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G = C3×C23.D10order 480 = 25·3·5

Direct product of C3 and C23.D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C23.D10, C4⋊Dic53C6, C23.3(C6×D5), (C4×Dic5)⋊10C6, C10.D48C6, (C22×C6).3D10, C23.D5.3C6, (C12×Dic5)⋊28C2, (C2×C12).271D10, C1519(C422C2), C6.114(C4○D20), C30.186(C4○D4), (C2×C60).262C22, (C2×C30).337C23, C6.108(D42D5), (C22×C30).95C22, (C6×Dic5).154C22, C52(C3×C422C2), C2.9(C3×C4○D20), (C2×C4).22(C6×D5), C22.40(D5×C2×C6), (C2×C20).48(C2×C6), (C3×C4⋊Dic5)⋊21C2, C10.20(C3×C4○D4), C2.7(C3×D42D5), (C5×C22⋊C4).2C6, C22⋊C4.2(C3×D5), (C3×C22⋊C4).5D5, (C15×C22⋊C4).5C2, (C2×Dic5).6(C2×C6), (C3×C23.D5).9C2, (C3×C10.D4)⋊24C2, (C22×C10).14(C2×C6), (C2×C10).20(C22×C6), (C2×C6).333(C22×D5), SmallGroup(480,672)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C23.D10
C1C5C10C2×C10C2×C30C6×Dic5C12×Dic5 — C3×C23.D10
C5C2×C10 — C3×C23.D10
C1C2×C6C3×C22⋊C4

Generators and relations for C3×C23.D10
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 320 in 120 conjugacy classes, 58 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, C12 [×6], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×C12 [×2], C2×C12 [×4], C22×C6, C30 [×3], C30, C422C2, C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C3×Dic5 [×4], C60 [×2], C2×C30, C2×C30 [×3], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C3×C422C2, C6×Dic5 [×4], C2×C60 [×2], C22×C30, C23.D10, C12×Dic5, C3×C10.D4 [×2], C3×C4⋊Dic5, C3×C23.D5 [×2], C15×C22⋊C4, C3×C23.D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, D5, C2×C6 [×7], C4○D4 [×3], D10 [×3], C22×C6, C3×D5, C422C2, C22×D5, C3×C4○D4 [×3], C6×D5 [×3], C4○D20, D42D5 [×2], C3×C422C2, D5×C2×C6, C23.D10, C3×C4○D20, C3×D42D5 [×2], C3×C23.D10

Smallest permutation representation of C3×C23.D10
On 240 points
Generators in S240
(1 201 22)(2 202 23)(3 203 24)(4 204 25)(5 205 26)(6 206 27)(7 207 28)(8 208 29)(9 209 30)(10 210 31)(11 211 32)(12 212 33)(13 213 34)(14 214 35)(15 215 36)(16 216 37)(17 217 38)(18 218 39)(19 219 40)(20 220 21)(41 197 84)(42 198 85)(43 199 86)(44 200 87)(45 181 88)(46 182 89)(47 183 90)(48 184 91)(49 185 92)(50 186 93)(51 187 94)(52 188 95)(53 189 96)(54 190 97)(55 191 98)(56 192 99)(57 193 100)(58 194 81)(59 195 82)(60 196 83)(61 221 157)(62 222 158)(63 223 159)(64 224 160)(65 225 141)(66 226 142)(67 227 143)(68 228 144)(69 229 145)(70 230 146)(71 231 147)(72 232 148)(73 233 149)(74 234 150)(75 235 151)(76 236 152)(77 237 153)(78 238 154)(79 239 155)(80 240 156)(101 161 128)(102 162 129)(103 163 130)(104 164 131)(105 165 132)(106 166 133)(107 167 134)(108 168 135)(109 169 136)(110 170 137)(111 171 138)(112 172 139)(113 173 140)(114 174 121)(115 175 122)(116 176 123)(117 177 124)(118 178 125)(119 179 126)(120 180 127)
(2 103)(4 105)(6 107)(8 109)(10 111)(12 113)(14 115)(16 117)(18 119)(20 101)(21 128)(23 130)(25 132)(27 134)(29 136)(31 138)(33 140)(35 122)(37 124)(39 126)(41 51)(42 237)(43 53)(44 239)(45 55)(46 221)(47 57)(48 223)(49 59)(50 225)(52 227)(54 229)(56 231)(58 233)(60 235)(61 89)(62 72)(63 91)(64 74)(65 93)(66 76)(67 95)(68 78)(69 97)(70 80)(71 99)(73 81)(75 83)(77 85)(79 87)(82 92)(84 94)(86 96)(88 98)(90 100)(141 186)(142 152)(143 188)(144 154)(145 190)(146 156)(147 192)(148 158)(149 194)(150 160)(151 196)(153 198)(155 200)(157 182)(159 184)(161 220)(163 202)(165 204)(167 206)(169 208)(171 210)(173 212)(175 214)(177 216)(179 218)(181 191)(183 193)(185 195)(187 197)(189 199)(222 232)(224 234)(226 236)(228 238)(230 240)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 101)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 226)(42 227)(43 228)(44 229)(45 230)(46 231)(47 232)(48 233)(49 234)(50 235)(51 236)(52 237)(53 238)(54 239)(55 240)(56 221)(57 222)(58 223)(59 224)(60 225)(61 99)(62 100)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(141 196)(142 197)(143 198)(144 199)(145 200)(146 181)(147 182)(148 183)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 220)(162 201)(163 202)(164 203)(165 204)(166 205)(167 206)(168 207)(169 208)(170 209)(171 210)(172 211)(173 212)(174 213)(175 214)(176 215)(177 216)(178 217)(179 218)(180 219)
(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)
(1 148 112 193)(2 157 113 182)(3 146 114 191)(4 155 115 200)(5 144 116 189)(6 153 117 198)(7 142 118 187)(8 151 119 196)(9 160 120 185)(10 149 101 194)(11 158 102 183)(12 147 103 192)(13 156 104 181)(14 145 105 190)(15 154 106 199)(16 143 107 188)(17 152 108 197)(18 141 109 186)(19 150 110 195)(20 159 111 184)(21 223 138 48)(22 232 139 57)(23 221 140 46)(24 230 121 55)(25 239 122 44)(26 228 123 53)(27 237 124 42)(28 226 125 51)(29 235 126 60)(30 224 127 49)(31 233 128 58)(32 222 129 47)(33 231 130 56)(34 240 131 45)(35 229 132 54)(36 238 133 43)(37 227 134 52)(38 236 135 41)(39 225 136 50)(40 234 137 59)(61 173 89 202)(62 162 90 211)(63 171 91 220)(64 180 92 209)(65 169 93 218)(66 178 94 207)(67 167 95 216)(68 176 96 205)(69 165 97 214)(70 174 98 203)(71 163 99 212)(72 172 100 201)(73 161 81 210)(74 170 82 219)(75 179 83 208)(76 168 84 217)(77 177 85 206)(78 166 86 215)(79 175 87 204)(80 164 88 213)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E4F4G4H4I5A5B6A···6F6G6H10A···10F10G10H10I10J12A12B12C12D12E12F12G···12N12O12P12Q12R15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order1222233444444444556···66610···101010101012121212121212···12121212121515151520···2030···3030···3060···60
size1111411224101010102020221···1442···2444422224410···102020202022224···42···24···44···4

102 irreducible representations

dim111111111111222222222244
type+++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6D5C4○D4D10D10C3×D5C3×C4○D4C6×D5C6×D5C4○D20C3×C4○D20D42D5C3×D42D5
kernelC3×C23.D10C12×Dic5C3×C10.D4C3×C4⋊Dic5C3×C23.D5C15×C22⋊C4C23.D10C4×Dic5C10.D4C4⋊Dic5C23.D5C5×C22⋊C4C3×C22⋊C4C30C2×C12C22×C6C22⋊C4C10C2×C4C23C6C2C6C2
# reps11212122424226424128481648

Matrix representation of C3×C23.D10 in GL6(𝔽61)

1300000
0130000
0047000
0004700
0000470
0000047
,
100000
53600000
001000
0006000
000010
000001
,
100000
010000
0060000
0006000
000010
000001
,
6000000
0600000
001000
000100
000010
000001
,
1460000
0600000
0050000
0005000
00004317
0000430
,
1100000
0110000
0005000
0050000
000066
00004555

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47],[1,53,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,46,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,43,43,0,0,0,0,17,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,6,45,0,0,0,0,6,55] >;

C3×C23.D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3.D_{10}
% in TeX

G:=Group("C3xC2^3.D10");
// GroupNames label

G:=SmallGroup(480,672);
// by ID

G=gap.SmallGroup(480,672);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,1598,555,18822]);
// Polycyclic

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

׿
×
𝔽