Copied to
clipboard

G = C6×D42D5order 480 = 25·3·5

Direct product of C6 and D42D5

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

Aliases: C6×D42D5, C30.74C24, C60.209C23, D45(C6×D5), (D4×C10)⋊6C6, (C6×D4)⋊17D5, (C3×D4)⋊27D10, (D4×C30)⋊13C2, C3015(C4○D4), Dic107(C2×C6), C10.6(C23×C6), C23.24(C6×D5), C6.74(C23×D5), (C6×Dic10)⋊28C2, (C2×Dic10)⋊12C6, (C2×C12).370D10, (D4×C15)⋊30C22, (D5×C12)⋊23C22, C20.20(C22×C6), (C6×D5).53C23, D10.2(C22×C6), (C22×C6).78D10, (C2×C60).305C22, (C2×C30).383C23, (C6×Dic5)⋊36C22, (C22×Dic5)⋊11C6, C12.209(C22×D5), Dic5.3(C22×C6), (C3×Dic10)⋊34C22, (C3×Dic5).55C23, (C22×C30).109C22, (C2×C4×D5)⋊4C6, C52(C6×C4○D4), C4.20(D5×C2×C6), (D5×C2×C12)⋊14C2, (C2×D4)⋊8(C3×D5), (C4×D5)⋊4(C2×C6), (C5×D4)⋊6(C2×C6), C102(C3×C4○D4), C1524(C2×C4○D4), C5⋊D42(C2×C6), (C2×C5⋊D4)⋊10C6, (C6×C5⋊D4)⋊25C2, C22.1(D5×C2×C6), C2.7(D5×C22×C6), (C2×C6×Dic5)⋊19C2, (C2×C4).60(C6×D5), (C2×C20).42(C2×C6), (C2×Dic5)⋊9(C2×C6), (C3×C5⋊D4)⋊18C22, (C2×C10).1(C22×C6), (D5×C2×C6).139C22, (C2×C6).20(C22×D5), (C22×C10).28(C2×C6), (C22×D5).34(C2×C6), SmallGroup(480,1140)

Series: Derived Chief Lower central Upper central

C1C10 — C6×D42D5
C1C5C10C30C6×D5D5×C2×C6D5×C2×C12 — C6×D42D5
C5C10 — C6×D42D5
C1C2×C6C6×D4

Generators and relations for C6×D42D5
 G = < a,b,c,d,e | a6=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 848 in 328 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×4], C12 [×2], C12 [×6], C2×C6, C2×C6 [×4], C2×C6 [×8], C15, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12, C2×C12 [×15], C3×D4 [×4], C3×D4 [×8], C3×Q8 [×4], C22×C6 [×2], C22×C6, C3×D5 [×2], C30, C30 [×2], C30 [×4], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C22×C12 [×3], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×8], C3×Dic5 [×6], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C6×C4○D4, C3×Dic10 [×4], D5×C12 [×4], C6×Dic5, C6×Dic5 [×10], C3×C5⋊D4 [×8], C2×C60, D4×C15 [×4], D5×C2×C6, C22×C30 [×2], C2×D42D5, C6×Dic10, D5×C2×C12, C3×D42D5 [×8], C2×C6×Dic5 [×2], C6×C5⋊D4 [×2], D4×C30, C6×D42D5
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C4○D4 [×2], C24, D10 [×7], C22×C6 [×15], C3×D5, C2×C4○D4, C22×D5 [×7], C3×C4○D4 [×2], C23×C6, C6×D5 [×7], D42D5 [×2], C23×D5, C6×C4○D4, D5×C2×C6 [×7], C2×D42D5, C3×D42D5 [×2], D5×C22×C6, C6×D42D5

Smallest permutation representation of C6×D42D5
On 240 points
Generators in S240
(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)
(1 151 91 113)(2 152 92 114)(3 153 93 109)(4 154 94 110)(5 155 95 111)(6 156 96 112)(7 226 200 140)(8 227 201 141)(9 228 202 142)(10 223 203 143)(11 224 204 144)(12 225 199 139)(13 165 105 79)(14 166 106 80)(15 167 107 81)(16 168 108 82)(17 163 103 83)(18 164 104 84)(19 198 188 128)(20 193 189 129)(21 194 190 130)(22 195 191 131)(23 196 192 132)(24 197 187 127)(25 135 75 71)(26 136 76 72)(27 137 77 67)(28 138 78 68)(29 133 73 69)(30 134 74 70)(31 214 230 170)(32 215 231 171)(33 216 232 172)(34 211 233 173)(35 212 234 174)(36 213 229 169)(37 206 219 159)(38 207 220 160)(39 208 221 161)(40 209 222 162)(41 210 217 157)(42 205 218 158)(43 149 89 102)(44 150 90 97)(45 145 85 98)(46 146 86 99)(47 147 87 100)(48 148 88 101)(49 182 236 176)(50 183 237 177)(51 184 238 178)(52 185 239 179)(53 186 240 180)(54 181 235 175)(55 121 61 115)(56 122 62 116)(57 123 63 117)(58 124 64 118)(59 125 65 119)(60 126 66 120)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 109)(7 143)(8 144)(9 139)(10 140)(11 141)(12 142)(13 82)(14 83)(15 84)(16 79)(17 80)(18 81)(19 131)(20 132)(21 127)(22 128)(23 129)(24 130)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 173)(32 174)(33 169)(34 170)(35 171)(36 172)(37 162)(38 157)(39 158)(40 159)(41 160)(42 161)(43 99)(44 100)(45 101)(46 102)(47 97)(48 98)(49 179)(50 180)(51 175)(52 176)(53 177)(54 178)(55 118)(56 119)(57 120)(58 115)(59 116)(60 117)(61 124)(62 125)(63 126)(64 121)(65 122)(66 123)(73 136)(74 137)(75 138)(76 133)(77 134)(78 135)(85 148)(86 149)(87 150)(88 145)(89 146)(90 147)(91 154)(92 155)(93 156)(94 151)(95 152)(96 153)(103 166)(104 167)(105 168)(106 163)(107 164)(108 165)(181 238)(182 239)(183 240)(184 235)(185 236)(186 237)(187 194)(188 195)(189 196)(190 197)(191 198)(192 193)(199 228)(200 223)(201 224)(202 225)(203 226)(204 227)(205 221)(206 222)(207 217)(208 218)(209 219)(210 220)(211 230)(212 231)(213 232)(214 233)(215 234)(216 229)
(1 103 29 65 45)(2 104 30 66 46)(3 105 25 61 47)(4 106 26 62 48)(5 107 27 63 43)(6 108 28 64 44)(7 230 221 49 192)(8 231 222 50 187)(9 232 217 51 188)(10 233 218 52 189)(11 234 219 53 190)(12 229 220 54 191)(13 75 55 87 93)(14 76 56 88 94)(15 77 57 89 95)(16 78 58 90 96)(17 73 59 85 91)(18 74 60 86 92)(19 202 33 41 238)(20 203 34 42 239)(21 204 35 37 240)(22 199 36 38 235)(23 200 31 39 236)(24 201 32 40 237)(67 123 102 111 167)(68 124 97 112 168)(69 125 98 113 163)(70 126 99 114 164)(71 121 100 109 165)(72 122 101 110 166)(79 135 115 147 153)(80 136 116 148 154)(81 137 117 149 155)(82 138 118 150 156)(83 133 119 145 151)(84 134 120 146 152)(127 227 171 162 183)(128 228 172 157 184)(129 223 173 158 185)(130 224 174 159 186)(131 225 169 160 181)(132 226 170 161 182)(139 213 207 175 195)(140 214 208 176 196)(141 215 209 177 197)(142 216 210 178 198)(143 211 205 179 193)(144 212 206 180 194)
(1 223)(2 224)(3 225)(4 226)(5 227)(6 228)(7 110)(8 111)(9 112)(10 113)(11 114)(12 109)(13 195)(14 196)(15 197)(16 198)(17 193)(18 194)(19 82)(20 83)(21 84)(22 79)(23 80)(24 81)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 148)(32 149)(33 150)(34 145)(35 146)(36 147)(37 120)(38 115)(39 116)(40 117)(41 118)(42 119)(43 171)(44 172)(45 173)(46 174)(47 169)(48 170)(49 72)(50 67)(51 68)(52 69)(53 70)(54 71)(55 207)(56 208)(57 209)(58 210)(59 205)(60 206)(61 160)(62 161)(63 162)(64 157)(65 158)(66 159)(73 179)(74 180)(75 175)(76 176)(77 177)(78 178)(85 211)(86 212)(87 213)(88 214)(89 215)(90 216)(91 143)(92 144)(93 139)(94 140)(95 141)(96 142)(97 232)(98 233)(99 234)(100 229)(101 230)(102 231)(103 129)(104 130)(105 131)(106 132)(107 127)(108 128)(121 220)(122 221)(123 222)(124 217)(125 218)(126 219)(133 239)(134 240)(135 235)(136 236)(137 237)(138 238)(151 203)(152 204)(153 199)(154 200)(155 201)(156 202)(163 189)(164 190)(165 191)(166 192)(167 187)(168 188)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B4A4B4C4D4E4F4G4H4I4J5A5B6A···6F6G···6N6O6P6Q6R10A···10F10G···10N12A12B12C12D12E···12L12M···12T15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order1222222222334444444444556···66···6666610···1010···101212121212···1212···12151515152020202030···3030···3060···60
size1111222210101122555510101010221···12···2101010102···24···422225···510···10222244442···24···44···4

120 irreducible representations

dim11111111111111222222222244
type+++++++++++-
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6D5C4○D4D10D10D10C3×D5C3×C4○D4C6×D5C6×D5C6×D5D42D5C3×D42D5
kernelC6×D42D5C6×Dic10D5×C2×C12C3×D42D5C2×C6×Dic5C6×C5⋊D4D4×C30C2×D42D5C2×Dic10C2×C4×D5D42D5C22×Dic5C2×C5⋊D4D4×C10C6×D4C30C2×C12C3×D4C22×C6C2×D4C10C2×C4D4C23C6C2
# reps1118221222164422428448416848

Matrix representation of C6×D42D5 in GL5(𝔽61)

600000
013000
001300
000140
000014
,
10000
01000
00100
000159
000160
,
10000
01000
00100
000602
00001
,
10000
0176000
01000
00010
00001
,
600000
0555300
012600
0005022
0005011

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,14,0,0,0,0,0,14],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,59,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,2,1],[1,0,0,0,0,0,17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,55,12,0,0,0,53,6,0,0,0,0,0,50,50,0,0,0,22,11] >;

C6×D42D5 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes_2D_5
% in TeX

G:=Group("C6xD4:2D5");
// GroupNames label

G:=SmallGroup(480,1140);
// by ID

G=gap.SmallGroup(480,1140);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,409,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽