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## G = C3×Dic5⋊4D4order 480 = 25·3·5

### Direct product of C3 and Dic5⋊4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Dic5⋊4D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×C5⋊D4 — C3×Dic5⋊4D4
 Lower central C5 — C10 — C3×Dic5⋊4D4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×Dic54D4
G = < a,b,c,d,e | a3=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 560 in 188 conjugacy classes, 86 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C12 [×7], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×7], C3×D4 [×4], C22×C6, C22×C6, C3×D5 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12 [×2], C6×D4, C3×Dic5 [×4], C3×Dic5, C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C12, D5×C12 [×2], C6×Dic5 [×3], C6×Dic5 [×2], C3×C5⋊D4 [×4], C2×C60 [×2], D5×C2×C6, C22×C30, Dic54D4, C12×Dic5, C3×C10.D4, C3×D10⋊C4, C15×C22⋊C4, D5×C2×C12, C2×C6×Dic5, C6×C5⋊D4, C3×Dic54D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, D10 [×3], C2×C12 [×6], C3×D4 [×2], C22×C6, C3×D5, C4×D4, C4×D5 [×2], C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5 [×3], C2×C4×D5, D4×D5, D42D5, D4×C12, D5×C12 [×2], D5×C2×C6, Dic54D4, D5×C2×C12, C3×D4×D5, C3×D42D5, C3×Dic54D4

Smallest permutation representation of C3×Dic54D4
On 240 points
Generators in S240
(1 52 32)(2 53 33)(3 54 34)(4 55 35)(5 56 36)(6 57 37)(7 58 38)(8 59 39)(9 60 40)(10 51 31)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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 77 6 72)(2 76 7 71)(3 75 8 80)(4 74 9 79)(5 73 10 78)(11 189 16 184)(12 188 17 183)(13 187 18 182)(14 186 19 181)(15 185 20 190)(21 88 26 83)(22 87 27 82)(23 86 28 81)(24 85 29 90)(25 84 30 89)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 95 39 100)(35 94 40 99)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 105 49 110)(45 104 50 109)(51 118 56 113)(52 117 57 112)(53 116 58 111)(54 115 59 120)(55 114 60 119)(61 128 66 123)(62 127 67 122)(63 126 68 121)(64 125 69 130)(65 124 70 129)(131 198 136 193)(132 197 137 192)(133 196 138 191)(134 195 139 200)(135 194 140 199)(141 208 146 203)(142 207 147 202)(143 206 148 201)(144 205 149 210)(145 204 150 209)(151 218 156 213)(152 217 157 212)(153 216 158 211)(154 215 159 220)(155 214 160 219)(161 228 166 223)(162 227 167 222)(163 226 168 221)(164 225 169 230)(165 224 170 229)(171 238 176 233)(172 237 177 232)(173 236 178 231)(174 235 179 240)(175 234 180 239)
(1 147 27 132)(2 146 28 131)(3 145 29 140)(4 144 30 139)(5 143 21 138)(6 142 22 137)(7 141 23 136)(8 150 24 135)(9 149 25 134)(10 148 26 133)(11 129 240 114)(12 128 231 113)(13 127 232 112)(14 126 233 111)(15 125 234 120)(16 124 235 119)(17 123 236 118)(18 122 237 117)(19 121 238 116)(20 130 239 115)(31 168 46 153)(32 167 47 152)(33 166 48 151)(34 165 49 160)(35 164 50 159)(36 163 41 158)(37 162 42 157)(38 161 43 156)(39 170 44 155)(40 169 45 154)(51 188 66 173)(52 187 67 172)(53 186 68 171)(54 185 69 180)(55 184 70 179)(56 183 61 178)(57 182 62 177)(58 181 63 176)(59 190 64 175)(60 189 65 174)(71 208 86 193)(72 207 87 192)(73 206 88 191)(74 205 89 200)(75 204 90 199)(76 203 81 198)(77 202 82 197)(78 201 83 196)(79 210 84 195)(80 209 85 194)(91 228 106 213)(92 227 107 212)(93 226 108 211)(94 225 109 220)(95 224 110 219)(96 223 101 218)(97 222 102 217)(98 221 103 216)(99 230 104 215)(100 229 105 214)
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 139)(9 140)(10 131)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 121)(18 122)(19 123)(20 124)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)

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

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

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

120 conjugacy classes

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

120 irreducible representations

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

Matrix representation of C3×Dic54D4 in GL5(𝔽61)

 1 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 13 0 0 0 0 0 13
,
 60 0 0 0 0 0 0 60 0 0 0 1 43 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 0 0 0 0 0 18 43 0 0 0 1 43 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 43 18 0 0 0 60 18 0 0 0 0 0 0 1 0 0 0 60 0
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,13,0,0,0,0,0,13],[60,0,0,0,0,0,0,1,0,0,0,60,43,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,18,1,0,0,0,43,43,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,43,60,0,0,0,18,18,0,0,0,0,0,0,60,0,0,0,1,0],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×Dic54D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5\rtimes_4D_4
% in TeX

G:=Group("C3xDic5:4D4");
// GroupNames label

G:=SmallGroup(480,674);
// by ID

G=gap.SmallGroup(480,674);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,555,142,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=d^4=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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