Copied to
clipboard

G = (C4×Dic3)⋊D5order 480 = 25·3·5

10th semidirect product of C4×Dic3 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C4×Dic3)⋊10D5, (C2×C20).259D6, C32(C422D5), Dic155C47C2, C30.Q85C2, (C22×D5).8D6, C157(C422C2), (Dic3×C20)⋊21C2, D10⋊C4.1S3, C30.29(C4○D4), C6.66(C4○D20), (C2×C12).180D10, (C2×C30).53C23, C30.4Q829C2, (C2×Dic5).12D6, C51(C23.8D6), C10.27(C4○D12), C2.9(D205S3), (C2×C60).403C22, D10⋊Dic3.7C2, C10.21(D42S3), (C2×Dic3).138D10, (C6×Dic5).31C22, C2.16(D6.D10), C2.14(Dic5.D6), (C2×Dic15).55C22, (C10×Dic3).162C22, (C2×C4).71(S3×D5), (D5×C2×C6).6C22, C22.140(C2×S3×D5), (C2×C6).65(C22×D5), (C2×C10).65(C22×S3), (C3×D10⋊C4).14C2, SmallGroup(480,439)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C4×Dic3)⋊D5
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — (C4×Dic3)⋊D5
C15C2×C30 — (C4×Dic3)⋊D5
C1C22C2×C4

Generators and relations for (C4×Dic3)⋊D5
 G = < a,b,c,d,e | a4=b6=d5=e2=1, c2=b3, ab=ba, ac=ca, ad=da, eae=ab3, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a2b3c, ede=d-1 >

Subgroups: 556 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4, C2×C4 [×5], C23, D5, C10 [×3], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12, C22×C6, C3×D5, C30 [×3], C422C2, C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60, C6×D5 [×3], C2×C30, C10.D4 [×3], D10⋊C4, D10⋊C4 [×2], C4×C20, C23.8D6, C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, C422D5, D10⋊Dic3 [×2], C30.Q8, Dic155C4, C3×D10⋊C4, Dic3×C20, C30.4Q8, (C4×Dic3)⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3 [×2], S3×D5, C4○D20 [×3], C23.8D6, C2×S3×D5, C422D5, D205S3, D6.D10, Dic5.D6, (C4×Dic3)⋊D5

Smallest permutation representation of (C4×Dic3)⋊D5
On 240 points
Generators in S240
(1 119 59 86)(2 120 60 87)(3 115 55 88)(4 116 56 89)(5 117 57 90)(6 118 58 85)(7 202 229 169)(8 203 230 170)(9 204 231 171)(10 199 232 172)(11 200 233 173)(12 201 234 174)(13 198 225 165)(14 193 226 166)(15 194 227 167)(16 195 228 168)(17 196 223 163)(18 197 224 164)(19 188 221 161)(20 189 222 162)(21 190 217 157)(22 191 218 158)(23 192 219 159)(24 187 220 160)(25 184 211 151)(26 185 212 152)(27 186 213 153)(28 181 214 154)(29 182 215 155)(30 183 216 156)(31 129 64 91)(32 130 65 92)(33 131 66 93)(34 132 61 94)(35 127 62 95)(36 128 63 96)(37 125 70 97)(38 126 71 98)(39 121 72 99)(40 122 67 100)(41 123 68 101)(42 124 69 102)(43 136 76 103)(44 137 77 104)(45 138 78 105)(46 133 73 106)(47 134 74 107)(48 135 75 108)(49 142 82 109)(50 143 83 110)(51 144 84 111)(52 139 79 112)(53 140 80 113)(54 141 81 114)(145 238 178 205)(146 239 179 206)(147 240 180 207)(148 235 175 208)(149 236 176 209)(150 237 177 210)
(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)
(1 178 4 175)(2 177 5 180)(3 176 6 179)(7 111 10 114)(8 110 11 113)(9 109 12 112)(13 105 16 108)(14 104 17 107)(15 103 18 106)(19 94 22 91)(20 93 23 96)(21 92 24 95)(25 98 28 101)(26 97 29 100)(27 102 30 99)(31 188 34 191)(32 187 35 190)(33 192 36 189)(37 182 40 185)(38 181 41 184)(39 186 42 183)(43 197 46 194)(44 196 47 193)(45 195 48 198)(49 201 52 204)(50 200 53 203)(51 199 54 202)(55 149 58 146)(56 148 59 145)(57 147 60 150)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 152 70 155)(68 151 71 154)(69 156 72 153)(73 167 76 164)(74 166 77 163)(75 165 78 168)(79 171 82 174)(80 170 83 173)(81 169 84 172)(85 239 88 236)(86 238 89 235)(87 237 90 240)(115 209 118 206)(116 208 119 205)(117 207 120 210)(121 213 124 216)(122 212 125 215)(123 211 126 214)(127 217 130 220)(128 222 131 219)(129 221 132 218)(133 227 136 224)(134 226 137 223)(135 225 138 228)(139 231 142 234)(140 230 143 233)(141 229 144 232)
(1 36 44 53 38)(2 31 45 54 39)(3 32 46 49 40)(4 33 47 50 41)(5 34 48 51 42)(6 35 43 52 37)(7 27 237 19 16)(8 28 238 20 17)(9 29 239 21 18)(10 30 240 22 13)(11 25 235 23 14)(12 26 236 24 15)(55 65 73 82 67)(56 66 74 83 68)(57 61 75 84 69)(58 62 76 79 70)(59 63 77 80 71)(60 64 78 81 72)(85 95 103 112 97)(86 96 104 113 98)(87 91 105 114 99)(88 92 106 109 100)(89 93 107 110 101)(90 94 108 111 102)(115 130 133 142 122)(116 131 134 143 123)(117 132 135 144 124)(118 127 136 139 125)(119 128 137 140 126)(120 129 138 141 121)(145 162 163 170 154)(146 157 164 171 155)(147 158 165 172 156)(148 159 166 173 151)(149 160 167 174 152)(150 161 168 169 153)(175 192 193 200 184)(176 187 194 201 185)(177 188 195 202 186)(178 189 196 203 181)(179 190 197 204 182)(180 191 198 199 183)(205 222 223 230 214)(206 217 224 231 215)(207 218 225 232 216)(208 219 226 233 211)(209 220 227 234 212)(210 221 228 229 213)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 221)(8 222)(9 217)(10 218)(11 219)(12 220)(13 225)(14 226)(15 227)(16 228)(17 223)(18 224)(19 229)(20 230)(21 231)(22 232)(23 233)(24 234)(25 208)(26 209)(27 210)(28 205)(29 206)(30 207)(31 54)(32 49)(33 50)(34 51)(35 52)(36 53)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 84)(62 79)(63 80)(64 81)(65 82)(66 83)(85 100)(86 101)(87 102)(88 97)(89 98)(90 99)(91 111)(92 112)(93 113)(94 114)(95 109)(96 110)(103 106)(104 107)(105 108)(115 125)(116 126)(117 121)(118 122)(119 123)(120 124)(127 142)(128 143)(129 144)(130 139)(131 140)(132 141)(133 136)(134 137)(135 138)(145 184)(146 185)(147 186)(148 181)(149 182)(150 183)(151 178)(152 179)(153 180)(154 175)(155 176)(156 177)(157 201)(158 202)(159 203)(160 204)(161 199)(162 200)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 191)(170 192)(171 187)(172 188)(173 189)(174 190)(211 235)(212 236)(213 237)(214 238)(215 239)(216 240)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A···20H20I···20X30A···30F60A···60H
order122223444444444556666610···1012121212151520···2020···2030···3060···60
size11112022266662060602222220202···2442020442···26···64···44···4

72 irreducible representations

dim11111112222222222444444
type++++++++++++++-++-
imageC1C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10C4○D12C4○D20D42S3S3×D5C2×S3×D5D205S3D6.D10Dic5.D6
kernel(C4×Dic3)⋊D5D10⋊Dic3C30.Q8Dic155C4C3×D10⋊C4Dic3×C20C30.4Q8D10⋊C4C4×Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C10C6C10C2×C4C22C2C2C2
# reps121111112111642424222444

Matrix representation of (C4×Dic3)⋊D5 in GL4(𝔽61) generated by

36700
502500
00500
00050
,
60000
06000
001319
00047
,
11000
01100
0083
004053
,
18100
426000
0010
0001
,
606000
0100
00124
00060
G:=sub<GL(4,GF(61))| [36,50,0,0,7,25,0,0,0,0,50,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,13,0,0,0,19,47],[11,0,0,0,0,11,0,0,0,0,8,40,0,0,3,53],[18,42,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,24,60] >;

(C4×Dic3)⋊D5 in GAP, Magma, Sage, TeX

(C_4\times {\rm Dic}_3)\rtimes D_5
% in TeX

G:=Group("(C4xDic3):D5");
// GroupNames label

G:=SmallGroup(480,439);
// by ID

G=gap.SmallGroup(480,439);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^6=d^5=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^3,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*b^3*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽