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G = Dic5×C3⋊D4order 480 = 25·3·5

Direct product of Dic5 and C3⋊D4

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

Aliases: Dic5×C3⋊D4, C1525(C4×D4), C34(D4×Dic5), D63(C2×Dic5), C6.159(D4×D5), D6⋊Dic529C2, (C3×Dic5)⋊14D4, C30.231(C2×D4), C222(S3×Dic5), C23.41(S3×D5), Dic31(C2×Dic5), C6.Dic1034C2, (C22×Dic5)⋊8S3, (C22×C6).91D10, (C22×C10).48D6, (Dic3×Dic5)⋊33C2, C10.81(C4○D12), C30.146(C4○D4), C6.54(D42D5), C30.38D423C2, C30.144(C22×C4), (C2×C30).193C23, (C2×Dic5).193D6, (C22×S3).52D10, C6.18(C22×Dic5), (C2×Dic3).121D10, C2.6(Dic3.D10), (C22×C30).55C22, (C6×Dic5).222C22, (C2×Dic15).133C22, (C10×Dic3).112C22, C57(C4×C3⋊D4), (C5×C3⋊D4)⋊3C4, (C2×C6×Dic5)⋊5C2, C2.6(D5×C3⋊D4), (C2×C10)⋊10(C4×S3), (C2×C30)⋊15(C2×C4), C10.125(S3×C2×C4), (C2×S3×Dic5)⋊13C2, (C2×C6)⋊5(C2×Dic5), (C2×C3⋊D4).7D5, (S3×C10)⋊16(C2×C4), C2.18(C2×S3×Dic5), C22.84(C2×S3×D5), (C10×C3⋊D4).5C2, C10.63(C2×C3⋊D4), (C5×Dic3)⋊11(C2×C4), (S3×C2×C10).48C22, (C2×C6).205(C22×D5), (C2×C10).205(C22×S3), SmallGroup(480,627)

Series: Derived Chief Lower central Upper central

C1C30 — Dic5×C3⋊D4
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — Dic5×C3⋊D4
C15C30 — Dic5×C3⋊D4
C1C22C23

Generators and relations for Dic5×C3⋊D4
 G = < a,b,c,d,e | a10=c3=d4=e2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 732 in 188 conjugacy classes, 72 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×9], D4 [×4], C23, C23, C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×4], C22×S3, C22×C6, C5×S3 [×2], C30 [×3], C30 [×2], C4×D4, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C5×D4 [×4], C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5, C22×Dic5, D4×C10, C4×C3⋊D4, S3×Dic5 [×2], C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15 [×2], S3×C2×C10, C22×C30, D4×Dic5, Dic3×Dic5, D6⋊Dic5, C6.Dic10, C30.38D4, C2×S3×Dic5, C2×C6×Dic5, C10×C3⋊D4, Dic5×C3⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C2×Dic5 [×6], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, D4×D5, D42D5, C22×Dic5, C4×C3⋊D4, S3×Dic5 [×2], C2×S3×D5, D4×Dic5, C2×S3×Dic5, Dic3.D10, D5×C3⋊D4, Dic5×C3⋊D4

Smallest permutation representation of Dic5×C3⋊D4
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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 149 16 144)(12 148 17 143)(13 147 18 142)(14 146 19 141)(15 145 20 150)(21 154 26 159)(22 153 27 158)(23 152 28 157)(24 151 29 156)(25 160 30 155)(31 108 36 103)(32 107 37 102)(33 106 38 101)(34 105 39 110)(35 104 40 109)(41 184 46 189)(42 183 47 188)(43 182 48 187)(44 181 49 186)(45 190 50 185)(51 136 56 131)(52 135 57 140)(53 134 58 139)(54 133 59 138)(55 132 60 137)(61 169 66 164)(62 168 67 163)(63 167 68 162)(64 166 69 161)(65 165 70 170)(71 237 76 232)(72 236 77 231)(73 235 78 240)(74 234 79 239)(75 233 80 238)(81 214 86 219)(82 213 87 218)(83 212 88 217)(84 211 89 216)(85 220 90 215)(91 209 96 204)(92 208 97 203)(93 207 98 202)(94 206 99 201)(95 205 100 210)(111 196 116 191)(112 195 117 200)(113 194 118 199)(114 193 119 198)(115 192 120 197)(121 226 126 221)(122 225 127 230)(123 224 128 229)(124 223 129 228)(125 222 130 227)
(1 28 43)(2 29 44)(3 30 45)(4 21 46)(5 22 47)(6 23 48)(7 24 49)(8 25 50)(9 26 41)(10 27 42)(11 61 52)(12 62 53)(13 63 54)(14 64 55)(15 65 56)(16 66 57)(17 67 58)(18 68 59)(19 69 60)(20 70 51)(31 236 217)(32 237 218)(33 238 219)(34 239 220)(35 240 211)(36 231 212)(37 232 213)(38 233 214)(39 234 215)(40 235 216)(71 87 102)(72 88 103)(73 89 104)(74 90 105)(75 81 106)(76 82 107)(77 83 108)(78 84 109)(79 85 110)(80 86 101)(91 128 112)(92 129 113)(93 130 114)(94 121 115)(95 122 116)(96 123 117)(97 124 118)(98 125 119)(99 126 120)(100 127 111)(131 145 165)(132 146 166)(133 147 167)(134 148 168)(135 149 169)(136 150 170)(137 141 161)(138 142 162)(139 143 163)(140 144 164)(151 186 180)(152 187 171)(153 188 172)(154 189 173)(155 190 174)(156 181 175)(157 182 176)(158 183 177)(159 184 178)(160 185 179)(191 205 225)(192 206 226)(193 207 227)(194 208 228)(195 209 229)(196 210 230)(197 201 221)(198 202 222)(199 203 223)(200 204 224)
(1 72 59 119)(2 73 60 120)(3 74 51 111)(4 75 52 112)(5 76 53 113)(6 77 54 114)(7 78 55 115)(8 79 56 116)(9 80 57 117)(10 71 58 118)(11 128 21 106)(12 129 22 107)(13 130 23 108)(14 121 24 109)(15 122 25 110)(16 123 26 101)(17 124 27 102)(18 125 28 103)(19 126 29 104)(20 127 30 105)(31 142 222 157)(32 143 223 158)(33 144 224 159)(34 145 225 160)(35 146 226 151)(36 147 227 152)(37 148 228 153)(38 149 229 154)(39 150 230 155)(40 141 221 156)(41 86 66 96)(42 87 67 97)(43 88 68 98)(44 89 69 99)(45 90 70 100)(46 81 61 91)(47 82 62 92)(48 83 63 93)(49 84 64 94)(50 85 65 95)(131 191 179 239)(132 192 180 240)(133 193 171 231)(134 194 172 232)(135 195 173 233)(136 196 174 234)(137 197 175 235)(138 198 176 236)(139 199 177 237)(140 200 178 238)(161 201 181 216)(162 202 182 217)(163 203 183 218)(164 204 184 219)(165 205 185 220)(166 206 186 211)(167 207 187 212)(168 208 188 213)(169 209 189 214)(170 210 190 215)
(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 201)(71 118)(72 119)(73 120)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 128)(82 129)(83 130)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(141 161)(142 162)(143 163)(144 164)(145 165)(146 166)(147 167)(148 168)(149 169)(150 170)(151 186)(152 187)(153 188)(154 189)(155 190)(156 181)(157 182)(158 183)(159 184)(160 185)(191 239)(192 240)(193 231)(194 232)(195 233)(196 234)(197 235)(198 236)(199 237)(200 238)(211 226)(212 227)(213 228)(214 229)(215 230)(216 221)(217 222)(218 223)(219 224)(220 225)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A···6G10A···10F10G10H10I10J10K10L10M10N12A···12H15A15B20A20B20C20D30A···30N
order122222223444444444444556···610···10101010101010101012···1215152020202030···30
size111122662555566101030303030222···22···244441212121210···1044121212124···4

72 irreducible representations

dim11111111122222222222224444444
type++++++++++++++-++++--+
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6C4○D4D10Dic5D10D10C3⋊D4C4×S3C4○D12S3×D5D4×D5D42D5S3×Dic5C2×S3×D5Dic3.D10D5×C3⋊D4
kernelDic5×C3⋊D4Dic3×Dic5D6⋊Dic5C6.Dic10C30.38D4C2×S3×Dic5C2×C6×Dic5C10×C3⋊D4C5×C3⋊D4C22×Dic5C3×Dic5C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C3⋊D4C22×S3C22×C6Dic5C2×C10C10C23C6C6C22C22C2C2
# reps11111111812221228224442224244

Matrix representation of Dic5×C3⋊D4 in GL4(𝔽61) generated by

1000
0100
00060
00144
,
1000
0100
00837
005153
,
06000
16000
0010
0001
,
185200
94300
00600
00060
,
06000
60000
0010
0001
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,44],[1,0,0,0,0,1,0,0,0,0,8,51,0,0,37,53],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[18,9,0,0,52,43,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,0,1] >;

Dic5×C3⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_3\rtimes D_4
% in TeX

G:=Group("Dic5xC3:D4");
// GroupNames label

G:=SmallGroup(480,627);
// by ID

G=gap.SmallGroup(480,627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=c^3=d^4=e^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽