direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊Dic5, C30.50D8, C60.117D4, C30.30SD16, (C5×D4)⋊4C12, (C6×D4).6D5, C20.7(C3×D4), (D4×C15)⋊10C4, C4⋊Dic5⋊10C6, D4⋊1(C3×Dic5), (C3×D4)⋊4Dic5, (D4×C10).1C6, (D4×C30).6C2, C10.12(C3×D8), C4.1(C6×Dic5), C20.28(C2×C12), C60.162(C2×C4), C6.28(D4⋊D5), (C2×C30).159D4, C10.5(C3×SD16), C15⋊18(D4⋊C4), (C2×C12).354D10, C12.91(C5⋊D4), C6.12(D4.D5), C12.30(C2×Dic5), (C2×C60).279C22, C6.22(C23.D5), C30.110(C22⋊C4), (C2×C5⋊2C8)⋊2C6, C5⋊4(C3×D4⋊C4), C2.3(C3×D4⋊D5), (C6×C5⋊2C8)⋊16C2, (C2×D4).1(C3×D5), (C2×C4).33(C6×D5), C4.12(C3×C5⋊D4), C2.3(C3×D4.D5), (C2×C20).15(C2×C6), (C3×C4⋊Dic5)⋊28C2, (C2×C10).34(C3×D4), C2.3(C3×C23.D5), (C2×C6).89(C5⋊D4), C10.24(C3×C22⋊C4), C22.17(C3×C5⋊D4), SmallGroup(480,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊Dic5
G = < a,b,c,d,e | a3=b4=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 272 in 100 conjugacy classes, 50 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, D4⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, C3×C4⋊C4, C2×C24, C6×D4, C3×Dic5, C60, C2×C30, C2×C30, C2×C5⋊2C8, C4⋊Dic5, D4×C10, C3×D4⋊C4, C3×C5⋊2C8, C6×Dic5, C2×C60, D4×C15, D4×C15, C22×C30, D4⋊Dic5, C6×C5⋊2C8, C3×C4⋊Dic5, D4×C30, C3×D4⋊Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D8, SD16, Dic5, D10, C2×C12, C3×D4, C3×D5, D4⋊C4, C2×Dic5, C5⋊D4, C3×C22⋊C4, C3×D8, C3×SD16, C3×Dic5, C6×D5, D4⋊D5, D4.D5, C23.D5, C3×D4⋊C4, C6×Dic5, C3×C5⋊D4, D4⋊Dic5, C3×D4⋊D5, C3×D4.D5, C3×C23.D5, C3×D4⋊Dic5
►On 240 points
Generators in S240
(1 95 55)(2 96 56)(3 97 57)(4 98 58)(5 99 59)(6 100 60)(7 91 51)(8 92 52)(9 93 53)(10 94 54)(11 108 68)(12 109 69)(13 110 70)(14 101 61)(15 102 62)(16 103 63)(17 104 64)(18 105 65)(19 106 66)(20 107 67)(21 208 168)(22 209 169)(23 210 170)(24 201 161)(25 202 162)(26 203 163)(27 204 164)(28 205 165)(29 206 166)(30 207 167)(31 111 71)(32 112 72)(33 113 73)(34 114 74)(35 115 75)(36 116 76)(37 117 77)(38 118 78)(39 119 79)(40 120 80)(41 124 81)(42 125 82)(43 126 83)(44 127 84)(45 128 85)(46 129 86)(47 130 87)(48 121 88)(49 122 89)(50 123 90)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)
(1 47 31 15)(2 48 32 16)(3 49 33 17)(4 50 34 18)(5 41 35 19)(6 42 36 20)(7 43 37 11)(8 44 38 12)(9 45 39 13)(10 46 40 14)(21 212 228 239)(22 213 229 240)(23 214 230 231)(24 215 221 232)(25 216 222 233)(26 217 223 234)(27 218 224 235)(28 219 225 236)(29 220 226 237)(30 211 227 238)(51 83 77 68)(52 84 78 69)(53 85 79 70)(54 86 80 61)(55 87 71 62)(56 88 72 63)(57 89 73 64)(58 90 74 65)(59 81 75 66)(60 82 76 67)(91 126 117 108)(92 127 118 109)(93 128 119 110)(94 129 120 101)(95 130 111 102)(96 121 112 103)(97 122 113 104)(98 123 114 105)(99 124 115 106)(100 125 116 107)(131 147 158 167)(132 148 159 168)(133 149 160 169)(134 150 151 170)(135 141 152 161)(136 142 153 162)(137 143 154 163)(138 144 155 164)(139 145 156 165)(140 146 157 166)(171 187 198 207)(172 188 199 208)(173 189 200 209)(174 190 191 210)(175 181 192 201)(176 182 193 202)(177 183 194 203)(178 184 195 204)(179 185 196 205)(180 186 197 206)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 11)(8 12)(9 13)(10 14)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(111 130)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)(131 158)(132 159)(133 160)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)(171 198)(172 199)(173 200)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)(211 238)(212 239)(213 240)(214 231)(215 232)(216 233)(217 234)(218 235)(219 236)(220 237)
(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)
(1 137 6 132)(2 136 7 131)(3 135 8 140)(4 134 9 139)(5 133 10 138)(11 147 16 142)(12 146 17 141)(13 145 18 150)(14 144 19 149)(15 143 20 148)(21 130 26 125)(22 129 27 124)(23 128 28 123)(24 127 29 122)(25 126 30 121)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 171 56 176)(52 180 57 175)(53 179 58 174)(54 178 59 173)(55 177 60 172)(61 184 66 189)(62 183 67 188)(63 182 68 187)(64 181 69 186)(65 190 70 185)(71 194 76 199)(72 193 77 198)(73 192 78 197)(74 191 79 196)(75 200 80 195)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 211 96 216)(92 220 97 215)(93 219 98 214)(94 218 99 213)(95 217 100 212)(101 224 106 229)(102 223 107 228)(103 222 108 227)(104 221 109 226)(105 230 110 225)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)
G:=sub<Sym(240)| (1,95,55)(2,96,56)(3,97,57)(4,98,58)(5,99,59)(6,100,60)(7,91,51)(8,92,52)(9,93,53)(10,94,54)(11,108,68)(12,109,69)(13,110,70)(14,101,61)(15,102,62)(16,103,63)(17,104,64)(18,105,65)(19,106,66)(20,107,67)(21,208,168)(22,209,169)(23,210,170)(24,201,161)(25,202,162)(26,203,163)(27,204,164)(28,205,165)(29,206,166)(30,207,167)(31,111,71)(32,112,72)(33,113,73)(34,114,74)(35,115,75)(36,116,76)(37,117,77)(38,118,78)(39,119,79)(40,120,80)(41,124,81)(42,125,82)(43,126,83)(44,127,84)(45,128,85)(46,129,86)(47,130,87)(48,121,88)(49,122,89)(50,123,90)(131,211,171)(132,212,172)(133,213,173)(134,214,174)(135,215,175)(136,216,176)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (1,47,31,15)(2,48,32,16)(3,49,33,17)(4,50,34,18)(5,41,35,19)(6,42,36,20)(7,43,37,11)(8,44,38,12)(9,45,39,13)(10,46,40,14)(21,212,228,239)(22,213,229,240)(23,214,230,231)(24,215,221,232)(25,216,222,233)(26,217,223,234)(27,218,224,235)(28,219,225,236)(29,220,226,237)(30,211,227,238)(51,83,77,68)(52,84,78,69)(53,85,79,70)(54,86,80,61)(55,87,71,62)(56,88,72,63)(57,89,73,64)(58,90,74,65)(59,81,75,66)(60,82,76,67)(91,126,117,108)(92,127,118,109)(93,128,119,110)(94,129,120,101)(95,130,111,102)(96,121,112,103)(97,122,113,104)(98,123,114,105)(99,124,115,106)(100,125,116,107)(131,147,158,167)(132,148,159,168)(133,149,160,169)(134,150,151,170)(135,141,152,161)(136,142,153,162)(137,143,154,163)(138,144,155,164)(139,145,156,165)(140,146,157,166)(171,187,198,207)(172,188,199,208)(173,189,200,209)(174,190,191,210)(175,181,192,201)(176,182,193,202)(177,183,194,203)(178,184,195,204)(179,185,196,205)(180,186,197,206), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,158)(132,159)(133,160)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157)(171,198)(172,199)(173,200)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(211,238)(212,239)(213,240)(214,231)(215,232)(216,233)(217,234)(218,235)(219,236)(220,237), (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), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(11,147,16,142)(12,146,17,141)(13,145,18,150)(14,144,19,149)(15,143,20,148)(21,130,26,125)(22,129,27,124)(23,128,28,123)(24,127,29,122)(25,126,30,121)(31,154,36,159)(32,153,37,158)(33,152,38,157)(34,151,39,156)(35,160,40,155)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,171,56,176)(52,180,57,175)(53,179,58,174)(54,178,59,173)(55,177,60,172)(61,184,66,189)(62,183,67,188)(63,182,68,187)(64,181,69,186)(65,190,70,185)(71,194,76,199)(72,193,77,198)(73,192,78,197)(74,191,79,196)(75,200,80,195)(81,209,86,204)(82,208,87,203)(83,207,88,202)(84,206,89,201)(85,205,90,210)(91,211,96,216)(92,220,97,215)(93,219,98,214)(94,218,99,213)(95,217,100,212)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235)>;
G:=Group( (1,95,55)(2,96,56)(3,97,57)(4,98,58)(5,99,59)(6,100,60)(7,91,51)(8,92,52)(9,93,53)(10,94,54)(11,108,68)(12,109,69)(13,110,70)(14,101,61)(15,102,62)(16,103,63)(17,104,64)(18,105,65)(19,106,66)(20,107,67)(21,208,168)(22,209,169)(23,210,170)(24,201,161)(25,202,162)(26,203,163)(27,204,164)(28,205,165)(29,206,166)(30,207,167)(31,111,71)(32,112,72)(33,113,73)(34,114,74)(35,115,75)(36,116,76)(37,117,77)(38,118,78)(39,119,79)(40,120,80)(41,124,81)(42,125,82)(43,126,83)(44,127,84)(45,128,85)(46,129,86)(47,130,87)(48,121,88)(49,122,89)(50,123,90)(131,211,171)(132,212,172)(133,213,173)(134,214,174)(135,215,175)(136,216,176)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (1,47,31,15)(2,48,32,16)(3,49,33,17)(4,50,34,18)(5,41,35,19)(6,42,36,20)(7,43,37,11)(8,44,38,12)(9,45,39,13)(10,46,40,14)(21,212,228,239)(22,213,229,240)(23,214,230,231)(24,215,221,232)(25,216,222,233)(26,217,223,234)(27,218,224,235)(28,219,225,236)(29,220,226,237)(30,211,227,238)(51,83,77,68)(52,84,78,69)(53,85,79,70)(54,86,80,61)(55,87,71,62)(56,88,72,63)(57,89,73,64)(58,90,74,65)(59,81,75,66)(60,82,76,67)(91,126,117,108)(92,127,118,109)(93,128,119,110)(94,129,120,101)(95,130,111,102)(96,121,112,103)(97,122,113,104)(98,123,114,105)(99,124,115,106)(100,125,116,107)(131,147,158,167)(132,148,159,168)(133,149,160,169)(134,150,151,170)(135,141,152,161)(136,142,153,162)(137,143,154,163)(138,144,155,164)(139,145,156,165)(140,146,157,166)(171,187,198,207)(172,188,199,208)(173,189,200,209)(174,190,191,210)(175,181,192,201)(176,182,193,202)(177,183,194,203)(178,184,195,204)(179,185,196,205)(180,186,197,206), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,158)(132,159)(133,160)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157)(171,198)(172,199)(173,200)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(211,238)(212,239)(213,240)(214,231)(215,232)(216,233)(217,234)(218,235)(219,236)(220,237), (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), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(11,147,16,142)(12,146,17,141)(13,145,18,150)(14,144,19,149)(15,143,20,148)(21,130,26,125)(22,129,27,124)(23,128,28,123)(24,127,29,122)(25,126,30,121)(31,154,36,159)(32,153,37,158)(33,152,38,157)(34,151,39,156)(35,160,40,155)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,171,56,176)(52,180,57,175)(53,179,58,174)(54,178,59,173)(55,177,60,172)(61,184,66,189)(62,183,67,188)(63,182,68,187)(64,181,69,186)(65,190,70,185)(71,194,76,199)(72,193,77,198)(73,192,78,197)(74,191,79,196)(75,200,80,195)(81,209,86,204)(82,208,87,203)(83,207,88,202)(84,206,89,201)(85,205,90,210)(91,211,96,216)(92,220,97,215)(93,219,98,214)(94,218,99,213)(95,217,100,212)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235) );
G=PermutationGroup([[(1,95,55),(2,96,56),(3,97,57),(4,98,58),(5,99,59),(6,100,60),(7,91,51),(8,92,52),(9,93,53),(10,94,54),(11,108,68),(12,109,69),(13,110,70),(14,101,61),(15,102,62),(16,103,63),(17,104,64),(18,105,65),(19,106,66),(20,107,67),(21,208,168),(22,209,169),(23,210,170),(24,201,161),(25,202,162),(26,203,163),(27,204,164),(28,205,165),(29,206,166),(30,207,167),(31,111,71),(32,112,72),(33,113,73),(34,114,74),(35,115,75),(36,116,76),(37,117,77),(38,118,78),(39,119,79),(40,120,80),(41,124,81),(42,125,82),(43,126,83),(44,127,84),(45,128,85),(46,129,86),(47,130,87),(48,121,88),(49,122,89),(50,123,90),(131,211,171),(132,212,172),(133,213,173),(134,214,174),(135,215,175),(136,216,176),(137,217,177),(138,218,178),(139,219,179),(140,220,180),(141,221,181),(142,222,182),(143,223,183),(144,224,184),(145,225,185),(146,226,186),(147,227,187),(148,228,188),(149,229,189),(150,230,190),(151,231,191),(152,232,192),(153,233,193),(154,234,194),(155,235,195),(156,236,196),(157,237,197),(158,238,198),(159,239,199),(160,240,200)], [(1,47,31,15),(2,48,32,16),(3,49,33,17),(4,50,34,18),(5,41,35,19),(6,42,36,20),(7,43,37,11),(8,44,38,12),(9,45,39,13),(10,46,40,14),(21,212,228,239),(22,213,229,240),(23,214,230,231),(24,215,221,232),(25,216,222,233),(26,217,223,234),(27,218,224,235),(28,219,225,236),(29,220,226,237),(30,211,227,238),(51,83,77,68),(52,84,78,69),(53,85,79,70),(54,86,80,61),(55,87,71,62),(56,88,72,63),(57,89,73,64),(58,90,74,65),(59,81,75,66),(60,82,76,67),(91,126,117,108),(92,127,118,109),(93,128,119,110),(94,129,120,101),(95,130,111,102),(96,121,112,103),(97,122,113,104),(98,123,114,105),(99,124,115,106),(100,125,116,107),(131,147,158,167),(132,148,159,168),(133,149,160,169),(134,150,151,170),(135,141,152,161),(136,142,153,162),(137,143,154,163),(138,144,155,164),(139,145,156,165),(140,146,157,166),(171,187,198,207),(172,188,199,208),(173,189,200,209),(174,190,191,210),(175,181,192,201),(176,182,193,202),(177,183,194,203),(178,184,195,204),(179,185,196,205),(180,186,197,206)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,11),(8,12),(9,13),(10,14),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(51,68),(52,69),(53,70),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(91,108),(92,109),(93,110),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107),(111,130),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129),(131,158),(132,159),(133,160),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157),(171,198),(172,199),(173,200),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,197),(211,238),(212,239),(213,240),(214,231),(215,232),(216,233),(217,234),(218,235),(219,236),(220,237)], [(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)], [(1,137,6,132),(2,136,7,131),(3,135,8,140),(4,134,9,139),(5,133,10,138),(11,147,16,142),(12,146,17,141),(13,145,18,150),(14,144,19,149),(15,143,20,148),(21,130,26,125),(22,129,27,124),(23,128,28,123),(24,127,29,122),(25,126,30,121),(31,154,36,159),(32,153,37,158),(33,152,38,157),(34,151,39,156),(35,160,40,155),(41,169,46,164),(42,168,47,163),(43,167,48,162),(44,166,49,161),(45,165,50,170),(51,171,56,176),(52,180,57,175),(53,179,58,174),(54,178,59,173),(55,177,60,172),(61,184,66,189),(62,183,67,188),(63,182,68,187),(64,181,69,186),(65,190,70,185),(71,194,76,199),(72,193,77,198),(73,192,78,197),(74,191,79,196),(75,200,80,195),(81,209,86,204),(82,208,87,203),(83,207,88,202),(84,206,89,201),(85,205,90,210),(91,211,96,216),(92,220,97,215),(93,219,98,214),(94,218,99,213),(95,217,100,212),(101,224,106,229),(102,223,107,228),(103,222,108,227),(104,221,109,226),(105,230,110,225),(111,234,116,239),(112,233,117,238),(113,232,118,237),(114,231,119,236),(115,240,120,235)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D5 | D8 | SD16 | D10 | Dic5 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C3×D8 | C3×SD16 | C6×D5 | C3×Dic5 | C3×C5⋊D4 | C3×C5⋊D4 | D4⋊D5 | D4.D5 | C3×D4⋊D5 | C3×D4.D5 |
| kernel | C3×D4⋊Dic5 | C6×C5⋊2C8 | C3×C4⋊Dic5 | D4×C30 | D4⋊Dic5 | D4×C15 | C2×C5⋊2C8 | C4⋊Dic5 | D4×C10 | C5×D4 | C60 | C2×C30 | C6×D4 | C30 | C30 | C2×C12 | C3×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C10 | C10 | C2×C4 | D4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D4⋊Dic5 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 215 |
| 0 | 0 | 204 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 37 | 240 |
| 52 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 181 | 240 | 0 | 0 |
| 227 | 60 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 20 | 0 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,240,204,0,0,215,1],[1,0,0,0,0,1,0,0,0,0,1,37,0,0,0,240],[52,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[181,227,0,0,240,60,0,0,0,0,0,20,0,0,12,0] >;
C3×D4⋊Dic5 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("C3xD4:Dic5"); // GroupNames label
G:=SmallGroup(480,110);
// by ID
G=gap.SmallGroup(480,110);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,2524,1271,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations