direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4⋊Dic3, C30.57D8, C60.140D4, C30.32SD16, (C3×D4)⋊1C20, C6.12(C5×D8), C12.7(C5×D4), (D4×C15)⋊13C4, C12.7(C2×C20), D4⋊1(C5×Dic3), (C5×D4)⋊7Dic3, (D4×C10).6S3, (C6×D4).1C10, C6.5(C5×SD16), C60.177(C2×C4), C4⋊Dic3⋊10C10, (D4×C30).11C2, (C2×C20).350D6, (C2×C30).176D4, C4.1(C10×Dic3), C15⋊19(D4⋊C4), C10.28(D4⋊S3), C20.91(C3⋊D4), C20.51(C2×Dic3), C10.12(D4.S3), (C2×C60).345C22, C30.117(C22⋊C4), C10.33(C6.D4), (C2×C3⋊C8)⋊2C10, (C10×C3⋊C8)⋊16C2, C3⋊3(C5×D4⋊C4), C2.3(C5×D4⋊S3), (C2×D4).1(C5×S3), (C2×C6).33(C5×D4), C4.12(C5×C3⋊D4), C2.3(C5×D4.S3), (C2×C4).38(S3×C10), (C5×C4⋊Dic3)⋊28C2, C6.13(C5×C22⋊C4), (C2×C12).15(C2×C10), C2.3(C5×C6.D4), C22.17(C5×C3⋊D4), (C2×C10).89(C3⋊D4), SmallGroup(480,151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊Dic3
G = < a,b,c,d,e | a5=b4=c2=d6=1, e2=d3, 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: 228 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, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, D4⋊C4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C2×C3⋊C8, C4⋊Dic3, C6×D4, C5×Dic3, C60, C2×C30, C2×C30, C5×C4⋊C4, C2×C40, D4×C10, D4⋊Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, D4×C15, D4×C15, C22×C30, C5×D4⋊C4, C10×C3⋊C8, C5×C4⋊Dic3, D4×C30, C5×D4⋊Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, D8, SD16, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, D4⋊C4, C2×C20, C5×D4, D4⋊S3, D4.S3, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C5×D8, C5×SD16, D4⋊Dic3, C10×Dic3, C5×C3⋊D4, C5×D4⋊C4, C5×D4⋊S3, C5×D4.S3, C5×C6.D4, C5×D4⋊Dic3
►On 240 points
Generators in S240
(1 110 86 62 38)(2 111 87 63 39)(3 112 88 64 40)(4 113 89 65 41)(5 114 90 66 42)(6 109 85 61 37)(7 103 79 55 31)(8 104 80 56 32)(9 105 81 57 33)(10 106 82 58 34)(11 107 83 59 35)(12 108 84 60 36)(13 220 196 172 148)(14 221 197 173 149)(15 222 198 174 150)(16 217 193 169 145)(17 218 194 170 146)(18 219 195 171 147)(19 115 91 67 43)(20 116 92 68 44)(21 117 93 69 45)(22 118 94 70 46)(23 119 95 71 47)(24 120 96 72 48)(25 122 97 73 49)(26 123 98 74 50)(27 124 99 75 51)(28 125 100 76 52)(29 126 101 77 53)(30 121 102 78 54)(127 223 199 175 151)(128 224 200 176 152)(129 225 201 177 153)(130 226 202 178 154)(131 227 203 179 155)(132 228 204 180 156)(133 229 205 181 157)(134 230 206 182 158)(135 231 207 183 159)(136 232 208 184 160)(137 233 209 185 161)(138 234 210 186 162)(139 235 211 187 163)(140 236 212 188 164)(141 237 213 189 165)(142 238 214 190 166)(143 239 215 191 167)(144 240 216 192 168)
(1 29 19 12)(2 30 20 7)(3 25 21 8)(4 26 22 9)(5 27 23 10)(6 28 24 11)(13 231 223 239)(14 232 224 240)(15 233 225 235)(16 234 226 236)(17 229 227 237)(18 230 228 238)(31 39 54 44)(32 40 49 45)(33 41 50 46)(34 42 51 47)(35 37 52 48)(36 38 53 43)(55 63 78 68)(56 64 73 69)(57 65 74 70)(58 66 75 71)(59 61 76 72)(60 62 77 67)(79 87 102 92)(80 88 97 93)(81 89 98 94)(82 90 99 95)(83 85 100 96)(84 86 101 91)(103 111 121 116)(104 112 122 117)(105 113 123 118)(106 114 124 119)(107 109 125 120)(108 110 126 115)(127 143 148 135)(128 144 149 136)(129 139 150 137)(130 140 145 138)(131 141 146 133)(132 142 147 134)(151 167 172 159)(152 168 173 160)(153 163 174 161)(154 164 169 162)(155 165 170 157)(156 166 171 158)(175 191 196 183)(176 192 197 184)(177 187 198 185)(178 188 193 186)(179 189 194 181)(180 190 195 182)(199 215 220 207)(200 216 221 208)(201 211 222 209)(202 212 217 210)(203 213 218 205)(204 214 219 206)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)(43 53)(44 54)(45 49)(46 50)(47 51)(48 52)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 77)(68 78)(69 73)(70 74)(71 75)(72 76)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)(91 101)(92 102)(93 97)(94 98)(95 99)(96 100)(103 111)(104 112)(105 113)(106 114)(107 109)(108 110)(115 126)(116 121)(117 122)(118 123)(119 124)(120 125)(133 141)(134 142)(135 143)(136 144)(137 139)(138 140)(157 165)(158 166)(159 167)(160 168)(161 163)(162 164)(181 189)(182 190)(183 191)(184 192)(185 187)(186 188)(205 213)(206 214)(207 215)(208 216)(209 211)(210 212)(229 237)(230 238)(231 239)(232 240)(233 235)(234 236)
(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)
(1 138 4 135)(2 137 5 134)(3 136 6 133)(7 129 10 132)(8 128 11 131)(9 127 12 130)(13 126 16 123)(14 125 17 122)(15 124 18 121)(19 140 22 143)(20 139 23 142)(21 144 24 141)(25 149 28 146)(26 148 29 145)(27 147 30 150)(31 153 34 156)(32 152 35 155)(33 151 36 154)(37 157 40 160)(38 162 41 159)(39 161 42 158)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 173 52 170)(50 172 53 169)(51 171 54 174)(55 177 58 180)(56 176 59 179)(57 175 60 178)(61 181 64 184)(62 186 65 183)(63 185 66 182)(67 188 70 191)(68 187 71 190)(69 192 72 189)(73 197 76 194)(74 196 77 193)(75 195 78 198)(79 201 82 204)(80 200 83 203)(81 199 84 202)(85 205 88 208)(86 210 89 207)(87 209 90 206)(91 212 94 215)(92 211 95 214)(93 216 96 213)(97 221 100 218)(98 220 101 217)(99 219 102 222)(103 225 106 228)(104 224 107 227)(105 223 108 226)(109 229 112 232)(110 234 113 231)(111 233 114 230)(115 236 118 239)(116 235 119 238)(117 240 120 237)
G:=sub<Sym(240)| (1,110,86,62,38)(2,111,87,63,39)(3,112,88,64,40)(4,113,89,65,41)(5,114,90,66,42)(6,109,85,61,37)(7,103,79,55,31)(8,104,80,56,32)(9,105,81,57,33)(10,106,82,58,34)(11,107,83,59,35)(12,108,84,60,36)(13,220,196,172,148)(14,221,197,173,149)(15,222,198,174,150)(16,217,193,169,145)(17,218,194,170,146)(18,219,195,171,147)(19,115,91,67,43)(20,116,92,68,44)(21,117,93,69,45)(22,118,94,70,46)(23,119,95,71,47)(24,120,96,72,48)(25,122,97,73,49)(26,123,98,74,50)(27,124,99,75,51)(28,125,100,76,52)(29,126,101,77,53)(30,121,102,78,54)(127,223,199,175,151)(128,224,200,176,152)(129,225,201,177,153)(130,226,202,178,154)(131,227,203,179,155)(132,228,204,180,156)(133,229,205,181,157)(134,230,206,182,158)(135,231,207,183,159)(136,232,208,184,160)(137,233,209,185,161)(138,234,210,186,162)(139,235,211,187,163)(140,236,212,188,164)(141,237,213,189,165)(142,238,214,190,166)(143,239,215,191,167)(144,240,216,192,168), (1,29,19,12)(2,30,20,7)(3,25,21,8)(4,26,22,9)(5,27,23,10)(6,28,24,11)(13,231,223,239)(14,232,224,240)(15,233,225,235)(16,234,226,236)(17,229,227,237)(18,230,228,238)(31,39,54,44)(32,40,49,45)(33,41,50,46)(34,42,51,47)(35,37,52,48)(36,38,53,43)(55,63,78,68)(56,64,73,69)(57,65,74,70)(58,66,75,71)(59,61,76,72)(60,62,77,67)(79,87,102,92)(80,88,97,93)(81,89,98,94)(82,90,99,95)(83,85,100,96)(84,86,101,91)(103,111,121,116)(104,112,122,117)(105,113,123,118)(106,114,124,119)(107,109,125,120)(108,110,126,115)(127,143,148,135)(128,144,149,136)(129,139,150,137)(130,140,145,138)(131,141,146,133)(132,142,147,134)(151,167,172,159)(152,168,173,160)(153,163,174,161)(154,164,169,162)(155,165,170,157)(156,166,171,158)(175,191,196,183)(176,192,197,184)(177,187,198,185)(178,188,193,186)(179,189,194,181)(180,190,195,182)(199,215,220,207)(200,216,221,208)(201,211,222,209)(202,212,217,210)(203,213,218,205)(204,214,219,206), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,77)(68,78)(69,73)(70,74)(71,75)(72,76)(79,87)(80,88)(81,89)(82,90)(83,85)(84,86)(91,101)(92,102)(93,97)(94,98)(95,99)(96,100)(103,111)(104,112)(105,113)(106,114)(107,109)(108,110)(115,126)(116,121)(117,122)(118,123)(119,124)(120,125)(133,141)(134,142)(135,143)(136,144)(137,139)(138,140)(157,165)(158,166)(159,167)(160,168)(161,163)(162,164)(181,189)(182,190)(183,191)(184,192)(185,187)(186,188)(205,213)(206,214)(207,215)(208,216)(209,211)(210,212)(229,237)(230,238)(231,239)(232,240)(233,235)(234,236), (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), (1,138,4,135)(2,137,5,134)(3,136,6,133)(7,129,10,132)(8,128,11,131)(9,127,12,130)(13,126,16,123)(14,125,17,122)(15,124,18,121)(19,140,22,143)(20,139,23,142)(21,144,24,141)(25,149,28,146)(26,148,29,145)(27,147,30,150)(31,153,34,156)(32,152,35,155)(33,151,36,154)(37,157,40,160)(38,162,41,159)(39,161,42,158)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,173,52,170)(50,172,53,169)(51,171,54,174)(55,177,58,180)(56,176,59,179)(57,175,60,178)(61,181,64,184)(62,186,65,183)(63,185,66,182)(67,188,70,191)(68,187,71,190)(69,192,72,189)(73,197,76,194)(74,196,77,193)(75,195,78,198)(79,201,82,204)(80,200,83,203)(81,199,84,202)(85,205,88,208)(86,210,89,207)(87,209,90,206)(91,212,94,215)(92,211,95,214)(93,216,96,213)(97,221,100,218)(98,220,101,217)(99,219,102,222)(103,225,106,228)(104,224,107,227)(105,223,108,226)(109,229,112,232)(110,234,113,231)(111,233,114,230)(115,236,118,239)(116,235,119,238)(117,240,120,237)>;
G:=Group( (1,110,86,62,38)(2,111,87,63,39)(3,112,88,64,40)(4,113,89,65,41)(5,114,90,66,42)(6,109,85,61,37)(7,103,79,55,31)(8,104,80,56,32)(9,105,81,57,33)(10,106,82,58,34)(11,107,83,59,35)(12,108,84,60,36)(13,220,196,172,148)(14,221,197,173,149)(15,222,198,174,150)(16,217,193,169,145)(17,218,194,170,146)(18,219,195,171,147)(19,115,91,67,43)(20,116,92,68,44)(21,117,93,69,45)(22,118,94,70,46)(23,119,95,71,47)(24,120,96,72,48)(25,122,97,73,49)(26,123,98,74,50)(27,124,99,75,51)(28,125,100,76,52)(29,126,101,77,53)(30,121,102,78,54)(127,223,199,175,151)(128,224,200,176,152)(129,225,201,177,153)(130,226,202,178,154)(131,227,203,179,155)(132,228,204,180,156)(133,229,205,181,157)(134,230,206,182,158)(135,231,207,183,159)(136,232,208,184,160)(137,233,209,185,161)(138,234,210,186,162)(139,235,211,187,163)(140,236,212,188,164)(141,237,213,189,165)(142,238,214,190,166)(143,239,215,191,167)(144,240,216,192,168), (1,29,19,12)(2,30,20,7)(3,25,21,8)(4,26,22,9)(5,27,23,10)(6,28,24,11)(13,231,223,239)(14,232,224,240)(15,233,225,235)(16,234,226,236)(17,229,227,237)(18,230,228,238)(31,39,54,44)(32,40,49,45)(33,41,50,46)(34,42,51,47)(35,37,52,48)(36,38,53,43)(55,63,78,68)(56,64,73,69)(57,65,74,70)(58,66,75,71)(59,61,76,72)(60,62,77,67)(79,87,102,92)(80,88,97,93)(81,89,98,94)(82,90,99,95)(83,85,100,96)(84,86,101,91)(103,111,121,116)(104,112,122,117)(105,113,123,118)(106,114,124,119)(107,109,125,120)(108,110,126,115)(127,143,148,135)(128,144,149,136)(129,139,150,137)(130,140,145,138)(131,141,146,133)(132,142,147,134)(151,167,172,159)(152,168,173,160)(153,163,174,161)(154,164,169,162)(155,165,170,157)(156,166,171,158)(175,191,196,183)(176,192,197,184)(177,187,198,185)(178,188,193,186)(179,189,194,181)(180,190,195,182)(199,215,220,207)(200,216,221,208)(201,211,222,209)(202,212,217,210)(203,213,218,205)(204,214,219,206), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,77)(68,78)(69,73)(70,74)(71,75)(72,76)(79,87)(80,88)(81,89)(82,90)(83,85)(84,86)(91,101)(92,102)(93,97)(94,98)(95,99)(96,100)(103,111)(104,112)(105,113)(106,114)(107,109)(108,110)(115,126)(116,121)(117,122)(118,123)(119,124)(120,125)(133,141)(134,142)(135,143)(136,144)(137,139)(138,140)(157,165)(158,166)(159,167)(160,168)(161,163)(162,164)(181,189)(182,190)(183,191)(184,192)(185,187)(186,188)(205,213)(206,214)(207,215)(208,216)(209,211)(210,212)(229,237)(230,238)(231,239)(232,240)(233,235)(234,236), (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), (1,138,4,135)(2,137,5,134)(3,136,6,133)(7,129,10,132)(8,128,11,131)(9,127,12,130)(13,126,16,123)(14,125,17,122)(15,124,18,121)(19,140,22,143)(20,139,23,142)(21,144,24,141)(25,149,28,146)(26,148,29,145)(27,147,30,150)(31,153,34,156)(32,152,35,155)(33,151,36,154)(37,157,40,160)(38,162,41,159)(39,161,42,158)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,173,52,170)(50,172,53,169)(51,171,54,174)(55,177,58,180)(56,176,59,179)(57,175,60,178)(61,181,64,184)(62,186,65,183)(63,185,66,182)(67,188,70,191)(68,187,71,190)(69,192,72,189)(73,197,76,194)(74,196,77,193)(75,195,78,198)(79,201,82,204)(80,200,83,203)(81,199,84,202)(85,205,88,208)(86,210,89,207)(87,209,90,206)(91,212,94,215)(92,211,95,214)(93,216,96,213)(97,221,100,218)(98,220,101,217)(99,219,102,222)(103,225,106,228)(104,224,107,227)(105,223,108,226)(109,229,112,232)(110,234,113,231)(111,233,114,230)(115,236,118,239)(116,235,119,238)(117,240,120,237) );
G=PermutationGroup([[(1,110,86,62,38),(2,111,87,63,39),(3,112,88,64,40),(4,113,89,65,41),(5,114,90,66,42),(6,109,85,61,37),(7,103,79,55,31),(8,104,80,56,32),(9,105,81,57,33),(10,106,82,58,34),(11,107,83,59,35),(12,108,84,60,36),(13,220,196,172,148),(14,221,197,173,149),(15,222,198,174,150),(16,217,193,169,145),(17,218,194,170,146),(18,219,195,171,147),(19,115,91,67,43),(20,116,92,68,44),(21,117,93,69,45),(22,118,94,70,46),(23,119,95,71,47),(24,120,96,72,48),(25,122,97,73,49),(26,123,98,74,50),(27,124,99,75,51),(28,125,100,76,52),(29,126,101,77,53),(30,121,102,78,54),(127,223,199,175,151),(128,224,200,176,152),(129,225,201,177,153),(130,226,202,178,154),(131,227,203,179,155),(132,228,204,180,156),(133,229,205,181,157),(134,230,206,182,158),(135,231,207,183,159),(136,232,208,184,160),(137,233,209,185,161),(138,234,210,186,162),(139,235,211,187,163),(140,236,212,188,164),(141,237,213,189,165),(142,238,214,190,166),(143,239,215,191,167),(144,240,216,192,168)], [(1,29,19,12),(2,30,20,7),(3,25,21,8),(4,26,22,9),(5,27,23,10),(6,28,24,11),(13,231,223,239),(14,232,224,240),(15,233,225,235),(16,234,226,236),(17,229,227,237),(18,230,228,238),(31,39,54,44),(32,40,49,45),(33,41,50,46),(34,42,51,47),(35,37,52,48),(36,38,53,43),(55,63,78,68),(56,64,73,69),(57,65,74,70),(58,66,75,71),(59,61,76,72),(60,62,77,67),(79,87,102,92),(80,88,97,93),(81,89,98,94),(82,90,99,95),(83,85,100,96),(84,86,101,91),(103,111,121,116),(104,112,122,117),(105,113,123,118),(106,114,124,119),(107,109,125,120),(108,110,126,115),(127,143,148,135),(128,144,149,136),(129,139,150,137),(130,140,145,138),(131,141,146,133),(132,142,147,134),(151,167,172,159),(152,168,173,160),(153,163,174,161),(154,164,169,162),(155,165,170,157),(156,166,171,158),(175,191,196,183),(176,192,197,184),(177,187,198,185),(178,188,193,186),(179,189,194,181),(180,190,195,182),(199,215,220,207),(200,216,221,208),(201,211,222,209),(202,212,217,210),(203,213,218,205),(204,214,219,206)], [(1,12),(2,7),(3,8),(4,9),(5,10),(6,11),(19,29),(20,30),(21,25),(22,26),(23,27),(24,28),(31,39),(32,40),(33,41),(34,42),(35,37),(36,38),(43,53),(44,54),(45,49),(46,50),(47,51),(48,52),(55,63),(56,64),(57,65),(58,66),(59,61),(60,62),(67,77),(68,78),(69,73),(70,74),(71,75),(72,76),(79,87),(80,88),(81,89),(82,90),(83,85),(84,86),(91,101),(92,102),(93,97),(94,98),(95,99),(96,100),(103,111),(104,112),(105,113),(106,114),(107,109),(108,110),(115,126),(116,121),(117,122),(118,123),(119,124),(120,125),(133,141),(134,142),(135,143),(136,144),(137,139),(138,140),(157,165),(158,166),(159,167),(160,168),(161,163),(162,164),(181,189),(182,190),(183,191),(184,192),(185,187),(186,188),(205,213),(206,214),(207,215),(208,216),(209,211),(210,212),(229,237),(230,238),(231,239),(232,240),(233,235),(234,236)], [(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)], [(1,138,4,135),(2,137,5,134),(3,136,6,133),(7,129,10,132),(8,128,11,131),(9,127,12,130),(13,126,16,123),(14,125,17,122),(15,124,18,121),(19,140,22,143),(20,139,23,142),(21,144,24,141),(25,149,28,146),(26,148,29,145),(27,147,30,150),(31,153,34,156),(32,152,35,155),(33,151,36,154),(37,157,40,160),(38,162,41,159),(39,161,42,158),(43,164,46,167),(44,163,47,166),(45,168,48,165),(49,173,52,170),(50,172,53,169),(51,171,54,174),(55,177,58,180),(56,176,59,179),(57,175,60,178),(61,181,64,184),(62,186,65,183),(63,185,66,182),(67,188,70,191),(68,187,71,190),(69,192,72,189),(73,197,76,194),(74,196,77,193),(75,195,78,198),(79,201,82,204),(80,200,83,203),(81,199,84,202),(85,205,88,208),(86,210,89,207),(87,209,90,206),(91,212,94,215),(92,211,95,214),(93,216,96,213),(97,221,100,218),(98,220,101,217),(99,219,102,222),(103,225,106,228),(104,224,107,227),(105,223,108,226),(109,229,112,232),(110,234,113,231),(111,233,114,230),(115,236,118,239),(116,235,119,238),(117,240,120,237)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 30M | ··· | 30AB | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
120 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 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D4 | D6 | Dic3 | D8 | SD16 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | C5×Dic3 | C5×D8 | C5×SD16 | C5×C3⋊D4 | C5×C3⋊D4 | D4⋊S3 | D4.S3 | C5×D4⋊S3 | C5×D4.S3 |
| kernel | C5×D4⋊Dic3 | C10×C3⋊C8 | C5×C4⋊Dic3 | D4×C30 | D4×C15 | D4⋊Dic3 | C2×C3⋊C8 | C4⋊Dic3 | C6×D4 | C3×D4 | D4×C10 | C60 | C2×C30 | C2×C20 | C5×D4 | C30 | C30 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C6 | C6 | C4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×D4⋊Dic3 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 239 |
| 0 | 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 239 |
| 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 219 |
| 0 | 0 | 0 | 230 | 0 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,98,0,0,0,0,0,98,0,0,0,0,0,87,0,0,0,0,0,87],[1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,1,1,0,0,0,239,240],[240,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,239,240],[240,0,0,0,0,0,225,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[177,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,230,0,0,0,219,0] >;
C5×D4⋊Dic3 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C5xD4:Dic3"); // GroupNames label
G:=SmallGroup(480,151);
// by ID
G=gap.SmallGroup(480,151);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=1,e^2=d^3,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