direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5×C3⋊D4, C15⋊25(C4×D4), C3⋊4(D4×Dic5), D6⋊3(C2×Dic5), C6.159(D4×D5), D6⋊Dic5⋊29C2, (C3×Dic5)⋊14D4, C30.231(C2×D4), C22⋊2(S3×Dic5), C23.41(S3×D5), Dic3⋊1(C2×Dic5), C6.Dic10⋊34C2, (C22×Dic5)⋊8S3, (C22×C6).91D10, (C22×C10).48D6, (Dic3×Dic5)⋊33C2, C10.81(C4○D12), C30.146(C4○D4), C6.54(D4⋊2D5), C30.38D4⋊23C2, 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, C5⋊7(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
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, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4×D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, C22×Dic5, D4×C10, C4×C3⋊D4, S3×Dic5, C6×Dic5, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, 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, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×Dic5, C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C22×Dic5, C4×C3⋊D4, S3×Dic5, C2×S3×D5, D4×Dic5, C2×S3×Dic5, Dic3.D10, D5×C3⋊D4, 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)(11 145 16 150)(12 144 17 149)(13 143 18 148)(14 142 19 147)(15 141 20 146)(21 152 26 157)(22 151 27 156)(23 160 28 155)(24 159 29 154)(25 158 30 153)(31 110 36 105)(32 109 37 104)(33 108 38 103)(34 107 39 102)(35 106 40 101)(41 186 46 181)(42 185 47 190)(43 184 48 189)(44 183 49 188)(45 182 50 187)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 167 66 162)(62 166 67 161)(63 165 68 170)(64 164 69 169)(65 163 70 168)(71 239 76 234)(72 238 77 233)(73 237 78 232)(74 236 79 231)(75 235 80 240)(81 212 86 217)(82 211 87 216)(83 220 88 215)(84 219 89 214)(85 218 90 213)(91 201 96 206)(92 210 97 205)(93 209 98 204)(94 208 99 203)(95 207 100 202)(111 194 116 199)(112 193 117 198)(113 192 118 197)(114 191 119 196)(115 200 120 195)(121 224 126 229)(122 223 127 228)(123 222 128 227)(124 221 129 226)(125 230 130 225)
(1 28 45)(2 29 46)(3 30 47)(4 21 48)(5 22 49)(6 23 50)(7 24 41)(8 25 42)(9 26 43)(10 27 44)(11 65 56)(12 66 57)(13 67 58)(14 68 59)(15 69 60)(16 70 51)(17 61 52)(18 62 53)(19 63 54)(20 64 55)(31 236 215)(32 237 216)(33 238 217)(34 239 218)(35 240 219)(36 231 220)(37 232 211)(38 233 212)(39 234 213)(40 235 214)(71 85 102)(72 86 103)(73 87 104)(74 88 105)(75 89 106)(76 90 107)(77 81 108)(78 82 109)(79 83 110)(80 84 101)(91 126 120)(92 127 111)(93 128 112)(94 129 113)(95 130 114)(96 121 115)(97 122 116)(98 123 117)(99 124 118)(100 125 119)(131 147 165)(132 148 166)(133 149 167)(134 150 168)(135 141 169)(136 142 170)(137 143 161)(138 144 162)(139 145 163)(140 146 164)(151 188 172)(152 189 173)(153 190 174)(154 181 175)(155 182 176)(156 183 177)(157 184 178)(158 185 179)(159 186 180)(160 187 171)(191 207 225)(192 208 226)(193 209 227)(194 210 228)(195 201 229)(196 202 230)(197 203 221)(198 204 222)(199 205 223)(200 206 224)
(1 74 59 119)(2 75 60 120)(3 76 51 111)(4 77 52 112)(5 78 53 113)(6 79 54 114)(7 80 55 115)(8 71 56 116)(9 72 57 117)(10 73 58 118)(11 122 25 102)(12 123 26 103)(13 124 27 104)(14 125 28 105)(15 126 29 106)(16 127 30 107)(17 128 21 108)(18 129 22 109)(19 130 23 110)(20 121 24 101)(31 142 230 155)(32 143 221 156)(33 144 222 157)(34 145 223 158)(35 146 224 159)(36 147 225 160)(37 148 226 151)(38 149 227 152)(39 150 228 153)(40 141 229 154)(41 84 64 96)(42 85 65 97)(43 86 66 98)(44 87 67 99)(45 88 68 100)(46 89 69 91)(47 90 70 92)(48 81 61 93)(49 82 62 94)(50 83 63 95)(131 191 171 231)(132 192 172 232)(133 193 173 233)(134 194 174 234)(135 195 175 235)(136 196 176 236)(137 197 177 237)(138 198 178 238)(139 199 179 239)(140 200 180 240)(161 203 183 216)(162 204 184 217)(163 205 185 218)(164 206 186 219)(165 207 187 220)(166 208 188 211)(167 209 189 212)(168 210 190 213)(169 201 181 214)(170 202 182 215)
(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 48)(22 49)(23 50)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 201)(71 116)(72 117)(73 118)(74 119)(75 120)(76 111)(77 112)(78 113)(79 114)(80 115)(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)(141 169)(142 170)(143 161)(144 162)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 188)(152 189)(153 190)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(191 231)(192 232)(193 233)(194 234)(195 235)(196 236)(197 237)(198 238)(199 239)(200 240)(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)(11,145,16,150)(12,144,17,149)(13,143,18,148)(14,142,19,147)(15,141,20,146)(21,152,26,157)(22,151,27,156)(23,160,28,155)(24,159,29,154)(25,158,30,153)(31,110,36,105)(32,109,37,104)(33,108,38,103)(34,107,39,102)(35,106,40,101)(41,186,46,181)(42,185,47,190)(43,184,48,189)(44,183,49,188)(45,182,50,187)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,167,66,162)(62,166,67,161)(63,165,68,170)(64,164,69,169)(65,163,70,168)(71,239,76,234)(72,238,77,233)(73,237,78,232)(74,236,79,231)(75,235,80,240)(81,212,86,217)(82,211,87,216)(83,220,88,215)(84,219,89,214)(85,218,90,213)(91,201,96,206)(92,210,97,205)(93,209,98,204)(94,208,99,203)(95,207,100,202)(111,194,116,199)(112,193,117,198)(113,192,118,197)(114,191,119,196)(115,200,120,195)(121,224,126,229)(122,223,127,228)(123,222,128,227)(124,221,129,226)(125,230,130,225), (1,28,45)(2,29,46)(3,30,47)(4,21,48)(5,22,49)(6,23,50)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,65,56)(12,66,57)(13,67,58)(14,68,59)(15,69,60)(16,70,51)(17,61,52)(18,62,53)(19,63,54)(20,64,55)(31,236,215)(32,237,216)(33,238,217)(34,239,218)(35,240,219)(36,231,220)(37,232,211)(38,233,212)(39,234,213)(40,235,214)(71,85,102)(72,86,103)(73,87,104)(74,88,105)(75,89,106)(76,90,107)(77,81,108)(78,82,109)(79,83,110)(80,84,101)(91,126,120)(92,127,111)(93,128,112)(94,129,113)(95,130,114)(96,121,115)(97,122,116)(98,123,117)(99,124,118)(100,125,119)(131,147,165)(132,148,166)(133,149,167)(134,150,168)(135,141,169)(136,142,170)(137,143,161)(138,144,162)(139,145,163)(140,146,164)(151,188,172)(152,189,173)(153,190,174)(154,181,175)(155,182,176)(156,183,177)(157,184,178)(158,185,179)(159,186,180)(160,187,171)(191,207,225)(192,208,226)(193,209,227)(194,210,228)(195,201,229)(196,202,230)(197,203,221)(198,204,222)(199,205,223)(200,206,224), (1,74,59,119)(2,75,60,120)(3,76,51,111)(4,77,52,112)(5,78,53,113)(6,79,54,114)(7,80,55,115)(8,71,56,116)(9,72,57,117)(10,73,58,118)(11,122,25,102)(12,123,26,103)(13,124,27,104)(14,125,28,105)(15,126,29,106)(16,127,30,107)(17,128,21,108)(18,129,22,109)(19,130,23,110)(20,121,24,101)(31,142,230,155)(32,143,221,156)(33,144,222,157)(34,145,223,158)(35,146,224,159)(36,147,225,160)(37,148,226,151)(38,149,227,152)(39,150,228,153)(40,141,229,154)(41,84,64,96)(42,85,65,97)(43,86,66,98)(44,87,67,99)(45,88,68,100)(46,89,69,91)(47,90,70,92)(48,81,61,93)(49,82,62,94)(50,83,63,95)(131,191,171,231)(132,192,172,232)(133,193,173,233)(134,194,174,234)(135,195,175,235)(136,196,176,236)(137,197,177,237)(138,198,178,238)(139,199,179,239)(140,200,180,240)(161,203,183,216)(162,204,184,217)(163,205,185,218)(164,206,186,219)(165,207,187,220)(166,208,188,211)(167,209,189,212)(168,210,190,213)(169,201,181,214)(170,202,182,215), (11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,48)(22,49)(23,50)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,201)(71,116)(72,117)(73,118)(74,119)(75,120)(76,111)(77,112)(78,113)(79,114)(80,115)(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)(141,169)(142,170)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,188)(152,189)(153,190)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(191,231)(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(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)(11,145,16,150)(12,144,17,149)(13,143,18,148)(14,142,19,147)(15,141,20,146)(21,152,26,157)(22,151,27,156)(23,160,28,155)(24,159,29,154)(25,158,30,153)(31,110,36,105)(32,109,37,104)(33,108,38,103)(34,107,39,102)(35,106,40,101)(41,186,46,181)(42,185,47,190)(43,184,48,189)(44,183,49,188)(45,182,50,187)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,167,66,162)(62,166,67,161)(63,165,68,170)(64,164,69,169)(65,163,70,168)(71,239,76,234)(72,238,77,233)(73,237,78,232)(74,236,79,231)(75,235,80,240)(81,212,86,217)(82,211,87,216)(83,220,88,215)(84,219,89,214)(85,218,90,213)(91,201,96,206)(92,210,97,205)(93,209,98,204)(94,208,99,203)(95,207,100,202)(111,194,116,199)(112,193,117,198)(113,192,118,197)(114,191,119,196)(115,200,120,195)(121,224,126,229)(122,223,127,228)(123,222,128,227)(124,221,129,226)(125,230,130,225), (1,28,45)(2,29,46)(3,30,47)(4,21,48)(5,22,49)(6,23,50)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,65,56)(12,66,57)(13,67,58)(14,68,59)(15,69,60)(16,70,51)(17,61,52)(18,62,53)(19,63,54)(20,64,55)(31,236,215)(32,237,216)(33,238,217)(34,239,218)(35,240,219)(36,231,220)(37,232,211)(38,233,212)(39,234,213)(40,235,214)(71,85,102)(72,86,103)(73,87,104)(74,88,105)(75,89,106)(76,90,107)(77,81,108)(78,82,109)(79,83,110)(80,84,101)(91,126,120)(92,127,111)(93,128,112)(94,129,113)(95,130,114)(96,121,115)(97,122,116)(98,123,117)(99,124,118)(100,125,119)(131,147,165)(132,148,166)(133,149,167)(134,150,168)(135,141,169)(136,142,170)(137,143,161)(138,144,162)(139,145,163)(140,146,164)(151,188,172)(152,189,173)(153,190,174)(154,181,175)(155,182,176)(156,183,177)(157,184,178)(158,185,179)(159,186,180)(160,187,171)(191,207,225)(192,208,226)(193,209,227)(194,210,228)(195,201,229)(196,202,230)(197,203,221)(198,204,222)(199,205,223)(200,206,224), (1,74,59,119)(2,75,60,120)(3,76,51,111)(4,77,52,112)(5,78,53,113)(6,79,54,114)(7,80,55,115)(8,71,56,116)(9,72,57,117)(10,73,58,118)(11,122,25,102)(12,123,26,103)(13,124,27,104)(14,125,28,105)(15,126,29,106)(16,127,30,107)(17,128,21,108)(18,129,22,109)(19,130,23,110)(20,121,24,101)(31,142,230,155)(32,143,221,156)(33,144,222,157)(34,145,223,158)(35,146,224,159)(36,147,225,160)(37,148,226,151)(38,149,227,152)(39,150,228,153)(40,141,229,154)(41,84,64,96)(42,85,65,97)(43,86,66,98)(44,87,67,99)(45,88,68,100)(46,89,69,91)(47,90,70,92)(48,81,61,93)(49,82,62,94)(50,83,63,95)(131,191,171,231)(132,192,172,232)(133,193,173,233)(134,194,174,234)(135,195,175,235)(136,196,176,236)(137,197,177,237)(138,198,178,238)(139,199,179,239)(140,200,180,240)(161,203,183,216)(162,204,184,217)(163,205,185,218)(164,206,186,219)(165,207,187,220)(166,208,188,211)(167,209,189,212)(168,210,190,213)(169,201,181,214)(170,202,182,215), (11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,48)(22,49)(23,50)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,201)(71,116)(72,117)(73,118)(74,119)(75,120)(76,111)(77,112)(78,113)(79,114)(80,115)(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)(141,169)(142,170)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,188)(152,189)(153,190)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(191,231)(192,232)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(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),(11,145,16,150),(12,144,17,149),(13,143,18,148),(14,142,19,147),(15,141,20,146),(21,152,26,157),(22,151,27,156),(23,160,28,155),(24,159,29,154),(25,158,30,153),(31,110,36,105),(32,109,37,104),(33,108,38,103),(34,107,39,102),(35,106,40,101),(41,186,46,181),(42,185,47,190),(43,184,48,189),(44,183,49,188),(45,182,50,187),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,167,66,162),(62,166,67,161),(63,165,68,170),(64,164,69,169),(65,163,70,168),(71,239,76,234),(72,238,77,233),(73,237,78,232),(74,236,79,231),(75,235,80,240),(81,212,86,217),(82,211,87,216),(83,220,88,215),(84,219,89,214),(85,218,90,213),(91,201,96,206),(92,210,97,205),(93,209,98,204),(94,208,99,203),(95,207,100,202),(111,194,116,199),(112,193,117,198),(113,192,118,197),(114,191,119,196),(115,200,120,195),(121,224,126,229),(122,223,127,228),(123,222,128,227),(124,221,129,226),(125,230,130,225)], [(1,28,45),(2,29,46),(3,30,47),(4,21,48),(5,22,49),(6,23,50),(7,24,41),(8,25,42),(9,26,43),(10,27,44),(11,65,56),(12,66,57),(13,67,58),(14,68,59),(15,69,60),(16,70,51),(17,61,52),(18,62,53),(19,63,54),(20,64,55),(31,236,215),(32,237,216),(33,238,217),(34,239,218),(35,240,219),(36,231,220),(37,232,211),(38,233,212),(39,234,213),(40,235,214),(71,85,102),(72,86,103),(73,87,104),(74,88,105),(75,89,106),(76,90,107),(77,81,108),(78,82,109),(79,83,110),(80,84,101),(91,126,120),(92,127,111),(93,128,112),(94,129,113),(95,130,114),(96,121,115),(97,122,116),(98,123,117),(99,124,118),(100,125,119),(131,147,165),(132,148,166),(133,149,167),(134,150,168),(135,141,169),(136,142,170),(137,143,161),(138,144,162),(139,145,163),(140,146,164),(151,188,172),(152,189,173),(153,190,174),(154,181,175),(155,182,176),(156,183,177),(157,184,178),(158,185,179),(159,186,180),(160,187,171),(191,207,225),(192,208,226),(193,209,227),(194,210,228),(195,201,229),(196,202,230),(197,203,221),(198,204,222),(199,205,223),(200,206,224)], [(1,74,59,119),(2,75,60,120),(3,76,51,111),(4,77,52,112),(5,78,53,113),(6,79,54,114),(7,80,55,115),(8,71,56,116),(9,72,57,117),(10,73,58,118),(11,122,25,102),(12,123,26,103),(13,124,27,104),(14,125,28,105),(15,126,29,106),(16,127,30,107),(17,128,21,108),(18,129,22,109),(19,130,23,110),(20,121,24,101),(31,142,230,155),(32,143,221,156),(33,144,222,157),(34,145,223,158),(35,146,224,159),(36,147,225,160),(37,148,226,151),(38,149,227,152),(39,150,228,153),(40,141,229,154),(41,84,64,96),(42,85,65,97),(43,86,66,98),(44,87,67,99),(45,88,68,100),(46,89,69,91),(47,90,70,92),(48,81,61,93),(49,82,62,94),(50,83,63,95),(131,191,171,231),(132,192,172,232),(133,193,173,233),(134,194,174,234),(135,195,175,235),(136,196,176,236),(137,197,177,237),(138,198,178,238),(139,199,179,239),(140,200,180,240),(161,203,183,216),(162,204,184,217),(163,205,185,218),(164,206,186,219),(165,207,187,220),(166,208,188,211),(167,209,189,212),(168,210,190,213),(169,201,181,214),(170,202,182,215)], [(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,48),(22,49),(23,50),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,201),(71,116),(72,117),(73,118),(74,119),(75,120),(76,111),(77,112),(78,113),(79,114),(80,115),(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),(141,169),(142,170),(143,161),(144,162),(145,163),(146,164),(147,165),(148,166),(149,167),(150,168),(151,188),(152,189),(153,190),(154,181),(155,182),(156,183),(157,184),(158,185),(159,186),(160,187),(191,231),(192,232),(193,233),(194,234),(195,235),(196,236),(197,237),(198,238),(199,239),(200,240),(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | ··· | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | Dic5 | D10 | D10 | C3⋊D4 | C4×S3 | C4○D12 | S3×D5 | D4×D5 | D4⋊2D5 | S3×Dic5 | C2×S3×D5 | Dic3.D10 | D5×C3⋊D4 |
| kernel | Dic5×C3⋊D4 | Dic3×Dic5 | D6⋊Dic5 | C6.Dic10 | C30.38D4 | C2×S3×Dic5 | C2×C6×Dic5 | C10×C3⋊D4 | C5×C3⋊D4 | C22×Dic5 | C3×Dic5 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | Dic5 | C2×C10 | C10 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic5×C3⋊D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 37 |
| 0 | 0 | 51 | 53 |
| 0 | 60 | 0 | 0 |
| 1 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 52 | 0 | 0 |
| 9 | 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 |
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