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## G = C2×Dic5.D6order 480 = 25·3·5

### Direct product of C2 and Dic5.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Dic5.D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C2×Dic5.D6
 Lower central C15 — C30 — C2×Dic5.D6
 Upper central C1 — C22 — C23

Generators and relations for C2×Dic5.D6
G = < a,b,c,d,e | a2=b10=d6=1, c2=e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=ece-1=b5c, ede-1=d-1 >

Subgroups: 1532 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×2], C22 [×10], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×4], D10 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×4], C2×Dic3 [×5], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6, C22×C6, C3×D5 [×2], D15 [×2], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C5⋊D4 [×4], C2×C20 [×6], C22×D5, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4, C2×C5⋊D4, C22×C20, C2×D42S3, D5×Dic3 [×4], D30.C2 [×4], C3⋊D20 [×4], C15⋊Q8 [×4], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3 [×2], C10×Dic3 [×4], C2×Dic15, C157D4 [×4], D5×C2×C6, C22×D15, C22×C30, C2×C4○D20, C2×D5×Dic3, Dic5.D6 [×8], C2×D30.C2, C2×C3⋊D20, C2×C15⋊Q8, C6×C5⋊D4, Dic3×C2×C10, C2×C157D4, C2×Dic5.D6
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], D42S3 [×2], S3×C23, S3×D5, C4○D20 [×2], C23×D5, C2×D42S3, C2×S3×D5 [×3], C2×C4○D20, Dic5.D6 [×2], C22×S3×D5, C2×Dic5.D6

Smallest permutation representation of C2×Dic5.D6
On 240 points
Generators in S240
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 185)(12 186)(13 187)(14 188)(15 189)(16 190)(17 181)(18 182)(19 183)(20 184)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 71)(30 72)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(131 191)(132 192)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(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 51 6 56)(2 60 7 55)(3 59 8 54)(4 58 9 53)(5 57 10 52)(11 214 16 219)(12 213 17 218)(13 212 18 217)(14 211 19 216)(15 220 20 215)(21 41 26 46)(22 50 27 45)(23 49 28 44)(24 48 29 43)(25 47 30 42)(31 63 36 68)(32 62 37 67)(33 61 38 66)(34 70 39 65)(35 69 40 64)(71 103 76 108)(72 102 77 107)(73 101 78 106)(74 110 79 105)(75 109 80 104)(81 111 86 116)(82 120 87 115)(83 119 88 114)(84 118 89 113)(85 117 90 112)(91 123 96 128)(92 122 97 127)(93 121 98 126)(94 130 99 125)(95 129 100 124)(131 163 136 168)(132 162 137 167)(133 161 138 166)(134 170 139 165)(135 169 140 164)(141 171 146 176)(142 180 147 175)(143 179 148 174)(144 178 149 173)(145 177 150 172)(151 183 156 188)(152 182 157 187)(153 181 158 186)(154 190 159 185)(155 189 160 184)(191 223 196 228)(192 222 197 227)(193 221 198 226)(194 230 199 225)(195 229 200 224)(201 231 206 236)(202 240 207 235)(203 239 208 234)(204 238 209 233)(205 237 210 232)
(1 32 25)(2 31 26 10 33 24)(3 40 27 9 34 23)(4 39 28 8 35 22)(5 38 29 7 36 21)(6 37 30)(11 233 230 20 234 229)(12 232 221 19 235 228)(13 231 222 18 236 227)(14 240 223 17 237 226)(15 239 224 16 238 225)(41 52 66 48 55 63)(42 51 67 47 56 62)(43 60 68 46 57 61)(44 59 69 45 58 70)(49 54 64 50 53 65)(71 87 96 73 85 98)(72 86 97)(74 84 99 80 88 95)(75 83 100 79 89 94)(76 82 91 78 90 93)(77 81 92)(101 112 126 108 115 123)(102 111 127 107 116 122)(103 120 128 106 117 121)(104 119 129 105 118 130)(109 114 124 110 113 125)(131 158 145 133 156 147)(132 157 146)(134 155 148 140 159 144)(135 154 149 139 160 143)(136 153 150 138 151 142)(137 152 141)(161 183 175 168 186 172)(162 182 176 167 187 171)(163 181 177 166 188 180)(164 190 178 165 189 179)(169 185 173 170 184 174)(191 218 205 193 216 207)(192 217 206)(194 215 208 200 219 204)(195 214 209 199 220 203)(196 213 210 198 211 202)(197 212 201)
(1 157 6 152)(2 156 7 151)(3 155 8 160)(4 154 9 159)(5 153 10 158)(11 113 16 118)(12 112 17 117)(13 111 18 116)(14 120 19 115)(15 119 20 114)(21 150 26 145)(22 149 27 144)(23 148 28 143)(24 147 29 142)(25 146 30 141)(31 133 36 138)(32 132 37 137)(33 131 38 136)(34 140 39 135)(35 139 40 134)(41 177 46 172)(42 176 47 171)(43 175 48 180)(44 174 49 179)(45 173 50 178)(51 182 56 187)(52 181 57 186)(53 190 58 185)(54 189 59 184)(55 188 60 183)(61 168 66 163)(62 167 67 162)(63 166 68 161)(64 165 69 170)(65 164 70 169)(71 202 76 207)(72 201 77 206)(73 210 78 205)(74 209 79 204)(75 208 80 203)(81 217 86 212)(82 216 87 211)(83 215 88 220)(84 214 89 219)(85 213 90 218)(91 193 96 198)(92 192 97 197)(93 191 98 196)(94 200 99 195)(95 199 100 194)(101 237 106 232)(102 236 107 231)(103 235 108 240)(104 234 109 239)(105 233 110 238)(121 228 126 223)(122 227 127 222)(123 226 128 221)(124 225 129 230)(125 224 130 229)

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

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

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

78 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 6B 6C 6D 6E 6F 6G 10A ··· 10N 12A 12B 15A 15B 20A ··· 20P 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 6 6 6 6 6 10 ··· 10 12 12 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 10 10 30 30 2 3 3 3 3 6 6 10 10 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 20 20 4 4 6 ··· 6 4 ··· 4

78 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 D6 D6 C4○D4 D10 D10 C4○D20 D4⋊2S3 S3×D5 C2×S3×D5 Dic5.D6 kernel C2×Dic5.D6 C2×D5×Dic3 Dic5.D6 C2×D30.C2 C2×C3⋊D20 C2×C15⋊Q8 C6×C5⋊D4 Dic3×C2×C10 C2×C15⋊7D4 C2×C5⋊D4 C22×Dic3 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C30 C2×Dic3 C22×C6 C6 C10 C23 C22 C2 # reps 1 1 8 1 1 1 1 1 1 1 2 1 4 1 1 4 12 2 16 2 2 6 8

Matrix representation of C2×Dic5.D6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 60 0 0 0 0 19 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 15 0 0 0 0 20 59 0 0 0 0 0 0 43 44 0 0 0 0 19 18 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 16 60 0 0 0 0 0 0 43 44 0 0 0 0 19 18 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 11 0 0 0 0 0 54 50 0 0 0 0 0 0 43 44 0 0 0 0 19 18 0 0 0 0 0 0 60 0 0 0 0 0 60 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,19,0,0,0,0,60,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,20,0,0,0,0,15,59,0,0,0,0,0,0,43,19,0,0,0,0,44,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,16,0,0,0,0,0,60,0,0,0,0,0,0,43,19,0,0,0,0,44,18,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[11,54,0,0,0,0,0,50,0,0,0,0,0,0,43,19,0,0,0,0,44,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1] >;

C2×Dic5.D6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5.D_6
% in TeX

G:=Group("C2xDic5.D6");
// GroupNames label

G:=SmallGroup(480,1113);
// by ID

G=gap.SmallGroup(480,1113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=d^6=1,c^2=e^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=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations

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