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G = C2×Dic3.D10  order 480 = 25·3·5

Direct product of C2 and Dic3.D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic3.D10, C30.44C24, D30.22C23, Dic15.24C23, C3⋊D4⋊14D10, C15⋊Q8⋊17C22, C10⋊4(C4○D12), C30⋊10(C4○D4), C6⋊3(D4⋊2D5), (C2×Dic5)⋊23D6, C23.47(S3×D5), C6.44(C23×D5), C5⋊D12⋊18C22, C15⋊7D4⋊18C22, C10.44(S3×C23), D6.20(C22×D5), (C22×C10).84D6, (C22×C6).99D10, (S3×C10).22C23, (C2×C30).247C23, D30.C2⋊16C22, (C22×Dic5)⋊11S3, (S3×Dic5)⋊16C22, (C6×Dic5)⋊29C22, (C22×S3).62D10, (C2×Dic3).133D10, (C22×C30).85C22, (C5×Dic3).23C23, Dic5.46(C22×S3), Dic3.22(C22×D5), (C3×Dic5).51C23, (C22×D15).76C22, (C10×Dic3).133C22, (C2×Dic15).154C22, C5⋊5(C2×C4○D12), C15⋊19(C2×C4○D4), C3⋊4(C2×D4⋊2D5), (C2×C15⋊Q8)⋊25C2, (C2×C6×Dic5)⋊9C2, (C2×C3⋊D4)⋊14D5, (C2×S3×Dic5)⋊23C2, (C2×C15⋊7D4)⋊23C2, (C10×C3⋊D4)⋊12C2, (C2×C5⋊D12)⋊21C2, C2.45(C22×S3×D5), C22.11(C2×S3×D5), (C2×D30.C2)⋊23C2, (C5×C3⋊D4)⋊13C22, (S3×C2×C10).62C22, (C2×C10).15(C22×S3), (C2×C6).253(C22×D5), SmallGroup(480,1116)

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

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, D4⋊2D5 [×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, C15⋊7D4 [×4], S3×C2×C10, C22×D15, C22×C30, C2×D4⋊2D5, C2×S3×Dic5, Dic3.D10 [×8], C2×D30.C2, C2×C5⋊D12, C2×C15⋊Q8, C2×C6×Dic5, C10×C3⋊D4, C2×C15⋊7D4, 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, D4⋊2D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×D4⋊2D5, Dic3.D10 [×2], C22×S3×D5, C2×Dic3.D10

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽61)

100000
010000
0060000
0006000
000010
000001
,
110000
6000000
00606000
001000
000010
000001
,
52430000
5290000
0060000
001100
000010
000001
,
100000
60600000
001000
00606000
00001743
0000170
,
1100000
50500000
001000
00606000
0000601
000001

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] >;
 

72 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A···6G10A···10F10G10H10I10J10K10L10M10N12A···12H15A15B20A20B20C20D30A···30N
order122222222234444444444556···610···10101010101010101012···1215152020202030···30
size111122663030255556610103030222···22···244441212121210···1044121212124···4

72 irreducible representations

dim11111111122222222224444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2S3D5D6D6C4○D4D10D10D10D10C4○D12S3×D5D4⋊2D5C2×S3×D5Dic3.D10
kernelC2×Dic3.D10C2×S3×Dic5Dic3.D10C2×D30.C2C2×C5⋊D12C2×C15⋊Q8C2×C6×Dic5C10×C3⋊D4C2×C15⋊7D4C22×Dic5C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C3⋊D4C22×S3C22×C6C10C23C6C22C2
# reps11811111112614282282468

In GAP, Magma, Sage, TeX 

C_2\times 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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