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## G = C3×C23.D10order 480 = 25·3·5

### Direct product of C3 and C23.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C23.D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5 — C3×C23.D10
 Lower central C5 — C2×C10 — C3×C23.D10
 Upper central C1 — C2×C6 — C3×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, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C422C2, C2×Dic5, C2×C20, C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×Dic5, C60, C2×C30, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C3×C422C2, C6×Dic5, C2×C60, C22×C30, C23.D10, C12×Dic5, C3×C10.D4, C3×C4⋊Dic5, C3×C23.D5, C15×C22⋊C4, C3×C23.D10
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, D10, C22×C6, C3×D5, C422C2, C22×D5, C3×C4○D4, C6×D5, C4○D20, D42D5, C3×C422C2, D5×C2×C6, C23.D10, C3×C4○D20, C3×D42D5, C3×C23.D10

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

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D5 C4○D4 D10 D10 C3×D5 C3×C4○D4 C6×D5 C6×D5 C4○D20 C3×C4○D20 D4⋊2D5 C3×D4⋊2D5 kernel C3×C23.D10 C12×Dic5 C3×C10.D4 C3×C4⋊Dic5 C3×C23.D5 C15×C22⋊C4 C23.D10 C4×Dic5 C10.D4 C4⋊Dic5 C23.D5 C5×C22⋊C4 C3×C22⋊C4 C30 C2×C12 C22×C6 C22⋊C4 C10 C2×C4 C23 C6 C2 C6 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 6 4 2 4 12 8 4 8 16 4 8

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

 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 0 0 0 0 0 53 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 46 0 0 0 0 0 60 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 43 17 0 0 0 0 43 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 6 0 0 0 0 45 55

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

׿
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