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G = Dic54D12order 480 = 25·3·5

1st semidirect product of Dic5 and D12 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic54D12, C157(C4×D4), C53(C4×D12), D61(C4×D5), D6⋊C420D5, D309(C2×C4), C5⋊D123C4, C2.1(D5×D12), C6.12(D4×D5), Dic55(C4×S3), (C3×Dic5)⋊6D4, C30.35(C2×D4), (C4×Dic5)⋊15S3, (C2×C20).196D6, C10.12(C2×D12), D303C425C2, (C12×Dic5)⋊25C2, (C2×C12).265D10, C31(Dic54D4), (C2×C30).95C23, C30.56(C22×C4), C30.Q815C2, C30.116(C4○D4), C10.69(C4○D12), C6.44(D42D5), (C2×C60).388C22, (C2×Dic5).174D6, (C2×Dic3).96D10, (C22×S3).34D10, C2.4(Dic3.D10), (C2×Dic15).76C22, (C6×Dic5).199C22, (C10×Dic3).55C22, (C22×D15).30C22, C6.24(C2×C4×D5), C2.26(C4×S3×D5), C10.57(S3×C2×C4), (C2×S3×Dic5)⋊1C2, (S3×C10)⋊8(C2×C4), (C5×D6⋊C4)⋊25C2, C22.50(C2×S3×D5), (C2×D30.C2)⋊3C2, (C2×C4).128(S3×D5), (C2×C5⋊D12).5C2, (C3×Dic5)⋊15(C2×C4), (S3×C2×C10).12C22, (C2×C6).107(C22×D5), (C2×C10).107(C22×S3), SmallGroup(480,481)

Series: Derived Chief Lower central Upper central

C1C30 — Dic54D12
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — Dic54D12
C15C30 — Dic54D12
C1C22C2×C4

Generators and relations for Dic54D12
 G = < a,b,c,d | a10=c12=d2=1, b2=a5, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1004 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], C12 [×5], D6 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], Dic5, C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, S3×C2×C4 [×2], C2×D12, C5×Dic3, C3×Dic5 [×4], Dic15, C60, S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×D12, S3×Dic5 [×2], D30.C2 [×2], C5⋊D12 [×4], C6×Dic5 [×2], C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic54D4, C30.Q8, C12×Dic5, C5×D6⋊C4, D303C4, C2×S3×Dic5, C2×D30.C2, C2×C5⋊D12, Dic54D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], D12 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, D4×D5, D42D5, C4×D12, C2×S3×D5, Dic54D4, C4×S3×D5, D5×D12, Dic3.D10, Dic54D12

Smallest permutation representation of Dic54D12
On 240 points
Generators in S240
(1 79 209 60 106 27 72 203 139 43)(2 44 140 204 61 28 107 49 210 80)(3 81 211 50 108 29 62 193 141 45)(4 46 142 194 63 30 97 51 212 82)(5 83 213 52 98 31 64 195 143 47)(6 48 144 196 65 32 99 53 214 84)(7 73 215 54 100 33 66 197 133 37)(8 38 134 198 67 34 101 55 216 74)(9 75 205 56 102 35 68 199 135 39)(10 40 136 200 69 36 103 57 206 76)(11 77 207 58 104 25 70 201 137 41)(12 42 138 202 71 26 105 59 208 78)(13 151 132 115 173 182 237 163 86 217)(14 218 87 164 238 183 174 116 121 152)(15 153 122 117 175 184 239 165 88 219)(16 220 89 166 240 185 176 118 123 154)(17 155 124 119 177 186 229 167 90 221)(18 222 91 168 230 187 178 120 125 156)(19 145 126 109 179 188 231 157 92 223)(20 224 93 158 232 189 180 110 127 146)(21 147 128 111 169 190 233 159 94 225)(22 226 95 160 234 191 170 112 129 148)(23 149 130 113 171 192 235 161 96 227)(24 228 85 162 236 181 172 114 131 150)
(1 124 27 167)(2 125 28 168)(3 126 29 157)(4 127 30 158)(5 128 31 159)(6 129 32 160)(7 130 33 161)(8 131 34 162)(9 132 35 163)(10 121 36 164)(11 122 25 165)(12 123 26 166)(13 199 182 205)(14 200 183 206)(15 201 184 207)(16 202 185 208)(17 203 186 209)(18 204 187 210)(19 193 188 211)(20 194 189 212)(21 195 190 213)(22 196 191 214)(23 197 192 215)(24 198 181 216)(37 113 100 96)(38 114 101 85)(39 115 102 86)(40 116 103 87)(41 117 104 88)(42 118 105 89)(43 119 106 90)(44 120 107 91)(45 109 108 92)(46 110 97 93)(47 111 98 94)(48 112 99 95)(49 222 140 178)(50 223 141 179)(51 224 142 180)(52 225 143 169)(53 226 144 170)(54 227 133 171)(55 228 134 172)(56 217 135 173)(57 218 136 174)(58 219 137 175)(59 220 138 176)(60 221 139 177)(61 230 80 156)(62 231 81 145)(63 232 82 146)(64 233 83 147)(65 234 84 148)(66 235 73 149)(67 236 74 150)(68 237 75 151)(69 238 76 152)(70 239 77 153)(71 240 78 154)(72 229 79 155)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 36)(13 221)(14 220)(15 219)(16 218)(17 217)(18 228)(19 227)(20 226)(21 225)(22 224)(23 223)(24 222)(37 62)(38 61)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 216)(50 215)(51 214)(52 213)(53 212)(54 211)(55 210)(56 209)(57 208)(58 207)(59 206)(60 205)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 156)(86 155)(87 154)(88 153)(89 152)(90 151)(91 150)(92 149)(93 148)(94 147)(95 146)(96 145)(109 235)(110 234)(111 233)(112 232)(113 231)(114 230)(115 229)(116 240)(117 239)(118 238)(119 237)(120 236)(121 166)(122 165)(123 164)(124 163)(125 162)(126 161)(127 160)(128 159)(129 158)(130 157)(131 168)(132 167)(133 193)(134 204)(135 203)(136 202)(137 201)(138 200)(139 199)(140 198)(141 197)(142 196)(143 195)(144 194)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 192)(180 191)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E···12L15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222234444444444445566610···10101010101212121212···121515202020202020202030···3060···60
size111166303022255556610103030222222···212121212222210···10444444121212124···44···4

72 irreducible representations

dim11111111122222222222224444444
type+++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10D10C4×S3D12C4×D5C4○D12S3×D5D4×D5D42D5C2×S3×D5C4×S3×D5D5×D12Dic3.D10
kernelDic54D12C30.Q8C12×Dic5C5×D6⋊C4D303C4C2×S3×Dic5C2×D30.C2C2×C5⋊D12C5⋊D12C4×Dic5C3×Dic5D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5Dic5D6C10C2×C4C6C6C22C2C2C2
# reps11111111812221222244842222444

Matrix representation of Dic54D12 in GL4(𝔽61) generated by

1000
0100
00160
004517
,
1000
0100
005750
00574
,
233800
234600
00171
001744
,
606000
0100
004460
004417
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,45,0,0,60,17],[1,0,0,0,0,1,0,0,0,0,57,57,0,0,50,4],[23,23,0,0,38,46,0,0,0,0,17,17,0,0,1,44],[60,0,0,0,60,1,0,0,0,0,44,44,0,0,60,17] >;

Dic54D12 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4D_{12}
% in TeX

G:=Group("Dic5:4D12");
// GroupNames label

G:=SmallGroup(480,481);
// by ID

G=gap.SmallGroup(480,481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,135,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽