direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10⋊Q8, (C6×D5)⋊9Q8, D10⋊1(C3×Q8), C6.51(Q8×D5), C10.26(C6×D4), C6.180(D4×D5), C10.13(C6×Q8), (C2×Dic10)⋊4C6, C30.339(C2×D4), C15⋊30(C22⋊Q8), C30.103(C2×Q8), (C6×Dic10)⋊20C2, D10⋊C4.5C6, (C2×C12).235D10, C10.D4⋊12C6, Dic5.15(C3×D4), (C3×Dic5).76D4, C6.120(C4○D20), C30.190(C4○D4), (C2×C60).422C22, (C2×C30).354C23, (C6×Dic5).159C22, (C5×C4⋊C4)⋊7C6, C4⋊C4⋊4(C3×D5), C2.6(C3×Q8×D5), C2.14(C3×D4×D5), (C3×C4⋊C4)⋊13D5, C5⋊2(C3×C22⋊Q8), (C15×C4⋊C4)⋊16C2, (C2×C4×D5).11C6, (C2×C20).6(C2×C6), (D5×C2×C12).30C2, (C2×C4).10(C6×D5), C22.51(D5×C2×C6), C2.15(C3×C4○D20), C10.12(C3×C4○D4), (D5×C2×C6).131C22, (C3×D10⋊C4).7C2, (C3×C10.D4)⋊34C2, (C2×C10).37(C22×C6), (C2×Dic5).33(C2×C6), (C22×D5).26(C2×C6), (C2×C6).350(C22×D5), SmallGroup(480,689)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10⋊Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b3c, ece-1=b8c, ede-1=d-1 >
Subgroups: 480 in 148 conjugacy classes, 66 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C12, C2×C12, C3×Q8, C22×C6, C3×D5, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C22×D5, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×Dic5, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C3×C22⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D10⋊Q8, C3×C10.D4, C3×D10⋊C4, C15×C4⋊C4, C6×Dic10, D5×C2×C12, C3×D10⋊Q8
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, C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5, C4○D20, D4×D5, Q8×D5, C3×C22⋊Q8, D5×C2×C6, D10⋊Q8, C3×C4○D20, C3×D4×D5, C3×Q8×D5, C3×D10⋊Q8
►On 240 points
Generators in S240
(1 70 50)(2 61 41)(3 62 42)(4 63 43)(5 64 44)(6 65 45)(7 66 46)(8 67 47)(9 68 48)(10 69 49)(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 51 31)(22 52 32)(23 53 33)(24 54 34)(25 55 35)(26 56 36)(27 57 37)(28 58 38)(29 59 39)(30 60 40)(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 21)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 231)(12 240)(13 239)(14 238)(15 237)(16 236)(17 235)(18 234)(19 233)(20 232)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(71 85)(72 84)(73 83)(74 82)(75 81)(76 90)(77 89)(78 88)(79 87)(80 86)(91 105)(92 104)(93 103)(94 102)(95 101)(96 110)(97 109)(98 108)(99 107)(100 106)(111 125)(112 124)(113 123)(114 122)(115 121)(116 130)(117 129)(118 128)(119 127)(120 126)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(171 190)(172 189)(173 188)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(191 205)(192 204)(193 203)(194 202)(195 201)(196 210)(197 209)(198 208)(199 207)(200 206)(211 225)(212 224)(213 223)(214 222)(215 221)(216 230)(217 229)(218 228)(219 227)(220 226)
(1 85 27 72)(2 84 28 71)(3 83 29 80)(4 82 30 79)(5 81 21 78)(6 90 22 77)(7 89 23 76)(8 88 24 75)(9 87 25 74)(10 86 26 73)(11 185 232 172)(12 184 233 171)(13 183 234 180)(14 182 235 179)(15 181 236 178)(16 190 237 177)(17 189 238 176)(18 188 239 175)(19 187 240 174)(20 186 231 173)(31 98 44 101)(32 97 45 110)(33 96 46 109)(34 95 47 108)(35 94 48 107)(36 93 49 106)(37 92 50 105)(38 91 41 104)(39 100 42 103)(40 99 43 102)(51 118 64 121)(52 117 65 130)(53 116 66 129)(54 115 67 128)(55 114 68 127)(56 113 69 126)(57 112 70 125)(58 111 61 124)(59 120 62 123)(60 119 63 122)(131 206 144 193)(132 205 145 192)(133 204 146 191)(134 203 147 200)(135 202 148 199)(136 201 149 198)(137 210 150 197)(138 209 141 196)(139 208 142 195)(140 207 143 194)(151 226 164 213)(152 225 165 212)(153 224 166 211)(154 223 167 220)(155 222 168 219)(156 221 169 218)(157 230 170 217)(158 229 161 216)(159 228 162 215)(160 227 163 214)
(1 145 27 132)(2 144 28 131)(3 143 29 140)(4 142 30 139)(5 141 21 138)(6 150 22 137)(7 149 23 136)(8 148 24 135)(9 147 25 134)(10 146 26 133)(11 112 232 125)(12 111 233 124)(13 120 234 123)(14 119 235 122)(15 118 236 121)(16 117 237 130)(17 116 238 129)(18 115 239 128)(19 114 240 127)(20 113 231 126)(31 158 44 161)(32 157 45 170)(33 156 46 169)(34 155 47 168)(35 154 48 167)(36 153 49 166)(37 152 50 165)(38 151 41 164)(39 160 42 163)(40 159 43 162)(51 178 64 181)(52 177 65 190)(53 176 66 189)(54 175 67 188)(55 174 68 187)(56 173 69 186)(57 172 70 185)(58 171 61 184)(59 180 62 183)(60 179 63 182)(71 193 84 206)(72 192 85 205)(73 191 86 204)(74 200 87 203)(75 199 88 202)(76 198 89 201)(77 197 90 210)(78 196 81 209)(79 195 82 208)(80 194 83 207)(91 213 104 226)(92 212 105 225)(93 211 106 224)(94 220 107 223)(95 219 108 222)(96 218 109 221)(97 217 110 230)(98 216 101 229)(99 215 102 228)(100 214 103 227)
G:=sub<Sym(240)| (1,70,50)(2,61,41)(3,62,42)(4,63,43)(5,64,44)(6,65,45)(7,66,46)(8,67,47)(9,68,48)(10,69,49)(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)(21,51,31)(22,52,32)(23,53,33)(24,54,34)(25,55,35)(26,56,36)(27,57,37)(28,58,38)(29,59,39)(30,60,40)(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), (1,21)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,231)(12,240)(13,239)(14,238)(15,237)(16,236)(17,235)(18,234)(19,233)(20,232)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(71,85)(72,84)(73,83)(74,82)(75,81)(76,90)(77,89)(78,88)(79,87)(80,86)(91,105)(92,104)(93,103)(94,102)(95,101)(96,110)(97,109)(98,108)(99,107)(100,106)(111,125)(112,124)(113,123)(114,122)(115,121)(116,130)(117,129)(118,128)(119,127)(120,126)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(191,205)(192,204)(193,203)(194,202)(195,201)(196,210)(197,209)(198,208)(199,207)(200,206)(211,225)(212,224)(213,223)(214,222)(215,221)(216,230)(217,229)(218,228)(219,227)(220,226), (1,85,27,72)(2,84,28,71)(3,83,29,80)(4,82,30,79)(5,81,21,78)(6,90,22,77)(7,89,23,76)(8,88,24,75)(9,87,25,74)(10,86,26,73)(11,185,232,172)(12,184,233,171)(13,183,234,180)(14,182,235,179)(15,181,236,178)(16,190,237,177)(17,189,238,176)(18,188,239,175)(19,187,240,174)(20,186,231,173)(31,98,44,101)(32,97,45,110)(33,96,46,109)(34,95,47,108)(35,94,48,107)(36,93,49,106)(37,92,50,105)(38,91,41,104)(39,100,42,103)(40,99,43,102)(51,118,64,121)(52,117,65,130)(53,116,66,129)(54,115,67,128)(55,114,68,127)(56,113,69,126)(57,112,70,125)(58,111,61,124)(59,120,62,123)(60,119,63,122)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,200)(135,202,148,199)(136,201,149,198)(137,210,150,197)(138,209,141,196)(139,208,142,195)(140,207,143,194)(151,226,164,213)(152,225,165,212)(153,224,166,211)(154,223,167,220)(155,222,168,219)(156,221,169,218)(157,230,170,217)(158,229,161,216)(159,228,162,215)(160,227,163,214), (1,145,27,132)(2,144,28,131)(3,143,29,140)(4,142,30,139)(5,141,21,138)(6,150,22,137)(7,149,23,136)(8,148,24,135)(9,147,25,134)(10,146,26,133)(11,112,232,125)(12,111,233,124)(13,120,234,123)(14,119,235,122)(15,118,236,121)(16,117,237,130)(17,116,238,129)(18,115,239,128)(19,114,240,127)(20,113,231,126)(31,158,44,161)(32,157,45,170)(33,156,46,169)(34,155,47,168)(35,154,48,167)(36,153,49,166)(37,152,50,165)(38,151,41,164)(39,160,42,163)(40,159,43,162)(51,178,64,181)(52,177,65,190)(53,176,66,189)(54,175,67,188)(55,174,68,187)(56,173,69,186)(57,172,70,185)(58,171,61,184)(59,180,62,183)(60,179,63,182)(71,193,84,206)(72,192,85,205)(73,191,86,204)(74,200,87,203)(75,199,88,202)(76,198,89,201)(77,197,90,210)(78,196,81,209)(79,195,82,208)(80,194,83,207)(91,213,104,226)(92,212,105,225)(93,211,106,224)(94,220,107,223)(95,219,108,222)(96,218,109,221)(97,217,110,230)(98,216,101,229)(99,215,102,228)(100,214,103,227)>;
G:=Group( (1,70,50)(2,61,41)(3,62,42)(4,63,43)(5,64,44)(6,65,45)(7,66,46)(8,67,47)(9,68,48)(10,69,49)(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)(21,51,31)(22,52,32)(23,53,33)(24,54,34)(25,55,35)(26,56,36)(27,57,37)(28,58,38)(29,59,39)(30,60,40)(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), (1,21)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,231)(12,240)(13,239)(14,238)(15,237)(16,236)(17,235)(18,234)(19,233)(20,232)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(71,85)(72,84)(73,83)(74,82)(75,81)(76,90)(77,89)(78,88)(79,87)(80,86)(91,105)(92,104)(93,103)(94,102)(95,101)(96,110)(97,109)(98,108)(99,107)(100,106)(111,125)(112,124)(113,123)(114,122)(115,121)(116,130)(117,129)(118,128)(119,127)(120,126)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(191,205)(192,204)(193,203)(194,202)(195,201)(196,210)(197,209)(198,208)(199,207)(200,206)(211,225)(212,224)(213,223)(214,222)(215,221)(216,230)(217,229)(218,228)(219,227)(220,226), (1,85,27,72)(2,84,28,71)(3,83,29,80)(4,82,30,79)(5,81,21,78)(6,90,22,77)(7,89,23,76)(8,88,24,75)(9,87,25,74)(10,86,26,73)(11,185,232,172)(12,184,233,171)(13,183,234,180)(14,182,235,179)(15,181,236,178)(16,190,237,177)(17,189,238,176)(18,188,239,175)(19,187,240,174)(20,186,231,173)(31,98,44,101)(32,97,45,110)(33,96,46,109)(34,95,47,108)(35,94,48,107)(36,93,49,106)(37,92,50,105)(38,91,41,104)(39,100,42,103)(40,99,43,102)(51,118,64,121)(52,117,65,130)(53,116,66,129)(54,115,67,128)(55,114,68,127)(56,113,69,126)(57,112,70,125)(58,111,61,124)(59,120,62,123)(60,119,63,122)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,200)(135,202,148,199)(136,201,149,198)(137,210,150,197)(138,209,141,196)(139,208,142,195)(140,207,143,194)(151,226,164,213)(152,225,165,212)(153,224,166,211)(154,223,167,220)(155,222,168,219)(156,221,169,218)(157,230,170,217)(158,229,161,216)(159,228,162,215)(160,227,163,214), (1,145,27,132)(2,144,28,131)(3,143,29,140)(4,142,30,139)(5,141,21,138)(6,150,22,137)(7,149,23,136)(8,148,24,135)(9,147,25,134)(10,146,26,133)(11,112,232,125)(12,111,233,124)(13,120,234,123)(14,119,235,122)(15,118,236,121)(16,117,237,130)(17,116,238,129)(18,115,239,128)(19,114,240,127)(20,113,231,126)(31,158,44,161)(32,157,45,170)(33,156,46,169)(34,155,47,168)(35,154,48,167)(36,153,49,166)(37,152,50,165)(38,151,41,164)(39,160,42,163)(40,159,43,162)(51,178,64,181)(52,177,65,190)(53,176,66,189)(54,175,67,188)(55,174,68,187)(56,173,69,186)(57,172,70,185)(58,171,61,184)(59,180,62,183)(60,179,63,182)(71,193,84,206)(72,192,85,205)(73,191,86,204)(74,200,87,203)(75,199,88,202)(76,198,89,201)(77,197,90,210)(78,196,81,209)(79,195,82,208)(80,194,83,207)(91,213,104,226)(92,212,105,225)(93,211,106,224)(94,220,107,223)(95,219,108,222)(96,218,109,221)(97,217,110,230)(98,216,101,229)(99,215,102,228)(100,214,103,227) );
G=PermutationGroup([[(1,70,50),(2,61,41),(3,62,42),(4,63,43),(5,64,44),(6,65,45),(7,66,46),(8,67,47),(9,68,48),(10,69,49),(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),(21,51,31),(22,52,32),(23,53,33),(24,54,34),(25,55,35),(26,56,36),(27,57,37),(28,58,38),(29,59,39),(30,60,40),(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)], [(1,21),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,231),(12,240),(13,239),(14,238),(15,237),(16,236),(17,235),(18,234),(19,233),(20,232),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(71,85),(72,84),(73,83),(74,82),(75,81),(76,90),(77,89),(78,88),(79,87),(80,86),(91,105),(92,104),(93,103),(94,102),(95,101),(96,110),(97,109),(98,108),(99,107),(100,106),(111,125),(112,124),(113,123),(114,122),(115,121),(116,130),(117,129),(118,128),(119,127),(120,126),(131,150),(132,149),(133,148),(134,147),(135,146),(136,145),(137,144),(138,143),(139,142),(140,141),(151,170),(152,169),(153,168),(154,167),(155,166),(156,165),(157,164),(158,163),(159,162),(160,161),(171,190),(172,189),(173,188),(174,187),(175,186),(176,185),(177,184),(178,183),(179,182),(180,181),(191,205),(192,204),(193,203),(194,202),(195,201),(196,210),(197,209),(198,208),(199,207),(200,206),(211,225),(212,224),(213,223),(214,222),(215,221),(216,230),(217,229),(218,228),(219,227),(220,226)], [(1,85,27,72),(2,84,28,71),(3,83,29,80),(4,82,30,79),(5,81,21,78),(6,90,22,77),(7,89,23,76),(8,88,24,75),(9,87,25,74),(10,86,26,73),(11,185,232,172),(12,184,233,171),(13,183,234,180),(14,182,235,179),(15,181,236,178),(16,190,237,177),(17,189,238,176),(18,188,239,175),(19,187,240,174),(20,186,231,173),(31,98,44,101),(32,97,45,110),(33,96,46,109),(34,95,47,108),(35,94,48,107),(36,93,49,106),(37,92,50,105),(38,91,41,104),(39,100,42,103),(40,99,43,102),(51,118,64,121),(52,117,65,130),(53,116,66,129),(54,115,67,128),(55,114,68,127),(56,113,69,126),(57,112,70,125),(58,111,61,124),(59,120,62,123),(60,119,63,122),(131,206,144,193),(132,205,145,192),(133,204,146,191),(134,203,147,200),(135,202,148,199),(136,201,149,198),(137,210,150,197),(138,209,141,196),(139,208,142,195),(140,207,143,194),(151,226,164,213),(152,225,165,212),(153,224,166,211),(154,223,167,220),(155,222,168,219),(156,221,169,218),(157,230,170,217),(158,229,161,216),(159,228,162,215),(160,227,163,214)], [(1,145,27,132),(2,144,28,131),(3,143,29,140),(4,142,30,139),(5,141,21,138),(6,150,22,137),(7,149,23,136),(8,148,24,135),(9,147,25,134),(10,146,26,133),(11,112,232,125),(12,111,233,124),(13,120,234,123),(14,119,235,122),(15,118,236,121),(16,117,237,130),(17,116,238,129),(18,115,239,128),(19,114,240,127),(20,113,231,126),(31,158,44,161),(32,157,45,170),(33,156,46,169),(34,155,47,168),(35,154,48,167),(36,153,49,166),(37,152,50,165),(38,151,41,164),(39,160,42,163),(40,159,43,162),(51,178,64,181),(52,177,65,190),(53,176,66,189),(54,175,67,188),(55,174,68,187),(56,173,69,186),(57,172,70,185),(58,171,61,184),(59,180,62,183),(60,179,63,182),(71,193,84,206),(72,192,85,205),(73,191,86,204),(74,200,87,203),(75,199,88,202),(76,198,89,201),(77,197,90,210),(78,196,81,209),(79,195,82,208),(80,194,83,207),(91,213,104,226),(92,212,105,225),(93,211,106,224),(94,220,107,223),(95,219,108,222),(96,218,109,221),(97,217,110,230),(98,216,101,229),(99,215,102,228),(100,214,103,227)]])
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 | C3×C4○D4 | C6×D5 | C4○D20 | C3×C4○D20 | D4×D5 | Q8×D5 | C3×D4×D5 | C3×Q8×D5 |
| kernel | C3×D10⋊Q8 | C3×C10.D4 | C3×D10⋊C4 | C15×C4⋊C4 | C6×Dic10 | D5×C2×C12 | D10⋊Q8 | C10.D4 | D10⋊C4 | C5×C4⋊C4 | C2×Dic10 | C2×C4×D5 | C3×Dic5 | C6×D5 | C3×C4⋊C4 | C30 | C2×C12 | Dic5 | D10 | C4⋊C4 | C10 | C2×C4 | C6 | C2 | 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 | 4 | 12 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D10⋊Q8 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 47 |
| 60 | 17 | 0 | 0 | 0 | 0 |
| 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 41 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 0 | 58 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 7 | 50 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47],[60,44,0,0,0,0,17,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,17,1,0,0,0,0,0,0,1,6,0,0,0,0,0,60,0,0,0,0,0,0,60,16,0,0,0,0,0,1],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,41,1,0,0,0,0,0,0,40,58,0,0,0,0,5,21],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,7,0,0,0,0,0,50] >;
C3×D10⋊Q8 in GAP, Magma, Sage, TeX
C_3\times D_{10}\rtimes Q_8 % in TeX
G:=Group("C3xD10:Q8"); // GroupNames label
G:=SmallGroup(480,689);
// by ID
G=gap.SmallGroup(480,689);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,176,1598,555,268,18822]);
// 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=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^3*c,e*c*e^-1=b^8*c,e*d*e^-1=d^-1>;
// generators/relations