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## G = C3×C20.C23order 480 = 25·3·5

### Direct product of C3 and C20.C23

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C20.C23
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C4○D20 — C3×C20.C23
 Lower central C5 — C10 — C20 — C3×C20.C23
 Upper central C1 — C6 — C2×C12 — C6×Q8

Generators and relations for C3×C20.C23
G = < a,b,c,d,e | a3=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d >

Subgroups: 352 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], D5, C10, C10, C12 [×2], C12 [×3], C2×C6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8 [×2], C3×D5, C30, C30, C8.C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C3×Dic5, C60 [×2], C60 [×2], C6×D5, C2×C30, C4.Dic5, Q8⋊D5 [×2], C5⋊Q16 [×2], C4○D20, Q8×C10, C3×C8.C22, C3×C52C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C20.C23, C3×C4.Dic5, C3×Q8⋊D5 [×2], C3×C5⋊Q16 [×2], C3×C4○D20, Q8×C30, C3×C20.C23
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C8.C22, C5⋊D4 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×C5⋊D4, C3×C8.C22, C3×C5⋊D4 [×2], D5×C2×C6, C20.C23, C6×C5⋊D4, C3×C20.C23

Smallest permutation representation of C3×C20.C23
On 240 points
Generators in S240
(1 56 24)(2 57 25)(3 58 26)(4 59 27)(5 60 28)(6 41 29)(7 42 30)(8 43 31)(9 44 32)(10 45 33)(11 46 34)(12 47 35)(13 48 36)(14 49 37)(15 50 38)(16 51 39)(17 52 40)(18 53 21)(19 54 22)(20 55 23)(61 116 99)(62 117 100)(63 118 81)(64 119 82)(65 120 83)(66 101 84)(67 102 85)(68 103 86)(69 104 87)(70 105 88)(71 106 89)(72 107 90)(73 108 91)(74 109 92)(75 110 93)(76 111 94)(77 112 95)(78 113 96)(79 114 97)(80 115 98)(121 166 146)(122 167 147)(123 168 148)(124 169 149)(125 170 150)(126 171 151)(127 172 152)(128 173 153)(129 174 154)(130 175 155)(131 176 156)(132 177 157)(133 178 158)(134 179 159)(135 180 160)(136 161 141)(137 162 142)(138 163 143)(139 164 144)(140 165 145)(181 226 201)(182 227 202)(183 228 203)(184 229 204)(185 230 205)(186 231 206)(187 232 207)(188 233 208)(189 234 209)(190 235 210)(191 236 211)(192 237 212)(193 238 213)(194 239 214)(195 240 215)(196 221 216)(197 222 217)(198 223 218)(199 224 219)(200 225 220)
(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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 126)(7 125)(8 124)(9 123)(10 122)(11 121)(12 140)(13 139)(14 138)(15 137)(16 136)(17 135)(18 134)(19 133)(20 132)(21 159)(22 158)(23 157)(24 156)(25 155)(26 154)(27 153)(28 152)(29 151)(30 150)(31 149)(32 148)(33 147)(34 146)(35 145)(36 144)(37 143)(38 142)(39 141)(40 160)(41 171)(42 170)(43 169)(44 168)(45 167)(46 166)(47 165)(48 164)(49 163)(50 162)(51 161)(52 180)(53 179)(54 178)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(61 198)(62 197)(63 196)(64 195)(65 194)(66 193)(67 192)(68 191)(69 190)(70 189)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 182)(78 181)(79 200)(80 199)(81 216)(82 215)(83 214)(84 213)(85 212)(86 211)(87 210)(88 209)(89 208)(90 207)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 220)(98 219)(99 218)(100 217)(101 238)(102 237)(103 236)(104 235)(105 234)(106 233)(107 232)(108 231)(109 230)(110 229)(111 228)(112 227)(113 226)(114 225)(115 224)(116 223)(117 222)(118 221)(119 240)(120 239)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)(161 166 171 176)(162 167 172 177)(163 168 173 178)(164 169 174 179)(165 170 175 180)(181 186 191 196)(182 187 192 197)(183 188 193 198)(184 189 194 199)(185 190 195 200)(201 206 211 216)(202 207 212 217)(203 208 213 218)(204 209 214 219)(205 210 215 220)(221 226 231 236)(222 227 232 237)(223 228 233 238)(224 229 234 239)(225 230 235 240)
(1 73 11 63)(2 64 12 74)(3 75 13 65)(4 66 14 76)(5 77 15 67)(6 68 16 78)(7 79 17 69)(8 70 18 80)(9 61 19 71)(10 72 20 62)(21 98 31 88)(22 89 32 99)(23 100 33 90)(24 91 34 81)(25 82 35 92)(26 93 36 83)(27 84 37 94)(28 95 38 85)(29 86 39 96)(30 97 40 87)(41 103 51 113)(42 114 52 104)(43 105 53 115)(44 116 54 106)(45 107 55 117)(46 118 56 108)(47 109 57 119)(48 120 58 110)(49 111 59 101)(50 102 60 112)(121 191 131 181)(122 182 132 192)(123 193 133 183)(124 184 134 194)(125 195 135 185)(126 186 136 196)(127 197 137 187)(128 188 138 198)(129 199 139 189)(130 190 140 200)(141 216 151 206)(142 207 152 217)(143 218 153 208)(144 209 154 219)(145 220 155 210)(146 211 156 201)(147 202 157 212)(148 213 158 203)(149 204 159 214)(150 215 160 205)(161 221 171 231)(162 232 172 222)(163 223 173 233)(164 234 174 224)(165 225 175 235)(166 236 176 226)(167 227 177 237)(168 238 178 228)(169 229 179 239)(170 240 180 230)

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

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

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

93 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 8A 8B 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 15A 15B 15C 15D 20A ··· 20L 24A 24B 24C 24D 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 3 4 4 4 4 4 5 5 6 6 6 6 6 6 8 8 10 ··· 10 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 24 24 24 30 ··· 30 60 ··· 60 size 1 1 2 20 1 1 2 2 4 4 20 2 2 1 1 2 2 20 20 20 20 2 ··· 2 2 2 2 2 4 4 4 4 20 20 2 2 2 2 4 ··· 4 20 20 20 20 2 ··· 2 4 ··· 4

93 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 2 2 2 2 4 4 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 C8.C22 C3×C8.C22 C20.C23 C3×C20.C23 kernel C3×C20.C23 C3×C4.Dic5 C3×Q8⋊D5 C3×C5⋊Q16 C3×C4○D20 Q8×C30 C20.C23 C4.Dic5 Q8⋊D5 C5⋊Q16 C4○D20 Q8×C10 C60 C2×C30 C6×Q8 C2×C12 C3×Q8 C20 C2×C10 C2×Q8 C12 C2×C6 C2×C4 Q8 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 4 2 2 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of C3×C20.C23 in GL6(𝔽241)

 225 0 0 0 0 0 0 225 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
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 205 239 0 0 0 0 91 36 0 0 0 0 84 181 0 154 0 0 166 181 87 0
,
 68 238 0 0 0 0 95 173 0 0 0 0 0 0 159 0 67 0 0 0 0 0 240 1 0 0 209 0 82 0 0 0 209 1 82 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 174 0 0 0 0 36 1 0 0 0 0 0 159 0 240 0 0 181 159 1 0
,
 210 74 0 0 0 0 215 31 0 0 0 0 0 0 177 0 0 0 0 0 135 64 0 0 0 0 16 0 0 64 0 0 16 0 64 0

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,205,91,84,166,0,0,239,36,181,181,0,0,0,0,0,87,0,0,0,0,154,0],[68,95,0,0,0,0,238,173,0,0,0,0,0,0,159,0,209,209,0,0,0,0,0,1,0,0,67,240,82,82,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,36,0,181,0,0,174,1,159,159,0,0,0,0,0,1,0,0,0,0,240,0],[210,215,0,0,0,0,74,31,0,0,0,0,0,0,177,135,16,16,0,0,0,64,0,0,0,0,0,0,0,64,0,0,0,0,64,0] >;

C3×C20.C23 in GAP, Magma, Sage, TeX

C_3\times C_{20}.C_2^3
% in TeX

G:=Group("C3xC20.C2^3");
// GroupNames label

G:=SmallGroup(480,735);
// by ID

G=gap.SmallGroup(480,735);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,268,2524,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^20=c^2=1,d^2=e^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^10*d>;
// generators/relations

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