Copied to
clipboard

## G = C3×D4.10D10order 480 = 25·3·5

### Direct product of C3 and D4.10D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4.10D10
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12 — C3×Q8×D5 — C3×D4.10D10
 Lower central C5 — C10 — C3×D4.10D10
 Upper central C1 — C6 — C3×C4○D4

Generators and relations for C3×D4.10D10
G = < a,b,c,d,e | a3=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b2c, ce=ec, ede-1=d9 >

Subgroups: 752 in 292 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C12, C12, C12, C2×C6, C2×C6, C15, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C2×C30, C2×Dic10, C4○D20, D42D5, Q8×D5, C5×C4○D4, C3×2- 1+4, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, Q8×C15, D4.10D10, C6×Dic10, C3×C4○D20, C3×D42D5, C3×Q8×D5, C15×C4○D4, C3×D4.10D10
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, 2- 1+4, C22×D5, C23×C6, C6×D5, C23×D5, C3×2- 1+4, D5×C2×C6, D4.10D10, D5×C22×C6, C3×D4.10D10

Smallest permutation representation of C3×D4.10D10
On 240 points
Generators in S240
(1 184 227)(2 185 228)(3 186 229)(4 187 230)(5 188 231)(6 189 232)(7 190 233)(8 191 234)(9 192 235)(10 193 236)(11 194 237)(12 195 238)(13 196 239)(14 197 240)(15 198 221)(16 199 222)(17 200 223)(18 181 224)(19 182 225)(20 183 226)(21 73 211)(22 74 212)(23 75 213)(24 76 214)(25 77 215)(26 78 216)(27 79 217)(28 80 218)(29 61 219)(30 62 220)(31 63 201)(32 64 202)(33 65 203)(34 66 204)(35 67 205)(36 68 206)(37 69 207)(38 70 208)(39 71 209)(40 72 210)(41 130 89)(42 131 90)(43 132 91)(44 133 92)(45 134 93)(46 135 94)(47 136 95)(48 137 96)(49 138 97)(50 139 98)(51 140 99)(52 121 100)(53 122 81)(54 123 82)(55 124 83)(56 125 84)(57 126 85)(58 127 86)(59 128 87)(60 129 88)(101 156 173)(102 157 174)(103 158 175)(104 159 176)(105 160 177)(106 141 178)(107 142 179)(108 143 180)(109 144 161)(110 145 162)(111 146 163)(112 147 164)(113 148 165)(114 149 166)(115 150 167)(116 151 168)(117 152 169)(118 153 170)(119 154 171)(120 155 172)
(1 205 11 215)(2 216 12 206)(3 207 13 217)(4 218 14 208)(5 209 15 219)(6 220 16 210)(7 211 17 201)(8 202 18 212)(9 213 19 203)(10 204 20 214)(21 200 31 190)(22 191 32 181)(23 182 33 192)(24 193 34 183)(25 184 35 194)(26 195 36 185)(27 186 37 196)(28 197 38 187)(29 188 39 198)(30 199 40 189)(41 150 51 160)(42 141 52 151)(43 152 53 142)(44 143 54 153)(45 154 55 144)(46 145 56 155)(47 156 57 146)(48 147 58 157)(49 158 59 148)(50 149 60 159)(61 231 71 221)(62 222 72 232)(63 233 73 223)(64 224 74 234)(65 235 75 225)(66 226 76 236)(67 237 77 227)(68 228 78 238)(69 239 79 229)(70 230 80 240)(81 107 91 117)(82 118 92 108)(83 109 93 119)(84 120 94 110)(85 111 95 101)(86 102 96 112)(87 113 97 103)(88 104 98 114)(89 115 99 105)(90 106 100 116)(121 168 131 178)(122 179 132 169)(123 170 133 180)(124 161 134 171)(125 172 135 162)(126 163 136 173)(127 174 137 164)(128 165 138 175)(129 176 139 166)(130 167 140 177)
(1 120)(2 111)(3 102)(4 113)(5 104)(6 115)(7 106)(8 117)(9 108)(10 119)(11 110)(12 101)(13 112)(14 103)(15 114)(16 105)(17 116)(18 107)(19 118)(20 109)(21 42)(22 53)(23 44)(24 55)(25 46)(26 57)(27 48)(28 59)(29 50)(30 41)(31 52)(32 43)(33 54)(34 45)(35 56)(36 47)(37 58)(38 49)(39 60)(40 51)(61 139)(62 130)(63 121)(64 132)(65 123)(66 134)(67 125)(68 136)(69 127)(70 138)(71 129)(72 140)(73 131)(74 122)(75 133)(76 124)(77 135)(78 126)(79 137)(80 128)(81 212)(82 203)(83 214)(84 205)(85 216)(86 207)(87 218)(88 209)(89 220)(90 211)(91 202)(92 213)(93 204)(94 215)(95 206)(96 217)(97 208)(98 219)(99 210)(100 201)(141 190)(142 181)(143 192)(144 183)(145 194)(146 185)(147 196)(148 187)(149 198)(150 189)(151 200)(152 191)(153 182)(154 193)(155 184)(156 195)(157 186)(158 197)(159 188)(160 199)(161 226)(162 237)(163 228)(164 239)(165 230)(166 221)(167 232)(168 223)(169 234)(170 225)(171 236)(172 227)(173 238)(174 229)(175 240)(176 231)(177 222)(178 233)(179 224)(180 235)
(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 88 11 98)(2 97 12 87)(3 86 13 96)(4 95 14 85)(5 84 15 94)(6 93 16 83)(7 82 17 92)(8 91 18 81)(9 100 19 90)(10 89 20 99)(21 143 31 153)(22 152 32 142)(23 141 33 151)(24 150 34 160)(25 159 35 149)(26 148 36 158)(27 157 37 147)(28 146 38 156)(29 155 39 145)(30 144 40 154)(41 183 51 193)(42 192 52 182)(43 181 53 191)(44 190 54 200)(45 199 55 189)(46 188 56 198)(47 197 57 187)(48 186 58 196)(49 195 59 185)(50 184 60 194)(61 172 71 162)(62 161 72 171)(63 170 73 180)(64 179 74 169)(65 168 75 178)(66 177 76 167)(67 166 77 176)(68 175 78 165)(69 164 79 174)(70 173 80 163)(101 218 111 208)(102 207 112 217)(103 216 113 206)(104 205 114 215)(105 214 115 204)(106 203 116 213)(107 212 117 202)(108 201 118 211)(109 210 119 220)(110 219 120 209)(121 225 131 235)(122 234 132 224)(123 223 133 233)(124 232 134 222)(125 221 135 231)(126 230 136 240)(127 239 137 229)(128 228 138 238)(129 237 139 227)(130 226 140 236)

G:=sub<Sym(240)| (1,184,227)(2,185,228)(3,186,229)(4,187,230)(5,188,231)(6,189,232)(7,190,233)(8,191,234)(9,192,235)(10,193,236)(11,194,237)(12,195,238)(13,196,239)(14,197,240)(15,198,221)(16,199,222)(17,200,223)(18,181,224)(19,182,225)(20,183,226)(21,73,211)(22,74,212)(23,75,213)(24,76,214)(25,77,215)(26,78,216)(27,79,217)(28,80,218)(29,61,219)(30,62,220)(31,63,201)(32,64,202)(33,65,203)(34,66,204)(35,67,205)(36,68,206)(37,69,207)(38,70,208)(39,71,209)(40,72,210)(41,130,89)(42,131,90)(43,132,91)(44,133,92)(45,134,93)(46,135,94)(47,136,95)(48,137,96)(49,138,97)(50,139,98)(51,140,99)(52,121,100)(53,122,81)(54,123,82)(55,124,83)(56,125,84)(57,126,85)(58,127,86)(59,128,87)(60,129,88)(101,156,173)(102,157,174)(103,158,175)(104,159,176)(105,160,177)(106,141,178)(107,142,179)(108,143,180)(109,144,161)(110,145,162)(111,146,163)(112,147,164)(113,148,165)(114,149,166)(115,150,167)(116,151,168)(117,152,169)(118,153,170)(119,154,171)(120,155,172), (1,205,11,215)(2,216,12,206)(3,207,13,217)(4,218,14,208)(5,209,15,219)(6,220,16,210)(7,211,17,201)(8,202,18,212)(9,213,19,203)(10,204,20,214)(21,200,31,190)(22,191,32,181)(23,182,33,192)(24,193,34,183)(25,184,35,194)(26,195,36,185)(27,186,37,196)(28,197,38,187)(29,188,39,198)(30,199,40,189)(41,150,51,160)(42,141,52,151)(43,152,53,142)(44,143,54,153)(45,154,55,144)(46,145,56,155)(47,156,57,146)(48,147,58,157)(49,158,59,148)(50,149,60,159)(61,231,71,221)(62,222,72,232)(63,233,73,223)(64,224,74,234)(65,235,75,225)(66,226,76,236)(67,237,77,227)(68,228,78,238)(69,239,79,229)(70,230,80,240)(81,107,91,117)(82,118,92,108)(83,109,93,119)(84,120,94,110)(85,111,95,101)(86,102,96,112)(87,113,97,103)(88,104,98,114)(89,115,99,105)(90,106,100,116)(121,168,131,178)(122,179,132,169)(123,170,133,180)(124,161,134,171)(125,172,135,162)(126,163,136,173)(127,174,137,164)(128,165,138,175)(129,176,139,166)(130,167,140,177), (1,120)(2,111)(3,102)(4,113)(5,104)(6,115)(7,106)(8,117)(9,108)(10,119)(11,110)(12,101)(13,112)(14,103)(15,114)(16,105)(17,116)(18,107)(19,118)(20,109)(21,42)(22,53)(23,44)(24,55)(25,46)(26,57)(27,48)(28,59)(29,50)(30,41)(31,52)(32,43)(33,54)(34,45)(35,56)(36,47)(37,58)(38,49)(39,60)(40,51)(61,139)(62,130)(63,121)(64,132)(65,123)(66,134)(67,125)(68,136)(69,127)(70,138)(71,129)(72,140)(73,131)(74,122)(75,133)(76,124)(77,135)(78,126)(79,137)(80,128)(81,212)(82,203)(83,214)(84,205)(85,216)(86,207)(87,218)(88,209)(89,220)(90,211)(91,202)(92,213)(93,204)(94,215)(95,206)(96,217)(97,208)(98,219)(99,210)(100,201)(141,190)(142,181)(143,192)(144,183)(145,194)(146,185)(147,196)(148,187)(149,198)(150,189)(151,200)(152,191)(153,182)(154,193)(155,184)(156,195)(157,186)(158,197)(159,188)(160,199)(161,226)(162,237)(163,228)(164,239)(165,230)(166,221)(167,232)(168,223)(169,234)(170,225)(171,236)(172,227)(173,238)(174,229)(175,240)(176,231)(177,222)(178,233)(179,224)(180,235), (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), (1,88,11,98)(2,97,12,87)(3,86,13,96)(4,95,14,85)(5,84,15,94)(6,93,16,83)(7,82,17,92)(8,91,18,81)(9,100,19,90)(10,89,20,99)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,183,51,193)(42,192,52,182)(43,181,53,191)(44,190,54,200)(45,199,55,189)(46,188,56,198)(47,197,57,187)(48,186,58,196)(49,195,59,185)(50,184,60,194)(61,172,71,162)(62,161,72,171)(63,170,73,180)(64,179,74,169)(65,168,75,178)(66,177,76,167)(67,166,77,176)(68,175,78,165)(69,164,79,174)(70,173,80,163)(101,218,111,208)(102,207,112,217)(103,216,113,206)(104,205,114,215)(105,214,115,204)(106,203,116,213)(107,212,117,202)(108,201,118,211)(109,210,119,220)(110,219,120,209)(121,225,131,235)(122,234,132,224)(123,223,133,233)(124,232,134,222)(125,221,135,231)(126,230,136,240)(127,239,137,229)(128,228,138,238)(129,237,139,227)(130,226,140,236)>;

G:=Group( (1,184,227)(2,185,228)(3,186,229)(4,187,230)(5,188,231)(6,189,232)(7,190,233)(8,191,234)(9,192,235)(10,193,236)(11,194,237)(12,195,238)(13,196,239)(14,197,240)(15,198,221)(16,199,222)(17,200,223)(18,181,224)(19,182,225)(20,183,226)(21,73,211)(22,74,212)(23,75,213)(24,76,214)(25,77,215)(26,78,216)(27,79,217)(28,80,218)(29,61,219)(30,62,220)(31,63,201)(32,64,202)(33,65,203)(34,66,204)(35,67,205)(36,68,206)(37,69,207)(38,70,208)(39,71,209)(40,72,210)(41,130,89)(42,131,90)(43,132,91)(44,133,92)(45,134,93)(46,135,94)(47,136,95)(48,137,96)(49,138,97)(50,139,98)(51,140,99)(52,121,100)(53,122,81)(54,123,82)(55,124,83)(56,125,84)(57,126,85)(58,127,86)(59,128,87)(60,129,88)(101,156,173)(102,157,174)(103,158,175)(104,159,176)(105,160,177)(106,141,178)(107,142,179)(108,143,180)(109,144,161)(110,145,162)(111,146,163)(112,147,164)(113,148,165)(114,149,166)(115,150,167)(116,151,168)(117,152,169)(118,153,170)(119,154,171)(120,155,172), (1,205,11,215)(2,216,12,206)(3,207,13,217)(4,218,14,208)(5,209,15,219)(6,220,16,210)(7,211,17,201)(8,202,18,212)(9,213,19,203)(10,204,20,214)(21,200,31,190)(22,191,32,181)(23,182,33,192)(24,193,34,183)(25,184,35,194)(26,195,36,185)(27,186,37,196)(28,197,38,187)(29,188,39,198)(30,199,40,189)(41,150,51,160)(42,141,52,151)(43,152,53,142)(44,143,54,153)(45,154,55,144)(46,145,56,155)(47,156,57,146)(48,147,58,157)(49,158,59,148)(50,149,60,159)(61,231,71,221)(62,222,72,232)(63,233,73,223)(64,224,74,234)(65,235,75,225)(66,226,76,236)(67,237,77,227)(68,228,78,238)(69,239,79,229)(70,230,80,240)(81,107,91,117)(82,118,92,108)(83,109,93,119)(84,120,94,110)(85,111,95,101)(86,102,96,112)(87,113,97,103)(88,104,98,114)(89,115,99,105)(90,106,100,116)(121,168,131,178)(122,179,132,169)(123,170,133,180)(124,161,134,171)(125,172,135,162)(126,163,136,173)(127,174,137,164)(128,165,138,175)(129,176,139,166)(130,167,140,177), (1,120)(2,111)(3,102)(4,113)(5,104)(6,115)(7,106)(8,117)(9,108)(10,119)(11,110)(12,101)(13,112)(14,103)(15,114)(16,105)(17,116)(18,107)(19,118)(20,109)(21,42)(22,53)(23,44)(24,55)(25,46)(26,57)(27,48)(28,59)(29,50)(30,41)(31,52)(32,43)(33,54)(34,45)(35,56)(36,47)(37,58)(38,49)(39,60)(40,51)(61,139)(62,130)(63,121)(64,132)(65,123)(66,134)(67,125)(68,136)(69,127)(70,138)(71,129)(72,140)(73,131)(74,122)(75,133)(76,124)(77,135)(78,126)(79,137)(80,128)(81,212)(82,203)(83,214)(84,205)(85,216)(86,207)(87,218)(88,209)(89,220)(90,211)(91,202)(92,213)(93,204)(94,215)(95,206)(96,217)(97,208)(98,219)(99,210)(100,201)(141,190)(142,181)(143,192)(144,183)(145,194)(146,185)(147,196)(148,187)(149,198)(150,189)(151,200)(152,191)(153,182)(154,193)(155,184)(156,195)(157,186)(158,197)(159,188)(160,199)(161,226)(162,237)(163,228)(164,239)(165,230)(166,221)(167,232)(168,223)(169,234)(170,225)(171,236)(172,227)(173,238)(174,229)(175,240)(176,231)(177,222)(178,233)(179,224)(180,235), (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), (1,88,11,98)(2,97,12,87)(3,86,13,96)(4,95,14,85)(5,84,15,94)(6,93,16,83)(7,82,17,92)(8,91,18,81)(9,100,19,90)(10,89,20,99)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,183,51,193)(42,192,52,182)(43,181,53,191)(44,190,54,200)(45,199,55,189)(46,188,56,198)(47,197,57,187)(48,186,58,196)(49,195,59,185)(50,184,60,194)(61,172,71,162)(62,161,72,171)(63,170,73,180)(64,179,74,169)(65,168,75,178)(66,177,76,167)(67,166,77,176)(68,175,78,165)(69,164,79,174)(70,173,80,163)(101,218,111,208)(102,207,112,217)(103,216,113,206)(104,205,114,215)(105,214,115,204)(106,203,116,213)(107,212,117,202)(108,201,118,211)(109,210,119,220)(110,219,120,209)(121,225,131,235)(122,234,132,224)(123,223,133,233)(124,232,134,222)(125,221,135,231)(126,230,136,240)(127,239,137,229)(128,228,138,238)(129,237,139,227)(130,226,140,236) );

G=PermutationGroup([[(1,184,227),(2,185,228),(3,186,229),(4,187,230),(5,188,231),(6,189,232),(7,190,233),(8,191,234),(9,192,235),(10,193,236),(11,194,237),(12,195,238),(13,196,239),(14,197,240),(15,198,221),(16,199,222),(17,200,223),(18,181,224),(19,182,225),(20,183,226),(21,73,211),(22,74,212),(23,75,213),(24,76,214),(25,77,215),(26,78,216),(27,79,217),(28,80,218),(29,61,219),(30,62,220),(31,63,201),(32,64,202),(33,65,203),(34,66,204),(35,67,205),(36,68,206),(37,69,207),(38,70,208),(39,71,209),(40,72,210),(41,130,89),(42,131,90),(43,132,91),(44,133,92),(45,134,93),(46,135,94),(47,136,95),(48,137,96),(49,138,97),(50,139,98),(51,140,99),(52,121,100),(53,122,81),(54,123,82),(55,124,83),(56,125,84),(57,126,85),(58,127,86),(59,128,87),(60,129,88),(101,156,173),(102,157,174),(103,158,175),(104,159,176),(105,160,177),(106,141,178),(107,142,179),(108,143,180),(109,144,161),(110,145,162),(111,146,163),(112,147,164),(113,148,165),(114,149,166),(115,150,167),(116,151,168),(117,152,169),(118,153,170),(119,154,171),(120,155,172)], [(1,205,11,215),(2,216,12,206),(3,207,13,217),(4,218,14,208),(5,209,15,219),(6,220,16,210),(7,211,17,201),(8,202,18,212),(9,213,19,203),(10,204,20,214),(21,200,31,190),(22,191,32,181),(23,182,33,192),(24,193,34,183),(25,184,35,194),(26,195,36,185),(27,186,37,196),(28,197,38,187),(29,188,39,198),(30,199,40,189),(41,150,51,160),(42,141,52,151),(43,152,53,142),(44,143,54,153),(45,154,55,144),(46,145,56,155),(47,156,57,146),(48,147,58,157),(49,158,59,148),(50,149,60,159),(61,231,71,221),(62,222,72,232),(63,233,73,223),(64,224,74,234),(65,235,75,225),(66,226,76,236),(67,237,77,227),(68,228,78,238),(69,239,79,229),(70,230,80,240),(81,107,91,117),(82,118,92,108),(83,109,93,119),(84,120,94,110),(85,111,95,101),(86,102,96,112),(87,113,97,103),(88,104,98,114),(89,115,99,105),(90,106,100,116),(121,168,131,178),(122,179,132,169),(123,170,133,180),(124,161,134,171),(125,172,135,162),(126,163,136,173),(127,174,137,164),(128,165,138,175),(129,176,139,166),(130,167,140,177)], [(1,120),(2,111),(3,102),(4,113),(5,104),(6,115),(7,106),(8,117),(9,108),(10,119),(11,110),(12,101),(13,112),(14,103),(15,114),(16,105),(17,116),(18,107),(19,118),(20,109),(21,42),(22,53),(23,44),(24,55),(25,46),(26,57),(27,48),(28,59),(29,50),(30,41),(31,52),(32,43),(33,54),(34,45),(35,56),(36,47),(37,58),(38,49),(39,60),(40,51),(61,139),(62,130),(63,121),(64,132),(65,123),(66,134),(67,125),(68,136),(69,127),(70,138),(71,129),(72,140),(73,131),(74,122),(75,133),(76,124),(77,135),(78,126),(79,137),(80,128),(81,212),(82,203),(83,214),(84,205),(85,216),(86,207),(87,218),(88,209),(89,220),(90,211),(91,202),(92,213),(93,204),(94,215),(95,206),(96,217),(97,208),(98,219),(99,210),(100,201),(141,190),(142,181),(143,192),(144,183),(145,194),(146,185),(147,196),(148,187),(149,198),(150,189),(151,200),(152,191),(153,182),(154,193),(155,184),(156,195),(157,186),(158,197),(159,188),(160,199),(161,226),(162,237),(163,228),(164,239),(165,230),(166,221),(167,232),(168,223),(169,234),(170,225),(171,236),(172,227),(173,238),(174,229),(175,240),(176,231),(177,222),(178,233),(179,224),(180,235)], [(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)], [(1,88,11,98),(2,97,12,87),(3,86,13,96),(4,95,14,85),(5,84,15,94),(6,93,16,83),(7,82,17,92),(8,91,18,81),(9,100,19,90),(10,89,20,99),(21,143,31,153),(22,152,32,142),(23,141,33,151),(24,150,34,160),(25,159,35,149),(26,148,36,158),(27,157,37,147),(28,146,38,156),(29,155,39,145),(30,144,40,154),(41,183,51,193),(42,192,52,182),(43,181,53,191),(44,190,54,200),(45,199,55,189),(46,188,56,198),(47,197,57,187),(48,186,58,196),(49,195,59,185),(50,184,60,194),(61,172,71,162),(62,161,72,171),(63,170,73,180),(64,179,74,169),(65,168,75,178),(66,177,76,167),(67,166,77,176),(68,175,78,165),(69,164,79,174),(70,173,80,163),(101,218,111,208),(102,207,112,217),(103,216,113,206),(104,205,114,215),(105,214,115,204),(106,203,116,213),(107,212,117,202),(108,201,118,211),(109,210,119,220),(110,219,120,209),(121,225,131,235),(122,234,132,224),(123,223,133,233),(124,232,134,222),(125,221,135,231),(126,230,136,240),(127,239,137,229),(128,228,138,238),(129,237,139,227),(130,226,140,236)]])

111 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E ··· 4J 5A 5B 6A 6B 6C ··· 6H 6I 6J 6K 6L 10A 10B 10C ··· 10H 12A ··· 12H 12I ··· 12T 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 5 5 6 6 6 ··· 6 6 6 6 6 10 10 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 10 10 1 1 2 2 2 2 10 ··· 10 2 2 1 1 2 ··· 2 10 10 10 10 2 2 4 ··· 4 2 ··· 2 10 ··· 10 2 2 2 2 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

111 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 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 D5 D10 D10 D10 C3×D5 C6×D5 C6×D5 C6×D5 2- 1+4 C3×2- 1+4 D4.10D10 C3×D4.10D10 kernel C3×D4.10D10 C6×Dic10 C3×C4○D20 C3×D4⋊2D5 C3×Q8×D5 C15×C4○D4 D4.10D10 C2×Dic10 C4○D20 D4⋊2D5 Q8×D5 C5×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 6 2 1 2 6 6 12 4 2 2 6 6 2 4 12 12 4 1 2 4 8

Matrix representation of C3×D4.10D10 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 34 0 28 0 0 34 0 28 48 0 27 0 0 48 0 27
,
 25 57 40 18 4 36 43 21 45 5 36 4 56 16 57 25
,
 60 43 2 36 18 18 25 25 60 43 1 18 18 18 43 43
,
 36 34 50 54 57 25 8 11 36 34 25 27 57 25 4 36
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[34,0,48,0,0,34,0,48,28,0,27,0,0,28,0,27],[25,4,45,56,57,36,5,16,40,43,36,57,18,21,4,25],[60,18,60,18,43,18,43,18,2,25,1,43,36,25,18,43],[36,57,36,57,34,25,34,25,50,8,25,4,54,11,27,36] >;

C3×D4.10D10 in GAP, Magma, Sage, TeX

C_3\times D_4._{10}D_{10}
% in TeX

G:=Group("C3xD4.10D10");
// GroupNames label

G:=SmallGroup(480,1147);
// by ID

G=gap.SmallGroup(480,1147);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,344,555,268,1571,192,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^10=e^2=b^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^2*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations

׿
×
𝔽