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## G = C2×Dic3.D10order 480 = 25·3·5

### Direct product of C2 and Dic3.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Dic3.D10
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5 — C2×Dic3.D10
 Lower central C15 — C30 — C2×Dic3.D10
 Upper central C1 — C22 — C23

Generators and relations for C2×Dic3.D10
G = < a,b,c,d,e | a2=b6=d10=1, c2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 1500 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×2], C22 [×10], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×4], D6 [×2], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×6], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C3⋊D4 [×4], C2×C12 [×6], C22×S3, C22×S3, C22×C6, C5×S3 [×2], D15 [×2], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×5], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10, C22×C10, C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5, C22×Dic5, C2×C5⋊D4 [×2], D4×C10, C2×C4○D12, S3×Dic5 [×4], D30.C2 [×4], C5⋊D12 [×4], C15⋊Q8 [×4], C6×Dic5 [×2], C6×Dic5 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15, C157D4 [×4], S3×C2×C10, C22×D15, C22×C30, C2×D42D5, C2×S3×Dic5, Dic3.D10 [×8], C2×D30.C2, C2×C5⋊D12, C2×C15⋊Q8, C2×C6×Dic5, C10×C3⋊D4, C2×C157D4, C2×Dic3.D10
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, D42D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×D42D5, Dic3.D10 [×2], C22×S3×D5, C2×Dic3.D10

Smallest permutation representation of C2×Dic3.D10
On 240 points
Generators in S240
(1 27)(2 28)(3 29)(4 30)(5 26)(6 16)(7 17)(8 18)(9 19)(10 20)(11 32)(12 33)(13 34)(14 35)(15 31)(21 39)(22 40)(23 36)(24 37)(25 38)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 117)(52 118)(53 119)(54 120)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 150)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(81 168)(82 169)(83 170)(84 161)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(121 206)(122 207)(123 208)(124 209)(125 210)(126 201)(127 202)(128 203)(129 204)(130 205)(131 221)(132 222)(133 223)(134 224)(135 225)(136 226)(137 227)(138 228)(139 229)(140 230)(151 237)(152 238)(153 239)(154 240)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)(171 193)(172 194)(173 195)(174 196)(175 197)(176 198)(177 199)(178 200)(179 191)(180 192)(181 213)(182 214)(183 215)(184 216)(185 217)(186 218)(187 219)(188 220)(189 211)(190 212)
(1 194 72 38 77 199)(2 200 78 39 73 195)(3 196 74 40 79 191)(4 192 80 36 75 197)(5 198 76 37 71 193)(6 106 116 14 111 101)(7 102 112 15 117 107)(8 108 118 11 113 103)(9 104 114 12 119 109)(10 110 120 13 115 105)(16 94 60 35 55 99)(17 100 56 31 51 95)(18 96 52 32 57 91)(19 92 58 33 53 97)(20 98 54 34 59 93)(21 44 173 28 178 49)(22 50 179 29 174 45)(23 46 175 30 180 41)(24 42 171 26 176 47)(25 48 177 27 172 43)(61 81 236 231 86 66)(62 67 87 232 237 82)(63 83 238 233 88 68)(64 69 89 234 239 84)(65 85 240 235 90 70)(121 138 188 183 133 126)(122 127 134 184 189 139)(123 140 190 185 135 128)(124 129 136 186 181 131)(125 132 182 187 137 130)(141 146 164 156 151 169)(142 170 152 157 165 147)(143 148 166 158 153 161)(144 162 154 159 167 149)(145 150 168 160 155 163)(201 206 228 220 215 223)(202 224 216 211 229 207)(203 208 230 212 217 225)(204 226 218 213 221 209)(205 210 222 214 219 227)
(1 132 38 137)(2 138 39 133)(3 134 40 139)(4 140 36 135)(5 136 37 131)(6 166 14 161)(7 162 15 167)(8 168 11 163)(9 164 12 169)(10 170 13 165)(16 89 35 84)(17 85 31 90)(18 81 32 86)(19 87 33 82)(20 83 34 88)(21 223 28 228)(22 229 29 224)(23 225 30 230)(24 221 26 226)(25 227 27 222)(41 203 175 212)(42 213 176 204)(43 205 177 214)(44 215 178 206)(45 207 179 216)(46 217 180 208)(47 209 171 218)(48 219 172 210)(49 201 173 220)(50 211 174 202)(51 235 100 65)(52 66 91 236)(53 237 92 67)(54 68 93 238)(55 239 94 69)(56 70 95 240)(57 231 96 61)(58 62 97 232)(59 233 98 63)(60 64 99 234)(71 181 198 129)(72 130 199 182)(73 183 200 121)(74 122 191 184)(75 185 192 123)(76 124 193 186)(77 187 194 125)(78 126 195 188)(79 189 196 127)(80 128 197 190)(101 158 116 143)(102 144 117 159)(103 160 118 145)(104 146 119 151)(105 152 120 147)(106 148 111 153)(107 154 112 149)(108 150 113 155)(109 156 114 141)(110 142 115 157)
(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 17 38 31)(2 16 39 35)(3 20 40 34)(4 19 36 33)(5 18 37 32)(6 21 14 28)(7 25 15 27)(8 24 11 26)(9 23 12 30)(10 22 13 29)(41 119 175 104)(42 118 176 103)(43 117 177 102)(44 116 178 101)(45 115 179 110)(46 114 180 109)(47 113 171 108)(48 112 172 107)(49 111 173 106)(50 120 174 105)(51 199 100 72)(52 198 91 71)(53 197 92 80)(54 196 93 79)(55 195 94 78)(56 194 95 77)(57 193 96 76)(58 192 97 75)(59 191 98 74)(60 200 99 73)(61 186 231 124)(62 185 232 123)(63 184 233 122)(64 183 234 121)(65 182 235 130)(66 181 236 129)(67 190 237 128)(68 189 238 127)(69 188 239 126)(70 187 240 125)(81 136 86 131)(82 135 87 140)(83 134 88 139)(84 133 89 138)(85 132 90 137)(141 217 156 208)(142 216 157 207)(143 215 158 206)(144 214 159 205)(145 213 160 204)(146 212 151 203)(147 211 152 202)(148 220 153 201)(149 219 154 210)(150 218 155 209)(161 223 166 228)(162 222 167 227)(163 221 168 226)(164 230 169 225)(165 229 170 224)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D10 D10 C4○D12 S3×D5 D4⋊2D5 C2×S3×D5 Dic3.D10 kernel C2×Dic3.D10 C2×S3×Dic5 Dic3.D10 C2×D30.C2 C2×C5⋊D12 C2×C15⋊Q8 C2×C6×Dic5 C10×C3⋊D4 C2×C15⋊7D4 C22×Dic5 C2×C3⋊D4 C2×Dic5 C22×C10 C30 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C10 C23 C6 C22 C2 # reps 1 1 8 1 1 1 1 1 1 1 2 6 1 4 2 8 2 2 8 2 4 6 8

Matrix representation of C2×Dic3.D10 in GL6(𝔽61)

 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
,
 1 1 0 0 0 0 60 0 0 0 0 0 0 0 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 52 43 0 0 0 0 52 9 0 0 0 0 0 0 60 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 17 43 0 0 0 0 17 0
,
 11 0 0 0 0 0 50 50 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 60 1 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [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],[1,60,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,52,0,0,0,0,43,9,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,17,17,0,0,0,0,43,0],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,1,1] >;

C2×Dic3.D10 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.D_{10}
% in TeX

G:=Group("C2xDic3.D10");
// GroupNames label

G:=SmallGroup(480,1116);
// by ID

G=gap.SmallGroup(480,1116);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,346,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^10=1,c^2=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations

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