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## G = C3×C22.D20order 480 = 25·3·5

### Direct product of C3 and C22.D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C22.D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×C5⋊D4 — C3×C22.D20
 Lower central C5 — C2×C10 — C3×C22.D20
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C22.D20
G = < a,b,c,d,e | a3=b2=c2=d20=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 512 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, C22⋊C4 [×2], 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], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C6×D4, C3×Dic5 [×3], C60 [×2], C6×D5 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C3×C22.D4, C6×Dic5, C6×Dic5 [×2], C6×Dic5 [×2], C3×C5⋊D4 [×2], C2×C60 [×2], D5×C2×C6, C22×C30, C22.D20, C3×C4⋊Dic5 [×2], C3×D10⋊C4 [×2], C15×C22⋊C4, C2×C6×Dic5, C6×C5⋊D4, C3×C22.D20
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, D20 [×2], C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], C2×D20, D42D5 [×2], C3×C22.D4, C3×D20 [×2], D5×C2×C6, C22.D20, C6×D20, C3×D42D5 [×2], C3×C22.D20

Smallest permutation representation of C3×C22.D20
On 240 points
Generators in S240
(1 239 33)(2 240 34)(3 221 35)(4 222 36)(5 223 37)(6 224 38)(7 225 39)(8 226 40)(9 227 21)(10 228 22)(11 229 23)(12 230 24)(13 231 25)(14 232 26)(15 233 27)(16 234 28)(17 235 29)(18 236 30)(19 237 31)(20 238 32)(41 63 126)(42 64 127)(43 65 128)(44 66 129)(45 67 130)(46 68 131)(47 69 132)(48 70 133)(49 71 134)(50 72 135)(51 73 136)(52 74 137)(53 75 138)(54 76 139)(55 77 140)(56 78 121)(57 79 122)(58 80 123)(59 61 124)(60 62 125)(81 105 166)(82 106 167)(83 107 168)(84 108 169)(85 109 170)(86 110 171)(87 111 172)(88 112 173)(89 113 174)(90 114 175)(91 115 176)(92 116 177)(93 117 178)(94 118 179)(95 119 180)(96 120 161)(97 101 162)(98 102 163)(99 103 164)(100 104 165)(141 196 205)(142 197 206)(143 198 207)(144 199 208)(145 200 209)(146 181 210)(147 182 211)(148 183 212)(149 184 213)(150 185 214)(151 186 215)(152 187 216)(153 188 217)(154 189 218)(155 190 219)(156 191 220)(157 192 201)(158 193 202)(159 194 203)(160 195 204)
(2 209)(4 211)(6 213)(8 215)(10 217)(12 219)(14 201)(16 203)(18 205)(20 207)(22 188)(24 190)(26 192)(28 194)(30 196)(32 198)(34 200)(36 182)(38 184)(40 186)(42 90)(44 92)(46 94)(48 96)(50 98)(52 100)(54 82)(56 84)(58 86)(60 88)(62 112)(64 114)(66 116)(68 118)(70 120)(72 102)(74 104)(76 106)(78 108)(80 110)(121 169)(123 171)(125 173)(127 175)(129 177)(131 179)(133 161)(135 163)(137 165)(139 167)(141 236)(143 238)(145 240)(147 222)(149 224)(151 226)(153 228)(155 230)(157 232)(159 234)
(1 208)(2 209)(3 210)(4 211)(5 212)(6 213)(7 214)(8 215)(9 216)(10 217)(11 218)(12 219)(13 220)(14 201)(15 202)(16 203)(17 204)(18 205)(19 206)(20 207)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 181)(36 182)(37 183)(38 184)(39 185)(40 186)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(141 236)(142 237)(143 238)(144 239)(145 240)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)
(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 77 208 107)(2 106 209 76)(3 75 210 105)(4 104 211 74)(5 73 212 103)(6 102 213 72)(7 71 214 101)(8 120 215 70)(9 69 216 119)(10 118 217 68)(11 67 218 117)(12 116 219 66)(13 65 220 115)(14 114 201 64)(15 63 202 113)(16 112 203 62)(17 61 204 111)(18 110 205 80)(19 79 206 109)(20 108 207 78)(21 47 187 95)(22 94 188 46)(23 45 189 93)(24 92 190 44)(25 43 191 91)(26 90 192 42)(27 41 193 89)(28 88 194 60)(29 59 195 87)(30 86 196 58)(31 57 197 85)(32 84 198 56)(33 55 199 83)(34 82 200 54)(35 53 181 81)(36 100 182 52)(37 51 183 99)(38 98 184 50)(39 49 185 97)(40 96 186 48)(121 238 169 143)(122 142 170 237)(123 236 171 141)(124 160 172 235)(125 234 173 159)(126 158 174 233)(127 232 175 157)(128 156 176 231)(129 230 177 155)(130 154 178 229)(131 228 179 153)(132 152 180 227)(133 226 161 151)(134 150 162 225)(135 224 163 149)(136 148 164 223)(137 222 165 147)(138 146 166 221)(139 240 167 145)(140 144 168 239)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 4G 5A 5B 6A ··· 6F 6G 6H 6I 6J 6K 6L 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 12E ··· 12L 12M 12N 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 6 6 10 ··· 10 10 10 10 10 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 2 2 20 1 1 4 4 10 10 10 10 20 2 2 1 ··· 1 2 2 2 2 20 20 2 ··· 2 4 4 4 4 4 4 4 4 10 ··· 10 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 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D5 C4○D4 D10 D10 C3×D4 C3×D5 D20 C3×C4○D4 C6×D5 C6×D5 C3×D20 D4⋊2D5 C3×D4⋊2D5 kernel C3×C22.D20 C3×C4⋊Dic5 C3×D10⋊C4 C15×C22⋊C4 C2×C6×Dic5 C6×C5⋊D4 C22.D20 C4⋊Dic5 D10⋊C4 C5×C22⋊C4 C22×Dic5 C2×C5⋊D4 C2×C30 C3×C22⋊C4 C30 C2×C12 C22×C6 C2×C10 C22⋊C4 C2×C6 C10 C2×C4 C23 C22 C6 C2 # reps 1 2 2 1 1 1 2 4 4 2 2 2 2 2 4 4 2 4 4 8 8 8 4 16 4 8

Matrix representation of C3×C22.D20 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 20 60
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 2 29 0 0 32 7 0 0 0 0 1 6 0 0 0 60
,
 33 25 0 0 37 28 0 0 0 0 50 0 0 0 24 11
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[2,32,0,0,29,7,0,0,0,0,1,0,0,0,6,60],[33,37,0,0,25,28,0,0,0,0,50,24,0,0,0,11] >;

C3×C22.D20 in GAP, Magma, Sage, TeX

C_3\times C_2^2.D_{20}
% in TeX

G:=Group("C3xC2^2.D20");
// GroupNames label

G:=SmallGroup(480,679);
// by ID

G=gap.SmallGroup(480,679);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,555,394,18822]);
// Polycyclic

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

׿
×
𝔽