direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×C5⋊D4, C15⋊27(C4×D4), C5⋊5(D4×Dic3), D10⋊4(C2×Dic3), (C5×Dic3)⋊14D4, C30.233(C2×D4), C10.157(S3×D4), C22⋊2(D5×Dic3), C23.29(S3×D5), Dic5⋊2(C2×Dic3), C6.84(C4○D20), C30.Q8⋊33C2, (C22×Dic3)⋊5D5, (C22×D5).64D6, (C22×C6).32D10, (Dic3×Dic5)⋊35C2, C30.148(C4○D4), D10⋊Dic3⋊31C2, C30.38D4⋊24C2, (C2×C30).195C23, C30.146(C22×C4), (C2×Dic5).128D6, (C22×C10).107D6, C10.56(D4⋊2S3), (C2×Dic3).187D10, C2.6(Dic5.D6), (C22×C30).57C22, C10.32(C22×Dic3), (C6×Dic5).112C22, (C10×Dic3).202C22, (C2×Dic15).135C22, C3⋊6(C4×C5⋊D4), (C2×C6)⋊1(C4×D5), C6.95(C2×C4×D5), (C6×D5)⋊9(C2×C4), (C3×C5⋊D4)⋊3C4, C2.5(S3×C5⋊D4), (C2×C30)⋊17(C2×C4), (Dic3×C2×C10)⋊5C2, (C2×D5×Dic3)⋊17C2, (C2×C5⋊D4).9S3, (C6×C5⋊D4).6C2, C6.60(C2×C5⋊D4), C2.19(C2×D5×Dic3), C22.86(C2×S3×D5), (C3×Dic5)⋊5(C2×C4), (D5×C2×C6).51C22, (C2×C10)⋊12(C2×Dic3), (C2×C6).207(C22×D5), (C2×C10).207(C22×S3), SmallGroup(480,629)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3×C5⋊D4
G = < a,b,c,d,e | a6=c5=d4=e2=1, b2=a3, 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: 764 in 188 conjugacy classes, 72 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×Dic3, C6×D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, D4×Dic3, D5×Dic3, C6×Dic5, C3×C5⋊D4, C10×Dic3, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C4×C5⋊D4, Dic3×Dic5, D10⋊Dic3, C30.Q8, C30.38D4, C2×D5×Dic3, C6×C5⋊D4, Dic3×C2×C10, Dic3×C5⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22×C4, C2×D4, C4○D4, D10, C2×Dic3, C22×S3, C4×D4, C4×D5, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C22×Dic3, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, D4×Dic3, D5×Dic3, C2×S3×D5, C4×C5⋊D4, C2×D5×Dic3, Dic5.D6, S3×C5⋊D4, Dic3×C5⋊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)
(1 170 4 173)(2 169 5 172)(3 174 6 171)(7 100 10 97)(8 99 11 102)(9 98 12 101)(13 190 16 187)(14 189 17 192)(15 188 18 191)(19 216 22 213)(20 215 23 212)(21 214 24 211)(25 177 28 180)(26 176 29 179)(27 175 30 178)(31 74 34 77)(32 73 35 76)(33 78 36 75)(37 108 40 105)(38 107 41 104)(39 106 42 103)(43 186 46 183)(44 185 47 182)(45 184 48 181)(49 139 52 142)(50 144 53 141)(51 143 54 140)(55 150 58 147)(56 149 59 146)(57 148 60 145)(61 160 64 157)(62 159 65 162)(63 158 66 161)(67 163 70 166)(68 168 71 165)(69 167 72 164)(79 200 82 203)(80 199 83 202)(81 204 84 201)(85 231 88 234)(86 230 89 233)(87 229 90 232)(91 134 94 137)(92 133 95 136)(93 138 96 135)(109 240 112 237)(110 239 113 236)(111 238 114 235)(115 210 118 207)(116 209 119 206)(117 208 120 205)(121 224 124 227)(122 223 125 226)(123 228 126 225)(127 198 130 195)(128 197 131 194)(129 196 132 193)(151 220 154 217)(152 219 155 222)(153 218 156 221)
(1 59 23 15 43)(2 60 24 16 44)(3 55 19 17 45)(4 56 20 18 46)(5 57 21 13 47)(6 58 22 14 48)(7 35 41 232 208)(8 36 42 233 209)(9 31 37 234 210)(10 32 38 229 205)(11 33 39 230 206)(12 34 40 231 207)(25 66 94 67 53)(26 61 95 68 54)(27 62 96 69 49)(28 63 91 70 50)(29 64 92 71 51)(30 65 93 72 52)(73 107 90 117 97)(74 108 85 118 98)(75 103 86 119 99)(76 104 87 120 100)(77 105 88 115 101)(78 106 89 116 102)(79 113 151 129 123)(80 114 152 130 124)(81 109 153 131 125)(82 110 154 132 126)(83 111 155 127 121)(84 112 156 128 122)(133 165 143 179 157)(134 166 144 180 158)(135 167 139 175 159)(136 168 140 176 160)(137 163 141 177 161)(138 164 142 178 162)(145 211 187 185 169)(146 212 188 186 170)(147 213 189 181 171)(148 214 190 182 172)(149 215 191 183 173)(150 216 192 184 174)(193 225 203 239 217)(194 226 204 240 218)(195 227 199 235 219)(196 228 200 236 220)(197 223 201 237 221)(198 224 202 238 222)
(1 89 52 80)(2 90 53 81)(3 85 54 82)(4 86 49 83)(5 87 50 84)(6 88 51 79)(7 137 218 187)(8 138 219 188)(9 133 220 189)(10 134 221 190)(11 135 222 191)(12 136 217 192)(13 100 91 156)(14 101 92 151)(15 102 93 152)(16 97 94 153)(17 98 95 154)(18 99 96 155)(19 74 61 132)(20 75 62 127)(21 76 63 128)(22 77 64 129)(23 78 65 130)(24 73 66 131)(25 125 60 107)(26 126 55 108)(27 121 56 103)(28 122 57 104)(29 123 58 105)(30 124 59 106)(31 157 196 213)(32 158 197 214)(33 159 198 215)(34 160 193 216)(35 161 194 211)(36 162 195 212)(37 179 228 147)(38 180 223 148)(39 175 224 149)(40 176 225 150)(41 177 226 145)(42 178 227 146)(43 116 72 114)(44 117 67 109)(45 118 68 110)(46 119 69 111)(47 120 70 112)(48 115 71 113)(139 202 173 230)(140 203 174 231)(141 204 169 232)(142 199 170 233)(143 200 171 234)(144 201 172 229)(163 240 185 208)(164 235 186 209)(165 236 181 210)(166 237 182 205)(167 238 183 206)(168 239 184 207)
(1 4)(2 5)(3 6)(7 197)(8 198)(9 193)(10 194)(11 195)(12 196)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)(25 70)(26 71)(27 72)(28 67)(29 68)(30 69)(31 217)(32 218)(33 219)(34 220)(35 221)(36 222)(37 239)(38 240)(39 235)(40 236)(41 237)(42 238)(43 56)(44 57)(45 58)(46 59)(47 60)(48 55)(49 52)(50 53)(51 54)(61 92)(62 93)(63 94)(64 95)(65 96)(66 91)(73 156)(74 151)(75 152)(76 153)(77 154)(78 155)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(97 128)(98 129)(99 130)(100 131)(101 132)(102 127)(103 114)(104 109)(105 110)(106 111)(107 112)(108 113)(115 126)(116 121)(117 122)(118 123)(119 124)(120 125)(133 160)(134 161)(135 162)(136 157)(137 158)(138 159)(139 142)(140 143)(141 144)(145 182)(146 183)(147 184)(148 185)(149 186)(150 181)(163 180)(164 175)(165 176)(166 177)(167 178)(168 179)(169 172)(170 173)(171 174)(187 214)(188 215)(189 216)(190 211)(191 212)(192 213)(199 230)(200 231)(201 232)(202 233)(203 234)(204 229)(205 226)(206 227)(207 228)(208 223)(209 224)(210 225)
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), (1,170,4,173)(2,169,5,172)(3,174,6,171)(7,100,10,97)(8,99,11,102)(9,98,12,101)(13,190,16,187)(14,189,17,192)(15,188,18,191)(19,216,22,213)(20,215,23,212)(21,214,24,211)(25,177,28,180)(26,176,29,179)(27,175,30,178)(31,74,34,77)(32,73,35,76)(33,78,36,75)(37,108,40,105)(38,107,41,104)(39,106,42,103)(43,186,46,183)(44,185,47,182)(45,184,48,181)(49,139,52,142)(50,144,53,141)(51,143,54,140)(55,150,58,147)(56,149,59,146)(57,148,60,145)(61,160,64,157)(62,159,65,162)(63,158,66,161)(67,163,70,166)(68,168,71,165)(69,167,72,164)(79,200,82,203)(80,199,83,202)(81,204,84,201)(85,231,88,234)(86,230,89,233)(87,229,90,232)(91,134,94,137)(92,133,95,136)(93,138,96,135)(109,240,112,237)(110,239,113,236)(111,238,114,235)(115,210,118,207)(116,209,119,206)(117,208,120,205)(121,224,124,227)(122,223,125,226)(123,228,126,225)(127,198,130,195)(128,197,131,194)(129,196,132,193)(151,220,154,217)(152,219,155,222)(153,218,156,221), (1,59,23,15,43)(2,60,24,16,44)(3,55,19,17,45)(4,56,20,18,46)(5,57,21,13,47)(6,58,22,14,48)(7,35,41,232,208)(8,36,42,233,209)(9,31,37,234,210)(10,32,38,229,205)(11,33,39,230,206)(12,34,40,231,207)(25,66,94,67,53)(26,61,95,68,54)(27,62,96,69,49)(28,63,91,70,50)(29,64,92,71,51)(30,65,93,72,52)(73,107,90,117,97)(74,108,85,118,98)(75,103,86,119,99)(76,104,87,120,100)(77,105,88,115,101)(78,106,89,116,102)(79,113,151,129,123)(80,114,152,130,124)(81,109,153,131,125)(82,110,154,132,126)(83,111,155,127,121)(84,112,156,128,122)(133,165,143,179,157)(134,166,144,180,158)(135,167,139,175,159)(136,168,140,176,160)(137,163,141,177,161)(138,164,142,178,162)(145,211,187,185,169)(146,212,188,186,170)(147,213,189,181,171)(148,214,190,182,172)(149,215,191,183,173)(150,216,192,184,174)(193,225,203,239,217)(194,226,204,240,218)(195,227,199,235,219)(196,228,200,236,220)(197,223,201,237,221)(198,224,202,238,222), (1,89,52,80)(2,90,53,81)(3,85,54,82)(4,86,49,83)(5,87,50,84)(6,88,51,79)(7,137,218,187)(8,138,219,188)(9,133,220,189)(10,134,221,190)(11,135,222,191)(12,136,217,192)(13,100,91,156)(14,101,92,151)(15,102,93,152)(16,97,94,153)(17,98,95,154)(18,99,96,155)(19,74,61,132)(20,75,62,127)(21,76,63,128)(22,77,64,129)(23,78,65,130)(24,73,66,131)(25,125,60,107)(26,126,55,108)(27,121,56,103)(28,122,57,104)(29,123,58,105)(30,124,59,106)(31,157,196,213)(32,158,197,214)(33,159,198,215)(34,160,193,216)(35,161,194,211)(36,162,195,212)(37,179,228,147)(38,180,223,148)(39,175,224,149)(40,176,225,150)(41,177,226,145)(42,178,227,146)(43,116,72,114)(44,117,67,109)(45,118,68,110)(46,119,69,111)(47,120,70,112)(48,115,71,113)(139,202,173,230)(140,203,174,231)(141,204,169,232)(142,199,170,233)(143,200,171,234)(144,201,172,229)(163,240,185,208)(164,235,186,209)(165,236,181,210)(166,237,182,205)(167,238,183,206)(168,239,184,207), (1,4)(2,5)(3,6)(7,197)(8,198)(9,193)(10,194)(11,195)(12,196)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,217)(32,218)(33,219)(34,220)(35,221)(36,222)(37,239)(38,240)(39,235)(40,236)(41,237)(42,238)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,52)(50,53)(51,54)(61,92)(62,93)(63,94)(64,95)(65,96)(66,91)(73,156)(74,151)(75,152)(76,153)(77,154)(78,155)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(97,128)(98,129)(99,130)(100,131)(101,132)(102,127)(103,114)(104,109)(105,110)(106,111)(107,112)(108,113)(115,126)(116,121)(117,122)(118,123)(119,124)(120,125)(133,160)(134,161)(135,162)(136,157)(137,158)(138,159)(139,142)(140,143)(141,144)(145,182)(146,183)(147,184)(148,185)(149,186)(150,181)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(169,172)(170,173)(171,174)(187,214)(188,215)(189,216)(190,211)(191,212)(192,213)(199,230)(200,231)(201,232)(202,233)(203,234)(204,229)(205,226)(206,227)(207,228)(208,223)(209,224)(210,225)>;
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), (1,170,4,173)(2,169,5,172)(3,174,6,171)(7,100,10,97)(8,99,11,102)(9,98,12,101)(13,190,16,187)(14,189,17,192)(15,188,18,191)(19,216,22,213)(20,215,23,212)(21,214,24,211)(25,177,28,180)(26,176,29,179)(27,175,30,178)(31,74,34,77)(32,73,35,76)(33,78,36,75)(37,108,40,105)(38,107,41,104)(39,106,42,103)(43,186,46,183)(44,185,47,182)(45,184,48,181)(49,139,52,142)(50,144,53,141)(51,143,54,140)(55,150,58,147)(56,149,59,146)(57,148,60,145)(61,160,64,157)(62,159,65,162)(63,158,66,161)(67,163,70,166)(68,168,71,165)(69,167,72,164)(79,200,82,203)(80,199,83,202)(81,204,84,201)(85,231,88,234)(86,230,89,233)(87,229,90,232)(91,134,94,137)(92,133,95,136)(93,138,96,135)(109,240,112,237)(110,239,113,236)(111,238,114,235)(115,210,118,207)(116,209,119,206)(117,208,120,205)(121,224,124,227)(122,223,125,226)(123,228,126,225)(127,198,130,195)(128,197,131,194)(129,196,132,193)(151,220,154,217)(152,219,155,222)(153,218,156,221), (1,59,23,15,43)(2,60,24,16,44)(3,55,19,17,45)(4,56,20,18,46)(5,57,21,13,47)(6,58,22,14,48)(7,35,41,232,208)(8,36,42,233,209)(9,31,37,234,210)(10,32,38,229,205)(11,33,39,230,206)(12,34,40,231,207)(25,66,94,67,53)(26,61,95,68,54)(27,62,96,69,49)(28,63,91,70,50)(29,64,92,71,51)(30,65,93,72,52)(73,107,90,117,97)(74,108,85,118,98)(75,103,86,119,99)(76,104,87,120,100)(77,105,88,115,101)(78,106,89,116,102)(79,113,151,129,123)(80,114,152,130,124)(81,109,153,131,125)(82,110,154,132,126)(83,111,155,127,121)(84,112,156,128,122)(133,165,143,179,157)(134,166,144,180,158)(135,167,139,175,159)(136,168,140,176,160)(137,163,141,177,161)(138,164,142,178,162)(145,211,187,185,169)(146,212,188,186,170)(147,213,189,181,171)(148,214,190,182,172)(149,215,191,183,173)(150,216,192,184,174)(193,225,203,239,217)(194,226,204,240,218)(195,227,199,235,219)(196,228,200,236,220)(197,223,201,237,221)(198,224,202,238,222), (1,89,52,80)(2,90,53,81)(3,85,54,82)(4,86,49,83)(5,87,50,84)(6,88,51,79)(7,137,218,187)(8,138,219,188)(9,133,220,189)(10,134,221,190)(11,135,222,191)(12,136,217,192)(13,100,91,156)(14,101,92,151)(15,102,93,152)(16,97,94,153)(17,98,95,154)(18,99,96,155)(19,74,61,132)(20,75,62,127)(21,76,63,128)(22,77,64,129)(23,78,65,130)(24,73,66,131)(25,125,60,107)(26,126,55,108)(27,121,56,103)(28,122,57,104)(29,123,58,105)(30,124,59,106)(31,157,196,213)(32,158,197,214)(33,159,198,215)(34,160,193,216)(35,161,194,211)(36,162,195,212)(37,179,228,147)(38,180,223,148)(39,175,224,149)(40,176,225,150)(41,177,226,145)(42,178,227,146)(43,116,72,114)(44,117,67,109)(45,118,68,110)(46,119,69,111)(47,120,70,112)(48,115,71,113)(139,202,173,230)(140,203,174,231)(141,204,169,232)(142,199,170,233)(143,200,171,234)(144,201,172,229)(163,240,185,208)(164,235,186,209)(165,236,181,210)(166,237,182,205)(167,238,183,206)(168,239,184,207), (1,4)(2,5)(3,6)(7,197)(8,198)(9,193)(10,194)(11,195)(12,196)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,217)(32,218)(33,219)(34,220)(35,221)(36,222)(37,239)(38,240)(39,235)(40,236)(41,237)(42,238)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,52)(50,53)(51,54)(61,92)(62,93)(63,94)(64,95)(65,96)(66,91)(73,156)(74,151)(75,152)(76,153)(77,154)(78,155)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(97,128)(98,129)(99,130)(100,131)(101,132)(102,127)(103,114)(104,109)(105,110)(106,111)(107,112)(108,113)(115,126)(116,121)(117,122)(118,123)(119,124)(120,125)(133,160)(134,161)(135,162)(136,157)(137,158)(138,159)(139,142)(140,143)(141,144)(145,182)(146,183)(147,184)(148,185)(149,186)(150,181)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(169,172)(170,173)(171,174)(187,214)(188,215)(189,216)(190,211)(191,212)(192,213)(199,230)(200,231)(201,232)(202,233)(203,234)(204,229)(205,226)(206,227)(207,228)(208,223)(209,224)(210,225) );
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)], [(1,170,4,173),(2,169,5,172),(3,174,6,171),(7,100,10,97),(8,99,11,102),(9,98,12,101),(13,190,16,187),(14,189,17,192),(15,188,18,191),(19,216,22,213),(20,215,23,212),(21,214,24,211),(25,177,28,180),(26,176,29,179),(27,175,30,178),(31,74,34,77),(32,73,35,76),(33,78,36,75),(37,108,40,105),(38,107,41,104),(39,106,42,103),(43,186,46,183),(44,185,47,182),(45,184,48,181),(49,139,52,142),(50,144,53,141),(51,143,54,140),(55,150,58,147),(56,149,59,146),(57,148,60,145),(61,160,64,157),(62,159,65,162),(63,158,66,161),(67,163,70,166),(68,168,71,165),(69,167,72,164),(79,200,82,203),(80,199,83,202),(81,204,84,201),(85,231,88,234),(86,230,89,233),(87,229,90,232),(91,134,94,137),(92,133,95,136),(93,138,96,135),(109,240,112,237),(110,239,113,236),(111,238,114,235),(115,210,118,207),(116,209,119,206),(117,208,120,205),(121,224,124,227),(122,223,125,226),(123,228,126,225),(127,198,130,195),(128,197,131,194),(129,196,132,193),(151,220,154,217),(152,219,155,222),(153,218,156,221)], [(1,59,23,15,43),(2,60,24,16,44),(3,55,19,17,45),(4,56,20,18,46),(5,57,21,13,47),(6,58,22,14,48),(7,35,41,232,208),(8,36,42,233,209),(9,31,37,234,210),(10,32,38,229,205),(11,33,39,230,206),(12,34,40,231,207),(25,66,94,67,53),(26,61,95,68,54),(27,62,96,69,49),(28,63,91,70,50),(29,64,92,71,51),(30,65,93,72,52),(73,107,90,117,97),(74,108,85,118,98),(75,103,86,119,99),(76,104,87,120,100),(77,105,88,115,101),(78,106,89,116,102),(79,113,151,129,123),(80,114,152,130,124),(81,109,153,131,125),(82,110,154,132,126),(83,111,155,127,121),(84,112,156,128,122),(133,165,143,179,157),(134,166,144,180,158),(135,167,139,175,159),(136,168,140,176,160),(137,163,141,177,161),(138,164,142,178,162),(145,211,187,185,169),(146,212,188,186,170),(147,213,189,181,171),(148,214,190,182,172),(149,215,191,183,173),(150,216,192,184,174),(193,225,203,239,217),(194,226,204,240,218),(195,227,199,235,219),(196,228,200,236,220),(197,223,201,237,221),(198,224,202,238,222)], [(1,89,52,80),(2,90,53,81),(3,85,54,82),(4,86,49,83),(5,87,50,84),(6,88,51,79),(7,137,218,187),(8,138,219,188),(9,133,220,189),(10,134,221,190),(11,135,222,191),(12,136,217,192),(13,100,91,156),(14,101,92,151),(15,102,93,152),(16,97,94,153),(17,98,95,154),(18,99,96,155),(19,74,61,132),(20,75,62,127),(21,76,63,128),(22,77,64,129),(23,78,65,130),(24,73,66,131),(25,125,60,107),(26,126,55,108),(27,121,56,103),(28,122,57,104),(29,123,58,105),(30,124,59,106),(31,157,196,213),(32,158,197,214),(33,159,198,215),(34,160,193,216),(35,161,194,211),(36,162,195,212),(37,179,228,147),(38,180,223,148),(39,175,224,149),(40,176,225,150),(41,177,226,145),(42,178,227,146),(43,116,72,114),(44,117,67,109),(45,118,68,110),(46,119,69,111),(47,120,70,112),(48,115,71,113),(139,202,173,230),(140,203,174,231),(141,204,169,232),(142,199,170,233),(143,200,171,234),(144,201,172,229),(163,240,185,208),(164,235,186,209),(165,236,181,210),(166,237,182,205),(167,238,183,206),(168,239,184,207)], [(1,4),(2,5),(3,6),(7,197),(8,198),(9,193),(10,194),(11,195),(12,196),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23),(25,70),(26,71),(27,72),(28,67),(29,68),(30,69),(31,217),(32,218),(33,219),(34,220),(35,221),(36,222),(37,239),(38,240),(39,235),(40,236),(41,237),(42,238),(43,56),(44,57),(45,58),(46,59),(47,60),(48,55),(49,52),(50,53),(51,54),(61,92),(62,93),(63,94),(64,95),(65,96),(66,91),(73,156),(74,151),(75,152),(76,153),(77,154),(78,155),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(97,128),(98,129),(99,130),(100,131),(101,132),(102,127),(103,114),(104,109),(105,110),(106,111),(107,112),(108,113),(115,126),(116,121),(117,122),(118,123),(119,124),(120,125),(133,160),(134,161),(135,162),(136,157),(137,158),(138,159),(139,142),(140,143),(141,144),(145,182),(146,183),(147,184),(148,185),(149,186),(150,181),(163,180),(164,175),(165,176),(166,177),(167,178),(168,179),(169,172),(170,173),(171,174),(187,214),(188,215),(189,216),(190,211),(191,212),(192,213),(199,230),(200,231),(201,232),(202,233),(203,234),(204,229),(205,226),(206,227),(207,228),(208,223),(209,224),(210,225)]])
78 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 | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20P | 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 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
78 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 | Dic3 | D6 | D6 | C4○D4 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D5×Dic3 | C2×S3×D5 | Dic5.D6 | S3×C5⋊D4 |
| kernel | Dic3×C5⋊D4 | Dic3×Dic5 | D10⋊Dic3 | C30.Q8 | C30.38D4 | C2×D5×Dic3 | C6×C5⋊D4 | Dic3×C2×C10 | C3×C5⋊D4 | C2×C5⋊D4 | C5×Dic3 | C22×Dic3 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | Dic3 | C2×C6 | C6 | C10 | C10 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic3×C5⋊D4 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 24 |
| 0 | 0 | 0 | 0 | 51 | 34 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 47 | 16 | 0 | 0 | 0 | 0 |
| 22 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 43 | 0 | 0 |
| 0 | 0 | 16 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 43 | 0 | 0 |
| 0 | 0 | 16 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,27,51,0,0,0,0,24,34],[17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,45,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,22,0,0,0,0,16,14,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3×C5⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_5\rtimes D_4 % in TeX
G:=Group("Dic3xC5:D4"); // GroupNames label
G:=SmallGroup(480,629);
// by ID
G=gap.SmallGroup(480,629);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^5=d^4=e^2=1,b^2=a^3,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