direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10⋊3Q8, C60.132D4, (Q8×C10)⋊7C6, (C6×Q8)⋊10D5, D10⋊3(C3×Q8), (C6×D5)⋊11Q8, C6.55(Q8×D5), C4⋊Dic5⋊15C6, (Q8×C30)⋊10C2, C10.57(C6×D4), C20.22(C3×D4), C10.17(C6×Q8), C30.414(C2×D4), C15⋊38(C22⋊Q8), C30.107(C2×Q8), D10⋊C4.6C6, (C2×C12).244D10, C10.D4⋊16C6, C12.97(C5⋊D4), C30.266(C4○D4), (C2×C60).426C22, (C2×C30).375C23, C6.54(Q8⋊2D5), (C6×Dic5).166C22, C2.9(C3×Q8×D5), (C2×C4×D5).4C6, C5⋊5(C3×C22⋊Q8), (C2×Q8)⋊5(C3×D5), (D5×C2×C12).15C2, (C2×C4).56(C6×D5), C2.21(C6×C5⋊D4), C4.18(C3×C5⋊D4), C22.64(D5×C2×C6), (C2×C20).37(C2×C6), (C3×C4⋊Dic5)⋊33C2, C10.36(C3×C4○D4), C6.142(C2×C5⋊D4), C2.8(C3×Q8⋊2D5), (D5×C2×C6).137C22, (C3×C10.D4)⋊38C2, (C2×C10).58(C22×C6), (C2×Dic5).18(C2×C6), (C22×D5).32(C2×C6), (C2×C6).371(C22×D5), (C3×D10⋊C4).16C2, SmallGroup(480,739)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10⋊3Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >
Subgroups: 448 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, 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, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C2×C4×D5, Q8×C10, C3×C22⋊Q8, D5×C12, C6×Dic5, C6×Dic5, C2×C60, C2×C60, Q8×C15, D5×C2×C6, D10⋊3Q8, C3×C10.D4, C3×C4⋊Dic5, C3×D10⋊C4, D5×C2×C12, Q8×C30, C3×D10⋊3Q8
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, C5⋊D4, C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C3×C22⋊Q8, C3×C5⋊D4, D5×C2×C6, D10⋊3Q8, C3×Q8×D5, C3×Q8⋊2D5, C6×C5⋊D4, C3×D10⋊3Q8
►On 240 points
Generators in S240
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(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)(11 238)(12 237)(13 236)(14 235)(15 234)(16 233)(17 232)(18 231)(19 240)(20 239)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 70)(59 69)(60 68)(71 82)(72 81)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(91 102)(92 101)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(111 122)(112 121)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 150)(139 149)(140 148)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 170)(159 169)(160 168)(171 187)(172 186)(173 185)(174 184)(175 183)(176 182)(177 181)(178 190)(179 189)(180 188)(191 202)(192 201)(193 210)(194 209)(195 208)(196 207)(197 206)(198 205)(199 204)(200 203)(211 222)(212 221)(213 230)(214 229)(215 228)(216 227)(217 226)(218 225)(219 224)(220 223)
(1 85 25 74)(2 86 26 75)(3 87 27 76)(4 88 28 77)(5 89 29 78)(6 90 30 79)(7 81 21 80)(8 82 22 71)(9 83 23 72)(10 84 24 73)(11 174 234 185)(12 175 235 186)(13 176 236 187)(14 177 237 188)(15 178 238 189)(16 179 239 190)(17 180 240 181)(18 171 231 182)(19 172 232 183)(20 173 233 184)(31 102 42 91)(32 103 43 92)(33 104 44 93)(34 105 45 94)(35 106 46 95)(36 107 47 96)(37 108 48 97)(38 109 49 98)(39 110 50 99)(40 101 41 100)(51 122 62 111)(52 123 63 112)(53 124 64 113)(54 125 65 114)(55 126 66 115)(56 127 67 116)(57 128 68 117)(58 129 69 118)(59 130 70 119)(60 121 61 120)(131 191 142 202)(132 192 143 203)(133 193 144 204)(134 194 145 205)(135 195 146 206)(136 196 147 207)(137 197 148 208)(138 198 149 209)(139 199 150 210)(140 200 141 201)(151 211 162 222)(152 212 163 223)(153 213 164 224)(154 214 165 225)(155 215 166 226)(156 216 167 227)(157 217 168 228)(158 218 169 229)(159 219 170 230)(160 220 161 221)
(1 145 25 134)(2 146 26 135)(3 147 27 136)(4 148 28 137)(5 149 29 138)(6 150 30 139)(7 141 21 140)(8 142 22 131)(9 143 23 132)(10 144 24 133)(11 125 234 114)(12 126 235 115)(13 127 236 116)(14 128 237 117)(15 129 238 118)(16 130 239 119)(17 121 240 120)(18 122 231 111)(19 123 232 112)(20 124 233 113)(31 162 42 151)(32 163 43 152)(33 164 44 153)(34 165 45 154)(35 166 46 155)(36 167 47 156)(37 168 48 157)(38 169 49 158)(39 170 50 159)(40 161 41 160)(51 182 62 171)(52 183 63 172)(53 184 64 173)(54 185 65 174)(55 186 66 175)(56 187 67 176)(57 188 68 177)(58 189 69 178)(59 190 70 179)(60 181 61 180)(71 202 82 191)(72 203 83 192)(73 204 84 193)(74 205 85 194)(75 206 86 195)(76 207 87 196)(77 208 88 197)(78 209 89 198)(79 210 90 199)(80 201 81 200)(91 222 102 211)(92 223 103 212)(93 224 104 213)(94 225 105 214)(95 226 106 215)(96 227 107 216)(97 228 108 217)(98 229 109 218)(99 230 110 219)(100 221 101 220)
G:=sub<Sym(240)| (1,54,34)(2,55,35)(3,56,36)(4,57,37)(5,58,38)(6,59,39)(7,60,40)(8,51,31)(9,52,32)(10,53,33)(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)(11,238)(12,237)(13,236)(14,235)(15,234)(16,233)(17,232)(18,231)(19,240)(20,239)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,70)(59,69)(60,68)(71,82)(72,81)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(91,102)(92,101)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,150)(139,149)(140,148)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,170)(159,169)(160,168)(171,187)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,190)(179,189)(180,188)(191,202)(192,201)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,203)(211,222)(212,221)(213,230)(214,229)(215,228)(216,227)(217,226)(218,225)(219,224)(220,223), (1,85,25,74)(2,86,26,75)(3,87,27,76)(4,88,28,77)(5,89,29,78)(6,90,30,79)(7,81,21,80)(8,82,22,71)(9,83,23,72)(10,84,24,73)(11,174,234,185)(12,175,235,186)(13,176,236,187)(14,177,237,188)(15,178,238,189)(16,179,239,190)(17,180,240,181)(18,171,231,182)(19,172,232,183)(20,173,233,184)(31,102,42,91)(32,103,43,92)(33,104,44,93)(34,105,45,94)(35,106,46,95)(36,107,47,96)(37,108,48,97)(38,109,49,98)(39,110,50,99)(40,101,41,100)(51,122,62,111)(52,123,63,112)(53,124,64,113)(54,125,65,114)(55,126,66,115)(56,127,67,116)(57,128,68,117)(58,129,69,118)(59,130,70,119)(60,121,61,120)(131,191,142,202)(132,192,143,203)(133,193,144,204)(134,194,145,205)(135,195,146,206)(136,196,147,207)(137,197,148,208)(138,198,149,209)(139,199,150,210)(140,200,141,201)(151,211,162,222)(152,212,163,223)(153,213,164,224)(154,214,165,225)(155,215,166,226)(156,216,167,227)(157,217,168,228)(158,218,169,229)(159,219,170,230)(160,220,161,221), (1,145,25,134)(2,146,26,135)(3,147,27,136)(4,148,28,137)(5,149,29,138)(6,150,30,139)(7,141,21,140)(8,142,22,131)(9,143,23,132)(10,144,24,133)(11,125,234,114)(12,126,235,115)(13,127,236,116)(14,128,237,117)(15,129,238,118)(16,130,239,119)(17,121,240,120)(18,122,231,111)(19,123,232,112)(20,124,233,113)(31,162,42,151)(32,163,43,152)(33,164,44,153)(34,165,45,154)(35,166,46,155)(36,167,47,156)(37,168,48,157)(38,169,49,158)(39,170,50,159)(40,161,41,160)(51,182,62,171)(52,183,63,172)(53,184,64,173)(54,185,65,174)(55,186,66,175)(56,187,67,176)(57,188,68,177)(58,189,69,178)(59,190,70,179)(60,181,61,180)(71,202,82,191)(72,203,83,192)(73,204,84,193)(74,205,85,194)(75,206,86,195)(76,207,87,196)(77,208,88,197)(78,209,89,198)(79,210,90,199)(80,201,81,200)(91,222,102,211)(92,223,103,212)(93,224,104,213)(94,225,105,214)(95,226,106,215)(96,227,107,216)(97,228,108,217)(98,229,109,218)(99,230,110,219)(100,221,101,220)>;
G:=Group( (1,54,34)(2,55,35)(3,56,36)(4,57,37)(5,58,38)(6,59,39)(7,60,40)(8,51,31)(9,52,32)(10,53,33)(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)(11,238)(12,237)(13,236)(14,235)(15,234)(16,233)(17,232)(18,231)(19,240)(20,239)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,70)(59,69)(60,68)(71,82)(72,81)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(91,102)(92,101)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,150)(139,149)(140,148)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,170)(159,169)(160,168)(171,187)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,190)(179,189)(180,188)(191,202)(192,201)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,203)(211,222)(212,221)(213,230)(214,229)(215,228)(216,227)(217,226)(218,225)(219,224)(220,223), (1,85,25,74)(2,86,26,75)(3,87,27,76)(4,88,28,77)(5,89,29,78)(6,90,30,79)(7,81,21,80)(8,82,22,71)(9,83,23,72)(10,84,24,73)(11,174,234,185)(12,175,235,186)(13,176,236,187)(14,177,237,188)(15,178,238,189)(16,179,239,190)(17,180,240,181)(18,171,231,182)(19,172,232,183)(20,173,233,184)(31,102,42,91)(32,103,43,92)(33,104,44,93)(34,105,45,94)(35,106,46,95)(36,107,47,96)(37,108,48,97)(38,109,49,98)(39,110,50,99)(40,101,41,100)(51,122,62,111)(52,123,63,112)(53,124,64,113)(54,125,65,114)(55,126,66,115)(56,127,67,116)(57,128,68,117)(58,129,69,118)(59,130,70,119)(60,121,61,120)(131,191,142,202)(132,192,143,203)(133,193,144,204)(134,194,145,205)(135,195,146,206)(136,196,147,207)(137,197,148,208)(138,198,149,209)(139,199,150,210)(140,200,141,201)(151,211,162,222)(152,212,163,223)(153,213,164,224)(154,214,165,225)(155,215,166,226)(156,216,167,227)(157,217,168,228)(158,218,169,229)(159,219,170,230)(160,220,161,221), (1,145,25,134)(2,146,26,135)(3,147,27,136)(4,148,28,137)(5,149,29,138)(6,150,30,139)(7,141,21,140)(8,142,22,131)(9,143,23,132)(10,144,24,133)(11,125,234,114)(12,126,235,115)(13,127,236,116)(14,128,237,117)(15,129,238,118)(16,130,239,119)(17,121,240,120)(18,122,231,111)(19,123,232,112)(20,124,233,113)(31,162,42,151)(32,163,43,152)(33,164,44,153)(34,165,45,154)(35,166,46,155)(36,167,47,156)(37,168,48,157)(38,169,49,158)(39,170,50,159)(40,161,41,160)(51,182,62,171)(52,183,63,172)(53,184,64,173)(54,185,65,174)(55,186,66,175)(56,187,67,176)(57,188,68,177)(58,189,69,178)(59,190,70,179)(60,181,61,180)(71,202,82,191)(72,203,83,192)(73,204,84,193)(74,205,85,194)(75,206,86,195)(76,207,87,196)(77,208,88,197)(78,209,89,198)(79,210,90,199)(80,201,81,200)(91,222,102,211)(92,223,103,212)(93,224,104,213)(94,225,105,214)(95,226,106,215)(96,227,107,216)(97,228,108,217)(98,229,109,218)(99,230,110,219)(100,221,101,220) );
G=PermutationGroup([[(1,54,34),(2,55,35),(3,56,36),(4,57,37),(5,58,38),(6,59,39),(7,60,40),(8,51,31),(9,52,32),(10,53,33),(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),(11,238),(12,237),(13,236),(14,235),(15,234),(16,233),(17,232),(18,231),(19,240),(20,239),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,50),(39,49),(40,48),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(57,61),(58,70),(59,69),(60,68),(71,82),(72,81),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(91,102),(92,101),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(111,122),(112,121),(113,130),(114,129),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,150),(139,149),(140,148),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162),(157,161),(158,170),(159,169),(160,168),(171,187),(172,186),(173,185),(174,184),(175,183),(176,182),(177,181),(178,190),(179,189),(180,188),(191,202),(192,201),(193,210),(194,209),(195,208),(196,207),(197,206),(198,205),(199,204),(200,203),(211,222),(212,221),(213,230),(214,229),(215,228),(216,227),(217,226),(218,225),(219,224),(220,223)], [(1,85,25,74),(2,86,26,75),(3,87,27,76),(4,88,28,77),(5,89,29,78),(6,90,30,79),(7,81,21,80),(8,82,22,71),(9,83,23,72),(10,84,24,73),(11,174,234,185),(12,175,235,186),(13,176,236,187),(14,177,237,188),(15,178,238,189),(16,179,239,190),(17,180,240,181),(18,171,231,182),(19,172,232,183),(20,173,233,184),(31,102,42,91),(32,103,43,92),(33,104,44,93),(34,105,45,94),(35,106,46,95),(36,107,47,96),(37,108,48,97),(38,109,49,98),(39,110,50,99),(40,101,41,100),(51,122,62,111),(52,123,63,112),(53,124,64,113),(54,125,65,114),(55,126,66,115),(56,127,67,116),(57,128,68,117),(58,129,69,118),(59,130,70,119),(60,121,61,120),(131,191,142,202),(132,192,143,203),(133,193,144,204),(134,194,145,205),(135,195,146,206),(136,196,147,207),(137,197,148,208),(138,198,149,209),(139,199,150,210),(140,200,141,201),(151,211,162,222),(152,212,163,223),(153,213,164,224),(154,214,165,225),(155,215,166,226),(156,216,167,227),(157,217,168,228),(158,218,169,229),(159,219,170,230),(160,220,161,221)], [(1,145,25,134),(2,146,26,135),(3,147,27,136),(4,148,28,137),(5,149,29,138),(6,150,30,139),(7,141,21,140),(8,142,22,131),(9,143,23,132),(10,144,24,133),(11,125,234,114),(12,126,235,115),(13,127,236,116),(14,128,237,117),(15,129,238,118),(16,130,239,119),(17,121,240,120),(18,122,231,111),(19,123,232,112),(20,124,233,113),(31,162,42,151),(32,163,43,152),(33,164,44,153),(34,165,45,154),(35,166,46,155),(36,167,47,156),(37,168,48,157),(38,169,49,158),(39,170,50,159),(40,161,41,160),(51,182,62,171),(52,183,63,172),(53,184,64,173),(54,185,65,174),(55,186,66,175),(56,187,67,176),(57,188,68,177),(58,189,69,178),(59,190,70,179),(60,181,61,180),(71,202,82,191),(72,203,83,192),(73,204,84,193),(74,205,85,194),(75,206,86,195),(76,207,87,196),(77,208,88,197),(78,209,89,198),(79,210,90,199),(80,201,81,200),(91,222,102,211),(92,223,103,212),(93,224,104,213),(94,225,105,214),(95,226,106,215),(96,227,107,216),(97,228,108,217),(98,229,109,218),(99,230,110,219),(100,221,101,220)]])
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 | C5⋊D4 | C3×C4○D4 | C6×D5 | C3×C5⋊D4 | Q8×D5 | Q8⋊2D5 | C3×Q8×D5 | C3×Q8⋊2D5 |
| kernel | C3×D10⋊3Q8 | C3×C10.D4 | C3×C4⋊Dic5 | C3×D10⋊C4 | D5×C2×C12 | Q8×C30 | D10⋊3Q8 | C10.D4 | C4⋊Dic5 | D10⋊C4 | C2×C4×D5 | Q8×C10 | C60 | C6×D5 | C6×Q8 | C30 | C2×C12 | C20 | D10 | C2×Q8 | C12 | C10 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 4 | 8 | 4 | 12 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D10⋊3Q8 ►in GL6(𝔽61)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 59 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 46 |
| 0 | 0 | 0 | 0 | 53 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 38 |
| 0 | 0 | 0 | 0 | 46 | 52 |
G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,18,0,0,0,0,43,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,53,0,0,0,0,46,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,46,0,0,0,0,38,52] >;
C3×D10⋊3Q8 in GAP, Magma, Sage, TeX
C_3\times D_{10}\rtimes_3Q_8 % in TeX
G:=Group("C3xD10:3Q8"); // GroupNames label
G:=SmallGroup(480,739);
// by ID
G=gap.SmallGroup(480,739);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,268,18822]);
// 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=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations