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G = C3×C23.23D10order 480 = 25·3·5

Direct product of C3 and C23.23D10

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

Aliases: C3×C23.23D10, (C22×C20)⋊6C6, (C22×C60)⋊2C2, C10.46(C6×D4), (C22×C12)⋊3D5, C23.D56C6, D10⋊C42C6, C10.D43C6, C30.400(C2×D4), (C2×C30).162D4, C23.28(C6×D5), (C2×C12).378D10, C30.195(C4○D4), C6.125(C4○D20), (C2×C60).448C22, (C2×C30).364C23, (C22×C6).108D10, C1535(C22.D4), (C22×C30).160C22, (C6×Dic5).162C22, C2.6(C6×C5⋊D4), (C2×C4).66(C6×D5), (C2×C5⋊D4).6C6, (C22×C4)⋊5(C3×D5), C22.55(D5×C2×C6), (C2×C20).80(C2×C6), C10.16(C3×C4○D4), C2.18(C3×C4○D20), (C2×C10).37(C3×D4), (C6×C5⋊D4).13C2, C6.127(C2×C5⋊D4), (C3×D10⋊C4)⋊2C2, (D5×C2×C6).81C22, C54(C3×C22.D4), C22.9(C3×C5⋊D4), (C3×C10.D4)⋊3C2, (C2×C6).62(C5⋊D4), (C3×C23.D5)⋊22C2, (C22×C10).47(C2×C6), (C2×C10).47(C22×C6), (C2×Dic5).14(C2×C6), (C22×D5).11(C2×C6), (C2×C6).360(C22×D5), SmallGroup(480,722)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C23.23D10
C1C5C10C2×C10C2×C30D5×C2×C6C6×C5⋊D4 — C3×C23.23D10
C5C2×C10 — C3×C23.23D10
C1C2×C6C22×C12

Generators and relations for C3×C23.23D10
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 480 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, C6, C6 [×2], C6 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×5], C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×2], C30 [×2], C22.D4, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C22×C12, C6×D4, C3×Dic5 [×3], C60 [×2], C6×D5 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C3×C22.D4, C6×Dic5, C6×Dic5 [×2], C3×C5⋊D4 [×2], C2×C60 [×2], C2×C60 [×2], D5×C2×C6, C22×C30, C23.23D10, C3×C10.D4 [×2], C3×D10⋊C4 [×2], C3×C23.D5, C6×C5⋊D4, C22×C60, C3×C23.23D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, C4○D4 [×2], D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C22.D4, C5⋊D4 [×2], C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], C4○D20 [×2], C2×C5⋊D4, C3×C22.D4, C3×C5⋊D4 [×2], D5×C2×C6, C23.23D10, C3×C4○D20 [×2], C6×C5⋊D4, C3×C23.23D10

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

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

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

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

138 conjugacy classes

class 1 2A2B2C2D2E2F3A3B4A4B4C4D4E4F4G5A5B6A···6F6G6H6I6J6K6L10A···10N12A···12H12I···12N15A15B15C15D20A···20P30A···30AB60A···60AF
order1222222334444444556···666666610···1012···1212···121515151520···2030···3060···60
size11112220112222202020221···1222220202···22···220···2022222···22···22···2

138 irreducible representations

dim11111111111122222222222222
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D5C4○D4D10D10C3×D4C3×D5C5⋊D4C3×C4○D4C6×D5C6×D5C4○D20C3×C5⋊D4C3×C4○D20
kernelC3×C23.23D10C3×C10.D4C3×D10⋊C4C3×C23.D5C6×C5⋊D4C22×C60C23.23D10C10.D4D10⋊C4C23.D5C2×C5⋊D4C22×C20C2×C30C22×C12C30C2×C12C22×C6C2×C10C22×C4C2×C6C10C2×C4C23C6C22C2
# reps12211124422222442448884161632

Matrix representation of C3×C23.23D10 in GL4(𝔽61) generated by

1000
0100
00130
00013
,
1000
466000
0010
0001
,
1000
0100
00600
00060
,
60000
06000
0010
0001
,
8000
423800
003953
005545
,
361700
172500
00352
003558
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,46,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[8,42,0,0,0,38,0,0,0,0,39,55,0,0,53,45],[36,17,0,0,17,25,0,0,0,0,3,35,0,0,52,58] >;

C3×C23.23D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{23}D_{10}
% in TeX

G:=Group("C3xC2^3.23D10");
// GroupNames label

G:=SmallGroup(480,722);
// by ID

G=gap.SmallGroup(480,722);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,268,18822]);
// Polycyclic

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

׿
×
𝔽