metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D5×C12)⋊1C4, C4⋊Dic3⋊16D5, (C4×D5)⋊1Dic3, C60⋊5C4⋊29C2, C12.72(C4×D5), C60.119(C2×C4), (C2×C20).110D6, C4.15(D5×Dic3), C15⋊9(C42⋊C2), (Dic3×Dic5)⋊8C2, C30.25(C4○D4), (C2×C12).295D10, (C2×C30).48C23, C6.6(Q8⋊2D5), C20.34(C2×Dic3), (C22×D5).82D6, C10.50(C4○D12), C6.19(D4⋊2D5), C2.2(C12.28D10), C2.2(D12⋊5D5), C30.120(C22×C4), (C2×C60).139C22, D10.11(C2×Dic3), (C2×Dic3).88D10, (C2×Dic5).211D6, D10⋊Dic3.2C2, C5⋊3(C23.26D6), Dic5.21(C2×Dic3), C10.22(C22×Dic3), (C10×Dic3).28C22, (C2×Dic15).50C22, (C6×Dic5).181C22, (C2×C4×D5).2S3, C6.85(C2×C4×D5), (D5×C2×C12).2C2, C3⋊5(C4⋊C4⋊7D5), (C5×C4⋊Dic3)⋊2C2, C2.11(C2×D5×Dic3), C22.33(C2×S3×D5), (C6×D5).51(C2×C4), (C2×C4).152(S3×D5), (D5×C2×C6).95C22, (C2×C6).60(C22×D5), (C2×C10).60(C22×S3), (C3×Dic5).59(C2×C4), SmallGroup(480,434)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C4×D5)⋊Dic3
G = < a,b,c,d,e | a4=b5=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=d-1 >
Subgroups: 604 in 152 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C23.26D6, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C4⋊C4⋊7D5, Dic3×Dic5, D10⋊Dic3, C5×C4⋊Dic3, C60⋊5C4, D5×C2×C12, (C4×D5)⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C4○D4, D10, C2×Dic3, C22×S3, C42⋊C2, C4×D5, C22×D5, C4○D12, C22×Dic3, S3×D5, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C23.26D6, D5×Dic3, C2×S3×D5, C4⋊C4⋊7D5, D12⋊5D5, C12.28D10, C2×D5×Dic3, (C4×D5)⋊Dic3
►On 240 points
Generators in S240
(1 74 14 86)(2 75 15 87)(3 76 16 88)(4 77 17 89)(5 78 18 90)(6 73 13 85)(7 170 230 214)(8 171 231 215)(9 172 232 216)(10 173 233 211)(11 174 234 212)(12 169 229 213)(19 128 63 79)(20 129 64 80)(21 130 65 81)(22 131 66 82)(23 132 61 83)(24 127 62 84)(25 122 67 105)(26 123 68 106)(27 124 69 107)(28 125 70 108)(29 126 71 103)(30 121 72 104)(31 168 228 188)(32 163 223 189)(33 164 224 190)(34 165 225 191)(35 166 226 192)(36 167 227 187)(37 147 207 184)(38 148 208 185)(39 149 209 186)(40 150 210 181)(41 145 205 182)(42 146 206 183)(43 119 59 97)(44 120 60 98)(45 115 55 99)(46 116 56 100)(47 117 57 101)(48 118 58 102)(49 155 95 109)(50 156 96 110)(51 151 91 111)(52 152 92 112)(53 153 93 113)(54 154 94 114)(133 193 178 238)(134 194 179 239)(135 195 180 240)(136 196 175 235)(137 197 176 236)(138 198 177 237)(139 199 162 222)(140 200 157 217)(141 201 158 218)(142 202 159 219)(143 203 160 220)(144 204 161 221)
(1 59 49 19 30)(2 60 50 20 25)(3 55 51 21 26)(4 56 52 22 27)(5 57 53 23 28)(6 58 54 24 29)(7 34 41 221 195)(8 35 42 222 196)(9 36 37 217 197)(10 31 38 218 198)(11 32 39 219 193)(12 33 40 220 194)(13 48 94 62 71)(14 43 95 63 72)(15 44 96 64 67)(16 45 91 65 68)(17 46 92 66 69)(18 47 93 61 70)(73 102 154 127 126)(74 97 155 128 121)(75 98 156 129 122)(76 99 151 130 123)(77 100 152 131 124)(78 101 153 132 125)(79 104 86 119 109)(80 105 87 120 110)(81 106 88 115 111)(82 107 89 116 112)(83 108 90 117 113)(84 103 85 118 114)(133 212 189 186 159)(134 213 190 181 160)(135 214 191 182 161)(136 215 192 183 162)(137 216 187 184 157)(138 211 188 185 158)(139 175 171 166 146)(140 176 172 167 147)(141 177 173 168 148)(142 178 174 163 149)(143 179 169 164 150)(144 180 170 165 145)(199 235 231 226 206)(200 236 232 227 207)(201 237 233 228 208)(202 238 234 223 209)(203 239 229 224 210)(204 240 230 225 205)
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 230)(8 231)(9 232)(10 233)(11 234)(12 229)(13 71)(14 72)(15 67)(16 68)(17 69)(18 70)(19 59)(20 60)(21 55)(22 56)(23 57)(24 58)(31 237)(32 238)(33 239)(34 240)(35 235)(36 236)(37 200)(38 201)(39 202)(40 203)(41 204)(42 199)(43 63)(44 64)(45 65)(46 66)(47 61)(48 62)(73 126)(74 121)(75 122)(76 123)(77 124)(78 125)(79 119)(80 120)(81 115)(82 116)(83 117)(84 118)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(97 128)(98 129)(99 130)(100 131)(101 132)(102 127)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 183)(140 184)(141 185)(142 186)(143 181)(144 182)(145 161)(146 162)(147 157)(148 158)(149 159)(150 160)(169 213)(170 214)(171 215)(172 216)(173 211)(174 212)(175 192)(176 187)(177 188)(178 189)(179 190)(180 191)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(205 221)(206 222)(207 217)(208 218)(209 219)(210 220)
(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)
(1 142 4 139)(2 141 5 144)(3 140 6 143)(7 156 10 153)(8 155 11 152)(9 154 12 151)(13 160 16 157)(14 159 17 162)(15 158 18 161)(19 163 22 166)(20 168 23 165)(21 167 24 164)(25 148 28 145)(26 147 29 150)(27 146 30 149)(31 132 34 129)(32 131 35 128)(33 130 36 127)(37 126 40 123)(38 125 41 122)(39 124 42 121)(43 133 46 136)(44 138 47 135)(45 137 48 134)(49 174 52 171)(50 173 53 170)(51 172 54 169)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 185 70 182)(68 184 71 181)(69 183 72 186)(73 220 76 217)(74 219 77 222)(75 218 78 221)(79 223 82 226)(80 228 83 225)(81 227 84 224)(85 203 88 200)(86 202 89 199)(87 201 90 204)(91 216 94 213)(92 215 95 212)(93 214 96 211)(97 193 100 196)(98 198 101 195)(99 197 102 194)(103 210 106 207)(104 209 107 206)(105 208 108 205)(109 234 112 231)(110 233 113 230)(111 232 114 229)(115 236 118 239)(116 235 119 238)(117 240 120 237)
G:=sub<Sym(240)| (1,74,14,86)(2,75,15,87)(3,76,16,88)(4,77,17,89)(5,78,18,90)(6,73,13,85)(7,170,230,214)(8,171,231,215)(9,172,232,216)(10,173,233,211)(11,174,234,212)(12,169,229,213)(19,128,63,79)(20,129,64,80)(21,130,65,81)(22,131,66,82)(23,132,61,83)(24,127,62,84)(25,122,67,105)(26,123,68,106)(27,124,69,107)(28,125,70,108)(29,126,71,103)(30,121,72,104)(31,168,228,188)(32,163,223,189)(33,164,224,190)(34,165,225,191)(35,166,226,192)(36,167,227,187)(37,147,207,184)(38,148,208,185)(39,149,209,186)(40,150,210,181)(41,145,205,182)(42,146,206,183)(43,119,59,97)(44,120,60,98)(45,115,55,99)(46,116,56,100)(47,117,57,101)(48,118,58,102)(49,155,95,109)(50,156,96,110)(51,151,91,111)(52,152,92,112)(53,153,93,113)(54,154,94,114)(133,193,178,238)(134,194,179,239)(135,195,180,240)(136,196,175,235)(137,197,176,236)(138,198,177,237)(139,199,162,222)(140,200,157,217)(141,201,158,218)(142,202,159,219)(143,203,160,220)(144,204,161,221), (1,59,49,19,30)(2,60,50,20,25)(3,55,51,21,26)(4,56,52,22,27)(5,57,53,23,28)(6,58,54,24,29)(7,34,41,221,195)(8,35,42,222,196)(9,36,37,217,197)(10,31,38,218,198)(11,32,39,219,193)(12,33,40,220,194)(13,48,94,62,71)(14,43,95,63,72)(15,44,96,64,67)(16,45,91,65,68)(17,46,92,66,69)(18,47,93,61,70)(73,102,154,127,126)(74,97,155,128,121)(75,98,156,129,122)(76,99,151,130,123)(77,100,152,131,124)(78,101,153,132,125)(79,104,86,119,109)(80,105,87,120,110)(81,106,88,115,111)(82,107,89,116,112)(83,108,90,117,113)(84,103,85,118,114)(133,212,189,186,159)(134,213,190,181,160)(135,214,191,182,161)(136,215,192,183,162)(137,216,187,184,157)(138,211,188,185,158)(139,175,171,166,146)(140,176,172,167,147)(141,177,173,168,148)(142,178,174,163,149)(143,179,169,164,150)(144,180,170,165,145)(199,235,231,226,206)(200,236,232,227,207)(201,237,233,228,208)(202,238,234,223,209)(203,239,229,224,210)(204,240,230,225,205), (1,30)(2,25)(3,26)(4,27)(5,28)(6,29)(7,230)(8,231)(9,232)(10,233)(11,234)(12,229)(13,71)(14,72)(15,67)(16,68)(17,69)(18,70)(19,59)(20,60)(21,55)(22,56)(23,57)(24,58)(31,237)(32,238)(33,239)(34,240)(35,235)(36,236)(37,200)(38,201)(39,202)(40,203)(41,204)(42,199)(43,63)(44,64)(45,65)(46,66)(47,61)(48,62)(73,126)(74,121)(75,122)(76,123)(77,124)(78,125)(79,119)(80,120)(81,115)(82,116)(83,117)(84,118)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(97,128)(98,129)(99,130)(100,131)(101,132)(102,127)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,183)(140,184)(141,185)(142,186)(143,181)(144,182)(145,161)(146,162)(147,157)(148,158)(149,159)(150,160)(169,213)(170,214)(171,215)(172,216)(173,211)(174,212)(175,192)(176,187)(177,188)(178,189)(179,190)(180,191)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(205,221)(206,222)(207,217)(208,218)(209,219)(210,220), (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), (1,142,4,139)(2,141,5,144)(3,140,6,143)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,160,16,157)(14,159,17,162)(15,158,18,161)(19,163,22,166)(20,168,23,165)(21,167,24,164)(25,148,28,145)(26,147,29,150)(27,146,30,149)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,220,76,217)(74,219,77,222)(75,218,78,221)(79,223,82,226)(80,228,83,225)(81,227,84,224)(85,203,88,200)(86,202,89,199)(87,201,90,204)(91,216,94,213)(92,215,95,212)(93,214,96,211)(97,193,100,196)(98,198,101,195)(99,197,102,194)(103,210,106,207)(104,209,107,206)(105,208,108,205)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237)>;
G:=Group( (1,74,14,86)(2,75,15,87)(3,76,16,88)(4,77,17,89)(5,78,18,90)(6,73,13,85)(7,170,230,214)(8,171,231,215)(9,172,232,216)(10,173,233,211)(11,174,234,212)(12,169,229,213)(19,128,63,79)(20,129,64,80)(21,130,65,81)(22,131,66,82)(23,132,61,83)(24,127,62,84)(25,122,67,105)(26,123,68,106)(27,124,69,107)(28,125,70,108)(29,126,71,103)(30,121,72,104)(31,168,228,188)(32,163,223,189)(33,164,224,190)(34,165,225,191)(35,166,226,192)(36,167,227,187)(37,147,207,184)(38,148,208,185)(39,149,209,186)(40,150,210,181)(41,145,205,182)(42,146,206,183)(43,119,59,97)(44,120,60,98)(45,115,55,99)(46,116,56,100)(47,117,57,101)(48,118,58,102)(49,155,95,109)(50,156,96,110)(51,151,91,111)(52,152,92,112)(53,153,93,113)(54,154,94,114)(133,193,178,238)(134,194,179,239)(135,195,180,240)(136,196,175,235)(137,197,176,236)(138,198,177,237)(139,199,162,222)(140,200,157,217)(141,201,158,218)(142,202,159,219)(143,203,160,220)(144,204,161,221), (1,59,49,19,30)(2,60,50,20,25)(3,55,51,21,26)(4,56,52,22,27)(5,57,53,23,28)(6,58,54,24,29)(7,34,41,221,195)(8,35,42,222,196)(9,36,37,217,197)(10,31,38,218,198)(11,32,39,219,193)(12,33,40,220,194)(13,48,94,62,71)(14,43,95,63,72)(15,44,96,64,67)(16,45,91,65,68)(17,46,92,66,69)(18,47,93,61,70)(73,102,154,127,126)(74,97,155,128,121)(75,98,156,129,122)(76,99,151,130,123)(77,100,152,131,124)(78,101,153,132,125)(79,104,86,119,109)(80,105,87,120,110)(81,106,88,115,111)(82,107,89,116,112)(83,108,90,117,113)(84,103,85,118,114)(133,212,189,186,159)(134,213,190,181,160)(135,214,191,182,161)(136,215,192,183,162)(137,216,187,184,157)(138,211,188,185,158)(139,175,171,166,146)(140,176,172,167,147)(141,177,173,168,148)(142,178,174,163,149)(143,179,169,164,150)(144,180,170,165,145)(199,235,231,226,206)(200,236,232,227,207)(201,237,233,228,208)(202,238,234,223,209)(203,239,229,224,210)(204,240,230,225,205), (1,30)(2,25)(3,26)(4,27)(5,28)(6,29)(7,230)(8,231)(9,232)(10,233)(11,234)(12,229)(13,71)(14,72)(15,67)(16,68)(17,69)(18,70)(19,59)(20,60)(21,55)(22,56)(23,57)(24,58)(31,237)(32,238)(33,239)(34,240)(35,235)(36,236)(37,200)(38,201)(39,202)(40,203)(41,204)(42,199)(43,63)(44,64)(45,65)(46,66)(47,61)(48,62)(73,126)(74,121)(75,122)(76,123)(77,124)(78,125)(79,119)(80,120)(81,115)(82,116)(83,117)(84,118)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(97,128)(98,129)(99,130)(100,131)(101,132)(102,127)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,183)(140,184)(141,185)(142,186)(143,181)(144,182)(145,161)(146,162)(147,157)(148,158)(149,159)(150,160)(169,213)(170,214)(171,215)(172,216)(173,211)(174,212)(175,192)(176,187)(177,188)(178,189)(179,190)(180,191)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(205,221)(206,222)(207,217)(208,218)(209,219)(210,220), (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), (1,142,4,139)(2,141,5,144)(3,140,6,143)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,160,16,157)(14,159,17,162)(15,158,18,161)(19,163,22,166)(20,168,23,165)(21,167,24,164)(25,148,28,145)(26,147,29,150)(27,146,30,149)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,220,76,217)(74,219,77,222)(75,218,78,221)(79,223,82,226)(80,228,83,225)(81,227,84,224)(85,203,88,200)(86,202,89,199)(87,201,90,204)(91,216,94,213)(92,215,95,212)(93,214,96,211)(97,193,100,196)(98,198,101,195)(99,197,102,194)(103,210,106,207)(104,209,107,206)(105,208,108,205)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237) );
G=PermutationGroup([[(1,74,14,86),(2,75,15,87),(3,76,16,88),(4,77,17,89),(5,78,18,90),(6,73,13,85),(7,170,230,214),(8,171,231,215),(9,172,232,216),(10,173,233,211),(11,174,234,212),(12,169,229,213),(19,128,63,79),(20,129,64,80),(21,130,65,81),(22,131,66,82),(23,132,61,83),(24,127,62,84),(25,122,67,105),(26,123,68,106),(27,124,69,107),(28,125,70,108),(29,126,71,103),(30,121,72,104),(31,168,228,188),(32,163,223,189),(33,164,224,190),(34,165,225,191),(35,166,226,192),(36,167,227,187),(37,147,207,184),(38,148,208,185),(39,149,209,186),(40,150,210,181),(41,145,205,182),(42,146,206,183),(43,119,59,97),(44,120,60,98),(45,115,55,99),(46,116,56,100),(47,117,57,101),(48,118,58,102),(49,155,95,109),(50,156,96,110),(51,151,91,111),(52,152,92,112),(53,153,93,113),(54,154,94,114),(133,193,178,238),(134,194,179,239),(135,195,180,240),(136,196,175,235),(137,197,176,236),(138,198,177,237),(139,199,162,222),(140,200,157,217),(141,201,158,218),(142,202,159,219),(143,203,160,220),(144,204,161,221)], [(1,59,49,19,30),(2,60,50,20,25),(3,55,51,21,26),(4,56,52,22,27),(5,57,53,23,28),(6,58,54,24,29),(7,34,41,221,195),(8,35,42,222,196),(9,36,37,217,197),(10,31,38,218,198),(11,32,39,219,193),(12,33,40,220,194),(13,48,94,62,71),(14,43,95,63,72),(15,44,96,64,67),(16,45,91,65,68),(17,46,92,66,69),(18,47,93,61,70),(73,102,154,127,126),(74,97,155,128,121),(75,98,156,129,122),(76,99,151,130,123),(77,100,152,131,124),(78,101,153,132,125),(79,104,86,119,109),(80,105,87,120,110),(81,106,88,115,111),(82,107,89,116,112),(83,108,90,117,113),(84,103,85,118,114),(133,212,189,186,159),(134,213,190,181,160),(135,214,191,182,161),(136,215,192,183,162),(137,216,187,184,157),(138,211,188,185,158),(139,175,171,166,146),(140,176,172,167,147),(141,177,173,168,148),(142,178,174,163,149),(143,179,169,164,150),(144,180,170,165,145),(199,235,231,226,206),(200,236,232,227,207),(201,237,233,228,208),(202,238,234,223,209),(203,239,229,224,210),(204,240,230,225,205)], [(1,30),(2,25),(3,26),(4,27),(5,28),(6,29),(7,230),(8,231),(9,232),(10,233),(11,234),(12,229),(13,71),(14,72),(15,67),(16,68),(17,69),(18,70),(19,59),(20,60),(21,55),(22,56),(23,57),(24,58),(31,237),(32,238),(33,239),(34,240),(35,235),(36,236),(37,200),(38,201),(39,202),(40,203),(41,204),(42,199),(43,63),(44,64),(45,65),(46,66),(47,61),(48,62),(73,126),(74,121),(75,122),(76,123),(77,124),(78,125),(79,119),(80,120),(81,115),(82,116),(83,117),(84,118),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(97,128),(98,129),(99,130),(100,131),(101,132),(102,127),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,183),(140,184),(141,185),(142,186),(143,181),(144,182),(145,161),(146,162),(147,157),(148,158),(149,159),(150,160),(169,213),(170,214),(171,215),(172,216),(173,211),(174,212),(175,192),(176,187),(177,188),(178,189),(179,190),(180,191),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(205,221),(206,222),(207,217),(208,218),(209,219),(210,220)], [(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)], [(1,142,4,139),(2,141,5,144),(3,140,6,143),(7,156,10,153),(8,155,11,152),(9,154,12,151),(13,160,16,157),(14,159,17,162),(15,158,18,161),(19,163,22,166),(20,168,23,165),(21,167,24,164),(25,148,28,145),(26,147,29,150),(27,146,30,149),(31,132,34,129),(32,131,35,128),(33,130,36,127),(37,126,40,123),(38,125,41,122),(39,124,42,121),(43,133,46,136),(44,138,47,135),(45,137,48,134),(49,174,52,171),(50,173,53,170),(51,172,54,169),(55,176,58,179),(56,175,59,178),(57,180,60,177),(61,191,64,188),(62,190,65,187),(63,189,66,192),(67,185,70,182),(68,184,71,181),(69,183,72,186),(73,220,76,217),(74,219,77,222),(75,218,78,221),(79,223,82,226),(80,228,83,225),(81,227,84,224),(85,203,88,200),(86,202,89,199),(87,201,90,204),(91,216,94,213),(92,215,95,212),(93,214,96,211),(97,193,100,196),(98,198,101,195),(99,197,102,194),(103,210,106,207),(104,209,107,206),(105,208,108,205),(109,234,112,231),(110,233,113,230),(111,232,114,229),(115,236,118,239),(116,235,119,238),(117,240,120,237)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 3 | 4 | 4 | 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 | 10 | 10 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 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 | 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 | C4 | S3 | D5 | Dic3 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4×D5 | C4○D12 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | D5×Dic3 | C2×S3×D5 | D12⋊5D5 | C12.28D10 |
| kernel | (C4×D5)⋊Dic3 | Dic3×Dic5 | D10⋊Dic3 | C5×C4⋊Dic3 | C60⋊5C4 | D5×C2×C12 | D5×C12 | C2×C4×D5 | C4⋊Dic3 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 8 | 8 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of (C4×D5)⋊Dic3 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 11 | 37 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 45 | 44 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 20 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 |
| 0 | 0 | 0 | 44 | 0 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 47 | 35 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 11 | 37 | 0 | 0 |
| 0 | 5 | 50 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,11,0,0,0,0,37,50,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,45,0,0,0,60,44],[60,0,0,0,0,0,1,0,0,0,0,20,60,0,0,0,0,0,0,44,0,0,0,43,0],[60,0,0,0,0,0,47,0,0,0,0,35,13,0,0,0,0,0,1,0,0,0,0,0,1],[50,0,0,0,0,0,11,5,0,0,0,37,50,0,0,0,0,0,60,0,0,0,0,0,60] >;
(C4×D5)⋊Dic3 in GAP, Magma, Sage, TeX
(C_4\times D_5)\rtimes {\rm Dic}_3 % in TeX
G:=Group("(C4xD5):Dic3"); // GroupNames label
G:=SmallGroup(480,434);
// by ID
G=gap.SmallGroup(480,434);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^5=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;
// generators/relations