direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4.10D10, C30.81C24, C60.216C23, C15⋊122- (1+4), (Q8×D5)⋊8C6, C4○D20⋊9C6, D4⋊2D5⋊5C6, D4.10(C6×D5), Q8.16(C6×D5), D20.14(C2×C6), (C3×D4).39D10, (C3×Q8).43D10, C6.81(C23×D5), (C6×Dic10)⋊30C2, (C2×Dic10)⋊14C6, (C2×C12).254D10, C5⋊2(C3×2- (1+4)), C10.13(C23×C6), C20.26(C22×C6), (C6×D5).59C23, D10.7(C22×C6), (C2×C30).257C23, (C2×C60).309C22, Dic10.14(C2×C6), (C3×D20).53C22, (D5×C12).81C22, (D4×C15).44C22, C12.216(C22×D5), (Q8×C15).48C22, Dic5.8(C22×C6), (C3×Dic5).61C23, (C6×Dic5).167C22, (C3×Dic10).56C22, C5⋊D4.(C2×C6), (C3×Q8×D5)⋊12C2, C4.34(D5×C2×C6), C4○D4⋊6(C3×D5), (C3×C4○D4)⋊9D5, (C5×C4○D4)⋊9C6, C22.4(D5×C2×C6), (C4×D5).6(C2×C6), (C2×C4).20(C6×D5), (C15×C4○D4)⋊10C2, (C3×C4○D20)⋊19C2, C2.14(D5×C22×C6), (C2×C20).46(C2×C6), (C5×D4).10(C2×C6), (C5×Q8).19(C2×C6), (C3×D4⋊2D5)⋊12C2, (C2×C10).5(C22×C6), (C2×C6).24(C22×D5), (C3×C5⋊D4).4C22, (C2×Dic5).19(C2×C6), SmallGroup(480,1147)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 752 in 292 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C6, C6 [×5], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], D5 [×2], C10, C10 [×3], C12, C12 [×3], C12 [×6], C2×C6 [×3], C2×C6 [×2], C15, C2×Q8 [×5], C4○D4, C4○D4 [×9], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], C2×C12 [×3], C2×C12 [×12], C3×D4 [×3], C3×D4 [×7], C3×Q8, C3×Q8 [×9], C3×D5 [×2], C30, C30 [×3], 2- (1+4), Dic10 [×9], C4×D5 [×6], D20, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C6×Q8 [×5], C3×C4○D4, C3×C4○D4 [×9], C3×Dic5 [×6], C60, C60 [×3], C6×D5 [×2], C2×C30 [×3], C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5 [×6], Q8×D5 [×2], C5×C4○D4, C3×2- (1+4), C3×Dic10 [×9], D5×C12 [×6], C3×D20, C6×Dic5 [×6], C3×C5⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, D4.10D10, C6×Dic10 [×3], C3×C4○D20 [×3], C3×D4⋊2D5 [×6], C3×Q8×D5 [×2], C15×C4○D4, C3×D4.10D10
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C24, D10 [×7], C22×C6 [×15], C3×D5, 2- (1+4), C22×D5 [×7], C23×C6, C6×D5 [×7], C23×D5, C3×2- (1+4), D5×C2×C6 [×7], D4.10D10, D5×C22×C6, C3×D4.10D10
Generators and relations
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 >
►On 240 points
Generators in S240
(1 103 71)(2 104 72)(3 105 73)(4 106 74)(5 107 75)(6 108 76)(7 109 77)(8 110 78)(9 111 79)(10 112 80)(11 113 61)(12 114 62)(13 115 63)(14 116 64)(15 117 65)(16 118 66)(17 119 67)(18 120 68)(19 101 69)(20 102 70)(21 181 50)(22 182 51)(23 183 52)(24 184 53)(25 185 54)(26 186 55)(27 187 56)(28 188 57)(29 189 58)(30 190 59)(31 191 60)(32 192 41)(33 193 42)(34 194 43)(35 195 44)(36 196 45)(37 197 46)(38 198 47)(39 199 48)(40 200 49)(81 239 217)(82 240 218)(83 221 219)(84 222 220)(85 223 201)(86 224 202)(87 225 203)(88 226 204)(89 227 205)(90 228 206)(91 229 207)(92 230 208)(93 231 209)(94 232 210)(95 233 211)(96 234 212)(97 235 213)(98 236 214)(99 237 215)(100 238 216)(121 149 173)(122 150 174)(123 151 175)(124 152 176)(125 153 177)(126 154 178)(127 155 179)(128 156 180)(129 157 161)(130 158 162)(131 159 163)(132 160 164)(133 141 165)(134 142 166)(135 143 167)(136 144 168)(137 145 169)(138 146 170)(139 147 171)(140 148 172)
(1 152 11 142)(2 143 12 153)(3 154 13 144)(4 145 14 155)(5 156 15 146)(6 147 16 157)(7 158 17 148)(8 149 18 159)(9 160 19 150)(10 151 20 141)(21 90 31 100)(22 81 32 91)(23 92 33 82)(24 83 34 93)(25 94 35 84)(26 85 36 95)(27 96 37 86)(28 87 38 97)(29 98 39 88)(30 89 40 99)(41 207 51 217)(42 218 52 208)(43 209 53 219)(44 220 54 210)(45 211 55 201)(46 202 56 212)(47 213 57 203)(48 204 58 214)(49 215 59 205)(50 206 60 216)(61 134 71 124)(62 125 72 135)(63 136 73 126)(64 127 74 137)(65 138 75 128)(66 129 76 139)(67 140 77 130)(68 131 78 121)(69 122 79 132)(70 133 80 123)(101 174 111 164)(102 165 112 175)(103 176 113 166)(104 167 114 177)(105 178 115 168)(106 169 116 179)(107 180 117 170)(108 171 118 161)(109 162 119 172)(110 173 120 163)(181 228 191 238)(182 239 192 229)(183 230 193 240)(184 221 194 231)(185 232 195 222)(186 223 196 233)(187 234 197 224)(188 225 198 235)(189 236 199 226)(190 227 200 237)
(1 93)(2 84)(3 95)(4 86)(5 97)(6 88)(7 99)(8 90)(9 81)(10 92)(11 83)(12 94)(13 85)(14 96)(15 87)(16 98)(17 89)(18 100)(19 91)(20 82)(21 149)(22 160)(23 151)(24 142)(25 153)(26 144)(27 155)(28 146)(29 157)(30 148)(31 159)(32 150)(33 141)(34 152)(35 143)(36 154)(37 145)(38 156)(39 147)(40 158)(41 122)(42 133)(43 124)(44 135)(45 126)(46 137)(47 128)(48 139)(49 130)(50 121)(51 132)(52 123)(53 134)(54 125)(55 136)(56 127)(57 138)(58 129)(59 140)(60 131)(61 219)(62 210)(63 201)(64 212)(65 203)(66 214)(67 205)(68 216)(69 207)(70 218)(71 209)(72 220)(73 211)(74 202)(75 213)(76 204)(77 215)(78 206)(79 217)(80 208)(101 229)(102 240)(103 231)(104 222)(105 233)(106 224)(107 235)(108 226)(109 237)(110 228)(111 239)(112 230)(113 221)(114 232)(115 223)(116 234)(117 225)(118 236)(119 227)(120 238)(161 189)(162 200)(163 191)(164 182)(165 193)(166 184)(167 195)(168 186)(169 197)(170 188)(171 199)(172 190)(173 181)(174 192)(175 183)(176 194)(177 185)(178 196)(179 187)(180 198)
(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 38 11 28)(2 27 12 37)(3 36 13 26)(4 25 14 35)(5 34 15 24)(6 23 16 33)(7 32 17 22)(8 21 18 31)(9 30 19 40)(10 39 20 29)(41 67 51 77)(42 76 52 66)(43 65 53 75)(44 74 54 64)(45 63 55 73)(46 72 56 62)(47 61 57 71)(48 70 58 80)(49 79 59 69)(50 68 60 78)(81 148 91 158)(82 157 92 147)(83 146 93 156)(84 155 94 145)(85 144 95 154)(86 153 96 143)(87 142 97 152)(88 151 98 141)(89 160 99 150)(90 149 100 159)(101 200 111 190)(102 189 112 199)(103 198 113 188)(104 187 114 197)(105 196 115 186)(106 185 116 195)(107 194 117 184)(108 183 118 193)(109 192 119 182)(110 181 120 191)(121 216 131 206)(122 205 132 215)(123 214 133 204)(124 203 134 213)(125 212 135 202)(126 201 136 211)(127 210 137 220)(128 219 138 209)(129 208 139 218)(130 217 140 207)(161 230 171 240)(162 239 172 229)(163 228 173 238)(164 237 174 227)(165 226 175 236)(166 235 176 225)(167 224 177 234)(168 233 178 223)(169 222 179 232)(170 231 180 221)
G:=sub<Sym(240)| (1,103,71)(2,104,72)(3,105,73)(4,106,74)(5,107,75)(6,108,76)(7,109,77)(8,110,78)(9,111,79)(10,112,80)(11,113,61)(12,114,62)(13,115,63)(14,116,64)(15,117,65)(16,118,66)(17,119,67)(18,120,68)(19,101,69)(20,102,70)(21,181,50)(22,182,51)(23,183,52)(24,184,53)(25,185,54)(26,186,55)(27,187,56)(28,188,57)(29,189,58)(30,190,59)(31,191,60)(32,192,41)(33,193,42)(34,194,43)(35,195,44)(36,196,45)(37,197,46)(38,198,47)(39,199,48)(40,200,49)(81,239,217)(82,240,218)(83,221,219)(84,222,220)(85,223,201)(86,224,202)(87,225,203)(88,226,204)(89,227,205)(90,228,206)(91,229,207)(92,230,208)(93,231,209)(94,232,210)(95,233,211)(96,234,212)(97,235,213)(98,236,214)(99,237,215)(100,238,216)(121,149,173)(122,150,174)(123,151,175)(124,152,176)(125,153,177)(126,154,178)(127,155,179)(128,156,180)(129,157,161)(130,158,162)(131,159,163)(132,160,164)(133,141,165)(134,142,166)(135,143,167)(136,144,168)(137,145,169)(138,146,170)(139,147,171)(140,148,172), (1,152,11,142)(2,143,12,153)(3,154,13,144)(4,145,14,155)(5,156,15,146)(6,147,16,157)(7,158,17,148)(8,149,18,159)(9,160,19,150)(10,151,20,141)(21,90,31,100)(22,81,32,91)(23,92,33,82)(24,83,34,93)(25,94,35,84)(26,85,36,95)(27,96,37,86)(28,87,38,97)(29,98,39,88)(30,89,40,99)(41,207,51,217)(42,218,52,208)(43,209,53,219)(44,220,54,210)(45,211,55,201)(46,202,56,212)(47,213,57,203)(48,204,58,214)(49,215,59,205)(50,206,60,216)(61,134,71,124)(62,125,72,135)(63,136,73,126)(64,127,74,137)(65,138,75,128)(66,129,76,139)(67,140,77,130)(68,131,78,121)(69,122,79,132)(70,133,80,123)(101,174,111,164)(102,165,112,175)(103,176,113,166)(104,167,114,177)(105,178,115,168)(106,169,116,179)(107,180,117,170)(108,171,118,161)(109,162,119,172)(110,173,120,163)(181,228,191,238)(182,239,192,229)(183,230,193,240)(184,221,194,231)(185,232,195,222)(186,223,196,233)(187,234,197,224)(188,225,198,235)(189,236,199,226)(190,227,200,237), (1,93)(2,84)(3,95)(4,86)(5,97)(6,88)(7,99)(8,90)(9,81)(10,92)(11,83)(12,94)(13,85)(14,96)(15,87)(16,98)(17,89)(18,100)(19,91)(20,82)(21,149)(22,160)(23,151)(24,142)(25,153)(26,144)(27,155)(28,146)(29,157)(30,148)(31,159)(32,150)(33,141)(34,152)(35,143)(36,154)(37,145)(38,156)(39,147)(40,158)(41,122)(42,133)(43,124)(44,135)(45,126)(46,137)(47,128)(48,139)(49,130)(50,121)(51,132)(52,123)(53,134)(54,125)(55,136)(56,127)(57,138)(58,129)(59,140)(60,131)(61,219)(62,210)(63,201)(64,212)(65,203)(66,214)(67,205)(68,216)(69,207)(70,218)(71,209)(72,220)(73,211)(74,202)(75,213)(76,204)(77,215)(78,206)(79,217)(80,208)(101,229)(102,240)(103,231)(104,222)(105,233)(106,224)(107,235)(108,226)(109,237)(110,228)(111,239)(112,230)(113,221)(114,232)(115,223)(116,234)(117,225)(118,236)(119,227)(120,238)(161,189)(162,200)(163,191)(164,182)(165,193)(166,184)(167,195)(168,186)(169,197)(170,188)(171,199)(172,190)(173,181)(174,192)(175,183)(176,194)(177,185)(178,196)(179,187)(180,198), (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), (1,38,11,28)(2,27,12,37)(3,36,13,26)(4,25,14,35)(5,34,15,24)(6,23,16,33)(7,32,17,22)(8,21,18,31)(9,30,19,40)(10,39,20,29)(41,67,51,77)(42,76,52,66)(43,65,53,75)(44,74,54,64)(45,63,55,73)(46,72,56,62)(47,61,57,71)(48,70,58,80)(49,79,59,69)(50,68,60,78)(81,148,91,158)(82,157,92,147)(83,146,93,156)(84,155,94,145)(85,144,95,154)(86,153,96,143)(87,142,97,152)(88,151,98,141)(89,160,99,150)(90,149,100,159)(101,200,111,190)(102,189,112,199)(103,198,113,188)(104,187,114,197)(105,196,115,186)(106,185,116,195)(107,194,117,184)(108,183,118,193)(109,192,119,182)(110,181,120,191)(121,216,131,206)(122,205,132,215)(123,214,133,204)(124,203,134,213)(125,212,135,202)(126,201,136,211)(127,210,137,220)(128,219,138,209)(129,208,139,218)(130,217,140,207)(161,230,171,240)(162,239,172,229)(163,228,173,238)(164,237,174,227)(165,226,175,236)(166,235,176,225)(167,224,177,234)(168,233,178,223)(169,222,179,232)(170,231,180,221)>;
G:=Group( (1,103,71)(2,104,72)(3,105,73)(4,106,74)(5,107,75)(6,108,76)(7,109,77)(8,110,78)(9,111,79)(10,112,80)(11,113,61)(12,114,62)(13,115,63)(14,116,64)(15,117,65)(16,118,66)(17,119,67)(18,120,68)(19,101,69)(20,102,70)(21,181,50)(22,182,51)(23,183,52)(24,184,53)(25,185,54)(26,186,55)(27,187,56)(28,188,57)(29,189,58)(30,190,59)(31,191,60)(32,192,41)(33,193,42)(34,194,43)(35,195,44)(36,196,45)(37,197,46)(38,198,47)(39,199,48)(40,200,49)(81,239,217)(82,240,218)(83,221,219)(84,222,220)(85,223,201)(86,224,202)(87,225,203)(88,226,204)(89,227,205)(90,228,206)(91,229,207)(92,230,208)(93,231,209)(94,232,210)(95,233,211)(96,234,212)(97,235,213)(98,236,214)(99,237,215)(100,238,216)(121,149,173)(122,150,174)(123,151,175)(124,152,176)(125,153,177)(126,154,178)(127,155,179)(128,156,180)(129,157,161)(130,158,162)(131,159,163)(132,160,164)(133,141,165)(134,142,166)(135,143,167)(136,144,168)(137,145,169)(138,146,170)(139,147,171)(140,148,172), (1,152,11,142)(2,143,12,153)(3,154,13,144)(4,145,14,155)(5,156,15,146)(6,147,16,157)(7,158,17,148)(8,149,18,159)(9,160,19,150)(10,151,20,141)(21,90,31,100)(22,81,32,91)(23,92,33,82)(24,83,34,93)(25,94,35,84)(26,85,36,95)(27,96,37,86)(28,87,38,97)(29,98,39,88)(30,89,40,99)(41,207,51,217)(42,218,52,208)(43,209,53,219)(44,220,54,210)(45,211,55,201)(46,202,56,212)(47,213,57,203)(48,204,58,214)(49,215,59,205)(50,206,60,216)(61,134,71,124)(62,125,72,135)(63,136,73,126)(64,127,74,137)(65,138,75,128)(66,129,76,139)(67,140,77,130)(68,131,78,121)(69,122,79,132)(70,133,80,123)(101,174,111,164)(102,165,112,175)(103,176,113,166)(104,167,114,177)(105,178,115,168)(106,169,116,179)(107,180,117,170)(108,171,118,161)(109,162,119,172)(110,173,120,163)(181,228,191,238)(182,239,192,229)(183,230,193,240)(184,221,194,231)(185,232,195,222)(186,223,196,233)(187,234,197,224)(188,225,198,235)(189,236,199,226)(190,227,200,237), (1,93)(2,84)(3,95)(4,86)(5,97)(6,88)(7,99)(8,90)(9,81)(10,92)(11,83)(12,94)(13,85)(14,96)(15,87)(16,98)(17,89)(18,100)(19,91)(20,82)(21,149)(22,160)(23,151)(24,142)(25,153)(26,144)(27,155)(28,146)(29,157)(30,148)(31,159)(32,150)(33,141)(34,152)(35,143)(36,154)(37,145)(38,156)(39,147)(40,158)(41,122)(42,133)(43,124)(44,135)(45,126)(46,137)(47,128)(48,139)(49,130)(50,121)(51,132)(52,123)(53,134)(54,125)(55,136)(56,127)(57,138)(58,129)(59,140)(60,131)(61,219)(62,210)(63,201)(64,212)(65,203)(66,214)(67,205)(68,216)(69,207)(70,218)(71,209)(72,220)(73,211)(74,202)(75,213)(76,204)(77,215)(78,206)(79,217)(80,208)(101,229)(102,240)(103,231)(104,222)(105,233)(106,224)(107,235)(108,226)(109,237)(110,228)(111,239)(112,230)(113,221)(114,232)(115,223)(116,234)(117,225)(118,236)(119,227)(120,238)(161,189)(162,200)(163,191)(164,182)(165,193)(166,184)(167,195)(168,186)(169,197)(170,188)(171,199)(172,190)(173,181)(174,192)(175,183)(176,194)(177,185)(178,196)(179,187)(180,198), (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), (1,38,11,28)(2,27,12,37)(3,36,13,26)(4,25,14,35)(5,34,15,24)(6,23,16,33)(7,32,17,22)(8,21,18,31)(9,30,19,40)(10,39,20,29)(41,67,51,77)(42,76,52,66)(43,65,53,75)(44,74,54,64)(45,63,55,73)(46,72,56,62)(47,61,57,71)(48,70,58,80)(49,79,59,69)(50,68,60,78)(81,148,91,158)(82,157,92,147)(83,146,93,156)(84,155,94,145)(85,144,95,154)(86,153,96,143)(87,142,97,152)(88,151,98,141)(89,160,99,150)(90,149,100,159)(101,200,111,190)(102,189,112,199)(103,198,113,188)(104,187,114,197)(105,196,115,186)(106,185,116,195)(107,194,117,184)(108,183,118,193)(109,192,119,182)(110,181,120,191)(121,216,131,206)(122,205,132,215)(123,214,133,204)(124,203,134,213)(125,212,135,202)(126,201,136,211)(127,210,137,220)(128,219,138,209)(129,208,139,218)(130,217,140,207)(161,230,171,240)(162,239,172,229)(163,228,173,238)(164,237,174,227)(165,226,175,236)(166,235,176,225)(167,224,177,234)(168,233,178,223)(169,222,179,232)(170,231,180,221) );
G=PermutationGroup([(1,103,71),(2,104,72),(3,105,73),(4,106,74),(5,107,75),(6,108,76),(7,109,77),(8,110,78),(9,111,79),(10,112,80),(11,113,61),(12,114,62),(13,115,63),(14,116,64),(15,117,65),(16,118,66),(17,119,67),(18,120,68),(19,101,69),(20,102,70),(21,181,50),(22,182,51),(23,183,52),(24,184,53),(25,185,54),(26,186,55),(27,187,56),(28,188,57),(29,189,58),(30,190,59),(31,191,60),(32,192,41),(33,193,42),(34,194,43),(35,195,44),(36,196,45),(37,197,46),(38,198,47),(39,199,48),(40,200,49),(81,239,217),(82,240,218),(83,221,219),(84,222,220),(85,223,201),(86,224,202),(87,225,203),(88,226,204),(89,227,205),(90,228,206),(91,229,207),(92,230,208),(93,231,209),(94,232,210),(95,233,211),(96,234,212),(97,235,213),(98,236,214),(99,237,215),(100,238,216),(121,149,173),(122,150,174),(123,151,175),(124,152,176),(125,153,177),(126,154,178),(127,155,179),(128,156,180),(129,157,161),(130,158,162),(131,159,163),(132,160,164),(133,141,165),(134,142,166),(135,143,167),(136,144,168),(137,145,169),(138,146,170),(139,147,171),(140,148,172)], [(1,152,11,142),(2,143,12,153),(3,154,13,144),(4,145,14,155),(5,156,15,146),(6,147,16,157),(7,158,17,148),(8,149,18,159),(9,160,19,150),(10,151,20,141),(21,90,31,100),(22,81,32,91),(23,92,33,82),(24,83,34,93),(25,94,35,84),(26,85,36,95),(27,96,37,86),(28,87,38,97),(29,98,39,88),(30,89,40,99),(41,207,51,217),(42,218,52,208),(43,209,53,219),(44,220,54,210),(45,211,55,201),(46,202,56,212),(47,213,57,203),(48,204,58,214),(49,215,59,205),(50,206,60,216),(61,134,71,124),(62,125,72,135),(63,136,73,126),(64,127,74,137),(65,138,75,128),(66,129,76,139),(67,140,77,130),(68,131,78,121),(69,122,79,132),(70,133,80,123),(101,174,111,164),(102,165,112,175),(103,176,113,166),(104,167,114,177),(105,178,115,168),(106,169,116,179),(107,180,117,170),(108,171,118,161),(109,162,119,172),(110,173,120,163),(181,228,191,238),(182,239,192,229),(183,230,193,240),(184,221,194,231),(185,232,195,222),(186,223,196,233),(187,234,197,224),(188,225,198,235),(189,236,199,226),(190,227,200,237)], [(1,93),(2,84),(3,95),(4,86),(5,97),(6,88),(7,99),(8,90),(9,81),(10,92),(11,83),(12,94),(13,85),(14,96),(15,87),(16,98),(17,89),(18,100),(19,91),(20,82),(21,149),(22,160),(23,151),(24,142),(25,153),(26,144),(27,155),(28,146),(29,157),(30,148),(31,159),(32,150),(33,141),(34,152),(35,143),(36,154),(37,145),(38,156),(39,147),(40,158),(41,122),(42,133),(43,124),(44,135),(45,126),(46,137),(47,128),(48,139),(49,130),(50,121),(51,132),(52,123),(53,134),(54,125),(55,136),(56,127),(57,138),(58,129),(59,140),(60,131),(61,219),(62,210),(63,201),(64,212),(65,203),(66,214),(67,205),(68,216),(69,207),(70,218),(71,209),(72,220),(73,211),(74,202),(75,213),(76,204),(77,215),(78,206),(79,217),(80,208),(101,229),(102,240),(103,231),(104,222),(105,233),(106,224),(107,235),(108,226),(109,237),(110,228),(111,239),(112,230),(113,221),(114,232),(115,223),(116,234),(117,225),(118,236),(119,227),(120,238),(161,189),(162,200),(163,191),(164,182),(165,193),(166,184),(167,195),(168,186),(169,197),(170,188),(171,199),(172,190),(173,181),(174,192),(175,183),(176,194),(177,185),(178,196),(179,187),(180,198)], [(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)], [(1,38,11,28),(2,27,12,37),(3,36,13,26),(4,25,14,35),(5,34,15,24),(6,23,16,33),(7,32,17,22),(8,21,18,31),(9,30,19,40),(10,39,20,29),(41,67,51,77),(42,76,52,66),(43,65,53,75),(44,74,54,64),(45,63,55,73),(46,72,56,62),(47,61,57,71),(48,70,58,80),(49,79,59,69),(50,68,60,78),(81,148,91,158),(82,157,92,147),(83,146,93,156),(84,155,94,145),(85,144,95,154),(86,153,96,143),(87,142,97,152),(88,151,98,141),(89,160,99,150),(90,149,100,159),(101,200,111,190),(102,189,112,199),(103,198,113,188),(104,187,114,197),(105,196,115,186),(106,185,116,195),(107,194,117,184),(108,183,118,193),(109,192,119,182),(110,181,120,191),(121,216,131,206),(122,205,132,215),(123,214,133,204),(124,203,134,213),(125,212,135,202),(126,201,136,211),(127,210,137,220),(128,219,138,209),(129,208,139,218),(130,217,140,207),(161,230,171,240),(162,239,172,229),(163,228,173,238),(164,237,174,227),(165,226,175,236),(166,235,176,225),(167,224,177,234),(168,233,178,223),(169,222,179,232),(170,231,180,221)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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