direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×Dic3⋊C4, D10.9Dic6, (C6×D5).5Q8, C6.31(Q8×D5), (D5×Dic3)⋊1C4, Dic3⋊5(C4×D5), (C6×D5).58D4, C6.127(D4×D5), C30.34(C2×Q8), C2.4(D5×Dic6), Dic15⋊7(C2×C4), D10.31(C4×S3), (C2×C20).189D6, C30.124(C2×D4), (C2×C12).262D10, C30.44(C22×C4), (C2×C30).82C23, C30.4Q8⋊24C2, C6.Dic10⋊11C2, Dic15⋊5C4⋊13C2, C10.13(C2×Dic6), D10.36(C3⋊D4), (C2×C60).385C22, (C2×Dic5).171D6, (C2×Dic3).89D10, (C22×D5).107D6, (C6×Dic5).194C22, (C10×Dic3).48C22, (C2×Dic15).69C22, C3⋊4(D5×C4⋊C4), C15⋊5(C2×C4⋊C4), C2.15(C4×S3×D5), C6.12(C2×C4×D5), C10.44(S3×C2×C4), C5⋊2(C2×Dic3⋊C4), (C2×C4×D5).13S3, C2.1(D5×C3⋊D4), (C3×D5)⋊2(C4⋊C4), (D5×C2×C12).27C2, (C2×D5×Dic3).1C2, C22.37(C2×S3×D5), (C5×Dic3)⋊7(C2×C4), (C6×D5).35(C2×C4), (C2×C4).125(S3×D5), C10.29(C2×C3⋊D4), (D5×C2×C6).99C22, (C5×Dic3⋊C4)⋊26C2, (C2×C6).94(C22×D5), (C2×C10).94(C22×S3), SmallGroup(480,468)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×Dic3⋊C4
G = < a,b,c,d,e | a5=b2=c6=e4=1, d2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c3d >
Subgroups: 796 in 184 conjugacy classes, 74 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2×C4⋊C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, Dic3⋊C4, C22×Dic3, C22×C12, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, C2×C30, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×Dic3⋊C4, D5×Dic3, D5×Dic3, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D5×C4⋊C4, Dic15⋊5C4, C6.Dic10, C5×Dic3⋊C4, C30.4Q8, C2×D5×Dic3, D5×C2×C12, D5×Dic3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C4×D5, C22×D5, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, S3×D5, C2×C4×D5, D4×D5, Q8×D5, C2×Dic3⋊C4, C2×S3×D5, D5×C4⋊C4, D5×Dic6, C4×S3×D5, D5×C3⋊D4, D5×Dic3⋊C4
►On 240 points
Generators in S240
(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 71)(2 72)(3 67)(4 68)(5 69)(6 70)(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 81)(32 82)(33 83)(34 84)(35 79)(36 80)(37 58)(38 59)(39 60)(40 55)(41 56)(42 57)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 65)(50 66)(51 61)(52 62)(53 63)(54 64)(85 125)(86 126)(87 121)(88 122)(89 123)(90 124)(91 141)(92 142)(93 143)(94 144)(95 139)(96 140)(97 118)(98 119)(99 120)(100 115)(101 116)(102 117)(103 136)(104 137)(105 138)(106 133)(107 134)(108 135)(109 130)(110 131)(111 132)(112 127)(113 128)(114 129)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(151 175)(152 176)(153 177)(154 178)(155 179)(156 180)(157 204)(158 199)(159 200)(160 201)(161 202)(162 203)(163 196)(164 197)(165 198)(166 193)(167 194)(168 195)(169 188)(170 189)(171 190)(172 191)(173 192)(174 187)(211 235)(212 236)(213 237)(214 238)(215 239)(216 240)
(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)
(1 175 4 178)(2 180 5 177)(3 179 6 176)(7 114 10 111)(8 113 11 110)(9 112 12 109)(13 108 16 105)(14 107 17 104)(15 106 18 103)(19 91 22 94)(20 96 23 93)(21 95 24 92)(25 101 28 98)(26 100 29 97)(27 99 30 102)(31 191 34 188)(32 190 35 187)(33 189 36 192)(37 185 40 182)(38 184 41 181)(39 183 42 186)(43 194 46 197)(44 193 47 196)(45 198 48 195)(49 204 52 201)(50 203 53 200)(51 202 54 199)(55 146 58 149)(56 145 59 148)(57 150 60 147)(61 161 64 158)(62 160 65 157)(63 159 66 162)(67 155 70 152)(68 154 71 151)(69 153 72 156)(73 164 76 167)(74 163 77 166)(75 168 78 165)(79 174 82 171)(80 173 83 170)(81 172 84 169)(85 236 88 239)(86 235 89 238)(87 240 90 237)(115 206 118 209)(116 205 119 208)(117 210 120 207)(121 216 124 213)(122 215 125 212)(123 214 126 211)(127 220 130 217)(128 219 131 222)(129 218 132 221)(133 224 136 227)(134 223 137 226)(135 228 138 225)(139 234 142 231)(140 233 143 230)(141 232 144 229)
(1 116 56 86)(2 117 57 87)(3 118 58 88)(4 119 59 89)(5 120 60 90)(6 115 55 85)(7 199 232 169)(8 200 233 170)(9 201 234 171)(10 202 229 172)(11 203 230 173)(12 204 231 174)(13 195 228 165)(14 196 223 166)(15 197 224 167)(16 198 225 168)(17 193 226 163)(18 194 227 164)(19 191 218 161)(20 192 219 162)(21 187 220 157)(22 188 221 158)(23 189 222 159)(24 190 217 160)(25 181 214 151)(26 182 215 152)(27 183 216 153)(28 184 211 154)(29 185 212 155)(30 186 213 156)(31 132 61 91)(32 127 62 92)(33 128 63 93)(34 129 64 94)(35 130 65 95)(36 131 66 96)(37 122 67 97)(38 123 68 98)(39 124 69 99)(40 125 70 100)(41 126 71 101)(42 121 72 102)(43 133 73 103)(44 134 74 104)(45 135 75 105)(46 136 76 106)(47 137 77 107)(48 138 78 108)(49 139 79 109)(50 140 80 110)(51 141 81 111)(52 142 82 112)(53 143 83 113)(54 144 84 114)(145 238 175 208)(146 239 176 209)(147 240 177 210)(148 235 178 205)(149 236 179 206)(150 237 180 207)
G:=sub<Sym(240)| (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), (1,71)(2,72)(3,67)(4,68)(5,69)(6,70)(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)(31,81)(32,82)(33,83)(34,84)(35,79)(36,80)(37,58)(38,59)(39,60)(40,55)(41,56)(42,57)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,65)(50,66)(51,61)(52,62)(53,63)(54,64)(85,125)(86,126)(87,121)(88,122)(89,123)(90,124)(91,141)(92,142)(93,143)(94,144)(95,139)(96,140)(97,118)(98,119)(99,120)(100,115)(101,116)(102,117)(103,136)(104,137)(105,138)(106,133)(107,134)(108,135)(109,130)(110,131)(111,132)(112,127)(113,128)(114,129)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,204)(158,199)(159,200)(160,201)(161,202)(162,203)(163,196)(164,197)(165,198)(166,193)(167,194)(168,195)(169,188)(170,189)(171,190)(172,191)(173,192)(174,187)(211,235)(212,236)(213,237)(214,238)(215,239)(216,240), (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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,108,16,105)(14,107,17,104)(15,106,18,103)(19,91,22,94)(20,96,23,93)(21,95,24,92)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,191,34,188)(32,190,35,187)(33,189,36,192)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,194,46,197)(44,193,47,196)(45,198,48,195)(49,204,52,201)(50,203,53,200)(51,202,54,199)(55,146,58,149)(56,145,59,148)(57,150,60,147)(61,161,64,158)(62,160,65,157)(63,159,66,162)(67,155,70,152)(68,154,71,151)(69,153,72,156)(73,164,76,167)(74,163,77,166)(75,168,78,165)(79,174,82,171)(80,173,83,170)(81,172,84,169)(85,236,88,239)(86,235,89,238)(87,240,90,237)(115,206,118,209)(116,205,119,208)(117,210,120,207)(121,216,124,213)(122,215,125,212)(123,214,126,211)(127,220,130,217)(128,219,131,222)(129,218,132,221)(133,224,136,227)(134,223,137,226)(135,228,138,225)(139,234,142,231)(140,233,143,230)(141,232,144,229), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,199,232,169)(8,200,233,170)(9,201,234,171)(10,202,229,172)(11,203,230,173)(12,204,231,174)(13,195,228,165)(14,196,223,166)(15,197,224,167)(16,198,225,168)(17,193,226,163)(18,194,227,164)(19,191,218,161)(20,192,219,162)(21,187,220,157)(22,188,221,158)(23,189,222,159)(24,190,217,160)(25,181,214,151)(26,182,215,152)(27,183,216,153)(28,184,211,154)(29,185,212,155)(30,186,213,156)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,133,73,103)(44,134,74,104)(45,135,75,105)(46,136,76,106)(47,137,77,107)(48,138,78,108)(49,139,79,109)(50,140,80,110)(51,141,81,111)(52,142,82,112)(53,143,83,113)(54,144,84,114)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,235,178,205)(149,236,179,206)(150,237,180,207)>;
G:=Group( (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), (1,71)(2,72)(3,67)(4,68)(5,69)(6,70)(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)(31,81)(32,82)(33,83)(34,84)(35,79)(36,80)(37,58)(38,59)(39,60)(40,55)(41,56)(42,57)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,65)(50,66)(51,61)(52,62)(53,63)(54,64)(85,125)(86,126)(87,121)(88,122)(89,123)(90,124)(91,141)(92,142)(93,143)(94,144)(95,139)(96,140)(97,118)(98,119)(99,120)(100,115)(101,116)(102,117)(103,136)(104,137)(105,138)(106,133)(107,134)(108,135)(109,130)(110,131)(111,132)(112,127)(113,128)(114,129)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,204)(158,199)(159,200)(160,201)(161,202)(162,203)(163,196)(164,197)(165,198)(166,193)(167,194)(168,195)(169,188)(170,189)(171,190)(172,191)(173,192)(174,187)(211,235)(212,236)(213,237)(214,238)(215,239)(216,240), (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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,108,16,105)(14,107,17,104)(15,106,18,103)(19,91,22,94)(20,96,23,93)(21,95,24,92)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,191,34,188)(32,190,35,187)(33,189,36,192)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,194,46,197)(44,193,47,196)(45,198,48,195)(49,204,52,201)(50,203,53,200)(51,202,54,199)(55,146,58,149)(56,145,59,148)(57,150,60,147)(61,161,64,158)(62,160,65,157)(63,159,66,162)(67,155,70,152)(68,154,71,151)(69,153,72,156)(73,164,76,167)(74,163,77,166)(75,168,78,165)(79,174,82,171)(80,173,83,170)(81,172,84,169)(85,236,88,239)(86,235,89,238)(87,240,90,237)(115,206,118,209)(116,205,119,208)(117,210,120,207)(121,216,124,213)(122,215,125,212)(123,214,126,211)(127,220,130,217)(128,219,131,222)(129,218,132,221)(133,224,136,227)(134,223,137,226)(135,228,138,225)(139,234,142,231)(140,233,143,230)(141,232,144,229), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,199,232,169)(8,200,233,170)(9,201,234,171)(10,202,229,172)(11,203,230,173)(12,204,231,174)(13,195,228,165)(14,196,223,166)(15,197,224,167)(16,198,225,168)(17,193,226,163)(18,194,227,164)(19,191,218,161)(20,192,219,162)(21,187,220,157)(22,188,221,158)(23,189,222,159)(24,190,217,160)(25,181,214,151)(26,182,215,152)(27,183,216,153)(28,184,211,154)(29,185,212,155)(30,186,213,156)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,133,73,103)(44,134,74,104)(45,135,75,105)(46,136,76,106)(47,137,77,107)(48,138,78,108)(49,139,79,109)(50,140,80,110)(51,141,81,111)(52,142,82,112)(53,143,83,113)(54,144,84,114)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,235,178,205)(149,236,179,206)(150,237,180,207) );
G=PermutationGroup([[(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)], [(1,71),(2,72),(3,67),(4,68),(5,69),(6,70),(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),(31,81),(32,82),(33,83),(34,84),(35,79),(36,80),(37,58),(38,59),(39,60),(40,55),(41,56),(42,57),(43,76),(44,77),(45,78),(46,73),(47,74),(48,75),(49,65),(50,66),(51,61),(52,62),(53,63),(54,64),(85,125),(86,126),(87,121),(88,122),(89,123),(90,124),(91,141),(92,142),(93,143),(94,144),(95,139),(96,140),(97,118),(98,119),(99,120),(100,115),(101,116),(102,117),(103,136),(104,137),(105,138),(106,133),(107,134),(108,135),(109,130),(110,131),(111,132),(112,127),(113,128),(114,129),(145,181),(146,182),(147,183),(148,184),(149,185),(150,186),(151,175),(152,176),(153,177),(154,178),(155,179),(156,180),(157,204),(158,199),(159,200),(160,201),(161,202),(162,203),(163,196),(164,197),(165,198),(166,193),(167,194),(168,195),(169,188),(170,189),(171,190),(172,191),(173,192),(174,187),(211,235),(212,236),(213,237),(214,238),(215,239),(216,240)], [(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)], [(1,175,4,178),(2,180,5,177),(3,179,6,176),(7,114,10,111),(8,113,11,110),(9,112,12,109),(13,108,16,105),(14,107,17,104),(15,106,18,103),(19,91,22,94),(20,96,23,93),(21,95,24,92),(25,101,28,98),(26,100,29,97),(27,99,30,102),(31,191,34,188),(32,190,35,187),(33,189,36,192),(37,185,40,182),(38,184,41,181),(39,183,42,186),(43,194,46,197),(44,193,47,196),(45,198,48,195),(49,204,52,201),(50,203,53,200),(51,202,54,199),(55,146,58,149),(56,145,59,148),(57,150,60,147),(61,161,64,158),(62,160,65,157),(63,159,66,162),(67,155,70,152),(68,154,71,151),(69,153,72,156),(73,164,76,167),(74,163,77,166),(75,168,78,165),(79,174,82,171),(80,173,83,170),(81,172,84,169),(85,236,88,239),(86,235,89,238),(87,240,90,237),(115,206,118,209),(116,205,119,208),(117,210,120,207),(121,216,124,213),(122,215,125,212),(123,214,126,211),(127,220,130,217),(128,219,131,222),(129,218,132,221),(133,224,136,227),(134,223,137,226),(135,228,138,225),(139,234,142,231),(140,233,143,230),(141,232,144,229)], [(1,116,56,86),(2,117,57,87),(3,118,58,88),(4,119,59,89),(5,120,60,90),(6,115,55,85),(7,199,232,169),(8,200,233,170),(9,201,234,171),(10,202,229,172),(11,203,230,173),(12,204,231,174),(13,195,228,165),(14,196,223,166),(15,197,224,167),(16,198,225,168),(17,193,226,163),(18,194,227,164),(19,191,218,161),(20,192,219,162),(21,187,220,157),(22,188,221,158),(23,189,222,159),(24,190,217,160),(25,181,214,151),(26,182,215,152),(27,183,216,153),(28,184,211,154),(29,185,212,155),(30,186,213,156),(31,132,61,91),(32,127,62,92),(33,128,63,93),(34,129,64,94),(35,130,65,95),(36,131,66,96),(37,122,67,97),(38,123,68,98),(39,124,69,99),(40,125,70,100),(41,126,71,101),(42,121,72,102),(43,133,73,103),(44,134,74,104),(45,135,75,105),(46,136,76,106),(47,137,77,107),(48,138,78,108),(49,139,79,109),(50,140,80,110),(51,141,81,111),(52,142,82,112),(53,143,83,113),(54,144,84,114),(145,238,175,208),(146,239,176,209),(147,240,177,210),(148,235,178,205),(149,236,179,206),(150,237,180,207)]])
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 | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 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 | C4 | S3 | D4 | Q8 | D5 | D6 | D6 | D6 | D10 | D10 | Dic6 | C4×S3 | C3⋊D4 | C4×D5 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 | C4×S3×D5 | D5×C3⋊D4 |
| kernel | D5×Dic3⋊C4 | Dic15⋊5C4 | C6.Dic10 | C5×Dic3⋊C4 | C30.4Q8 | C2×D5×Dic3 | D5×C2×C12 | D5×Dic3 | C2×C4×D5 | C6×D5 | C6×D5 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | D10 | D10 | D10 | Dic3 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D5×Dic3⋊C4 ►in GL4(𝔽61) generated by
| 60 | 1 | 0 | 0 |
| 16 | 44 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 45 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 20 | 8 |
| 0 | 0 | 49 | 41 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 23 | 46 |
| 0 | 0 | 15 | 38 |
G:=sub<GL(4,GF(61))| [60,16,0,0,1,44,0,0,0,0,1,0,0,0,0,1],[1,45,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,20,49,0,0,8,41],[50,0,0,0,0,50,0,0,0,0,23,15,0,0,46,38] >;
D5×Dic3⋊C4 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("D5xDic3:C4"); // GroupNames label
G:=SmallGroup(480,468);
// by ID
G=gap.SmallGroup(480,468);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^6=e^4=1,d^2=c^3,b*a*b=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=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations