,196),(136,202,143,195),(137,201,144,194),(138,210,145,193),(139,209,146,192),(140,208,147,191),(151,227,168,220),(152,226,169,219),(153,225,170,218),(154,224,161,217),(155,223,162,216),(156,222,163,215),(157,221,164,214),(158,230,165,213),(159,229,166,212),(160,228,167,211)], [(1,145,25,138),(2,146,26,139),(3,147,27,140),(4,148,28,131),(5,149,29,132),(6,150,30,133),(7,141,21,134),(8,142,22,135),(9,143,23,136),(10,144,24,137),(11,125,238,118),(12,126,239,119),(13,127,240,120),(14,128,231,111),(15,129,232,112),(16,130,233,113),(17,121,234,114),(18,122,235,115),(19,123,236,116),(20,124,237,117),(31,168,48,151),(32,169,49,152),(33,170,50,153),(34,161,41,154),(35,162,42,155),(36,163,43,156),(37,164,44,157),(38,165,45,158),(39,166,46,159),(40,167,47,160),(51,188,68,171),(52,189,69,172),(53,190,70,173),(54,181,61,174),(55,182,62,175),(56,183,63,176),(57,184,64,177),(58,185,65,178),(59,186,66,179),(60,187,67,180),(71,208,88,191),(72,209,89,192),(73,210,90,193),(74,201,81,194),(75,202,82,195),(76,203,83,196),(77,204,84,197),(78,205,85,198),(79,206,86,199),(80,207,87,200),(91,228,108,211),(92,229,109,212),(93,230,110,213),(94,221,101,214),(95,222,102,215),(96,223,103,216),(97,224,104,217),(98,225,105,218),(99,226,106,219),(100,227,107,220)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | D5 | C4○D4 | D10 | C3×D4 | C3×Q8 | C3×D5 | D20 | C3×C4○D4 | C6×D5 | C3×D20 | D4⋊2D5 | Q8×D5 | C3×D4⋊2D5 | C3×Q8×D5 |
| kernel | C3×D10⋊2Q8 | C3×C4⋊Dic5 | C3×D10⋊C4 | C15×C4⋊C4 | C6×Dic10 | D5×C2×C12 | D10⋊2Q8 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | C2×Dic10 | C2×C4×D5 | C60 | C6×D5 | C3×C4⋊C4 | C30 | C2×C12 | C20 | D10 | C4⋊C4 | C12 | C10 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 4 | 8 | 4 | 12 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D10⋊2Q8 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 44 | 0 | 0 |
| 17 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 44 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 32 | 7 | 0 | 0 |
| 2 | 29 | 0 | 0 |
| 0 | 0 | 39 | 15 |
| 0 | 0 | 49 | 22 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 17 | 8 |
| 0 | 0 | 40 | 44 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[1,17,0,0,44,17,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,44,60,0,0,0,0,60,0,0,0,0,60],[32,2,0,0,7,29,0,0,0,0,39,49,0,0,15,22],[60,0,0,0,0,60,0,0,0,0,17,40,0,0,8,44] >;
C3×D10⋊2Q8 in GAP, Magma, Sage, TeX
C_3\times D_{10}\rtimes_2Q_8 % in TeX
G:=Group("C3xD10:2Q8"); // GroupNames label
G:=SmallGroup(480,690);
// by ID
G=gap.SmallGroup(480,690);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,394,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations