direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12⋊D4, C20⋊9D12, C60⋊19D4, D6⋊2(C5×D4), C12⋊1(C5×D4), C4⋊2(C5×D12), D6⋊C4⋊7C10, C6.7(D4×C10), (S3×C10)⋊14D4, (C2×D12)⋊4C10, C2.9(C10×D12), (C10×D12)⋊20C2, C15⋊30(C4⋊D4), C10.179(S3×D4), C30.294(C2×D4), C10.78(C2×D12), (C2×C20).354D6, C30.270(C4○D4), (C2×C30).415C23, (C2×C60).334C22, C10.51(Q8⋊3S3), (C10×Dic3).221C22, C4⋊C4⋊3(C5×S3), (S3×C2×C4)⋊1C10, C3⋊2(C5×C4⋊D4), (C3×C4⋊C4)⋊6C10, (C5×C4⋊C4)⋊12S3, C2.13(C5×S3×D4), (S3×C2×C20)⋊11C2, (C15×C4⋊C4)⋊24C2, (C5×D6⋊C4)⋊23C2, C6.33(C5×C4○D4), (C2×C4).11(S3×C10), C22.50(S3×C2×C10), C2.6(C5×Q8⋊3S3), (C2×C12).24(C2×C10), (S3×C2×C10).112C22, (C22×S3).7(C2×C10), (C2×C6).36(C22×C10), (C2×C10).349(C22×S3), (C2×Dic3).28(C2×C10), SmallGroup(480,774)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12⋊D4
G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >
Subgroups: 532 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, C30, C4⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C12⋊D4, S3×C20, C5×D12, C10×Dic3, C2×C60, C2×C60, S3×C2×C10, S3×C2×C10, C5×C4⋊D4, C5×D6⋊C4, C15×C4⋊C4, S3×C2×C20, C10×D12, C10×D12, C5×C12⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, D12, C22×S3, C5×S3, C4⋊D4, C5×D4, C22×C10, C2×D12, S3×D4, Q8⋊3S3, S3×C10, D4×C10, C5×C4○D4, C12⋊D4, C5×D12, S3×C2×C10, C5×C4⋊D4, C10×D12, C5×S3×D4, C5×Q8⋊3S3, C5×C12⋊D4
►On 240 points
Generators in S240
(1 40 154 106 222)(2 41 155 107 223)(3 42 156 108 224)(4 43 145 97 225)(5 44 146 98 226)(6 45 147 99 227)(7 46 148 100 228)(8 47 149 101 217)(9 48 150 102 218)(10 37 151 103 219)(11 38 152 104 220)(12 39 153 105 221)(13 236 129 165 30)(14 237 130 166 31)(15 238 131 167 32)(16 239 132 168 33)(17 240 121 157 34)(18 229 122 158 35)(19 230 123 159 36)(20 231 124 160 25)(21 232 125 161 26)(22 233 126 162 27)(23 234 127 163 28)(24 235 128 164 29)(49 199 133 84 116)(50 200 134 73 117)(51 201 135 74 118)(52 202 136 75 119)(53 203 137 76 120)(54 204 138 77 109)(55 193 139 78 110)(56 194 140 79 111)(57 195 141 80 112)(58 196 142 81 113)(59 197 143 82 114)(60 198 144 83 115)(61 93 214 191 180)(62 94 215 192 169)(63 95 216 181 170)(64 96 205 182 171)(65 85 206 183 172)(66 86 207 184 173)(67 87 208 185 174)(68 88 209 186 175)(69 89 210 187 176)(70 90 211 188 177)(71 91 212 189 178)(72 92 213 190 179)
(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)
(1 169 159 138)(2 176 160 133)(3 171 161 140)(4 178 162 135)(5 173 163 142)(6 180 164 137)(7 175 165 144)(8 170 166 139)(9 177 167 134)(10 172 168 141)(11 179 157 136)(12 174 158 143)(13 115 148 88)(14 110 149 95)(15 117 150 90)(16 112 151 85)(17 119 152 92)(18 114 153 87)(19 109 154 94)(20 116 155 89)(21 111 156 96)(22 118 145 91)(23 113 146 86)(24 120 147 93)(25 84 41 69)(26 79 42 64)(27 74 43 71)(28 81 44 66)(29 76 45 61)(30 83 46 68)(31 78 47 63)(32 73 48 70)(33 80 37 65)(34 75 38 72)(35 82 39 67)(36 77 40 62)(49 107 210 231)(50 102 211 238)(51 97 212 233)(52 104 213 240)(53 99 214 235)(54 106 215 230)(55 101 216 237)(56 108 205 232)(57 103 206 239)(58 98 207 234)(59 105 208 229)(60 100 209 236)(121 202 220 190)(122 197 221 185)(123 204 222 192)(124 199 223 187)(125 194 224 182)(126 201 225 189)(127 196 226 184)(128 203 227 191)(129 198 228 186)(130 193 217 181)(131 200 218 188)(132 195 219 183)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 88)(14 87)(15 86)(16 85)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 84)(46 83)(47 82)(48 81)(49 99)(50 98)(51 97)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 100)(109 154)(110 153)(111 152)(112 151)(113 150)(114 149)(115 148)(116 147)(117 146)(118 145)(119 156)(120 155)(121 182)(122 181)(123 192)(124 191)(125 190)(126 189)(127 188)(128 187)(129 186)(130 185)(131 184)(132 183)(157 171)(158 170)(159 169)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(193 221)(194 220)(195 219)(196 218)(197 217)(198 228)(199 227)(200 226)(201 225)(202 224)(203 223)(204 222)(205 240)(206 239)(207 238)(208 237)(209 236)(210 235)(211 234)(212 233)(213 232)(214 231)(215 230)(216 229)
G:=sub<Sym(240)| (1,40,154,106,222)(2,41,155,107,223)(3,42,156,108,224)(4,43,145,97,225)(5,44,146,98,226)(6,45,147,99,227)(7,46,148,100,228)(8,47,149,101,217)(9,48,150,102,218)(10,37,151,103,219)(11,38,152,104,220)(12,39,153,105,221)(13,236,129,165,30)(14,237,130,166,31)(15,238,131,167,32)(16,239,132,168,33)(17,240,121,157,34)(18,229,122,158,35)(19,230,123,159,36)(20,231,124,160,25)(21,232,125,161,26)(22,233,126,162,27)(23,234,127,163,28)(24,235,128,164,29)(49,199,133,84,116)(50,200,134,73,117)(51,201,135,74,118)(52,202,136,75,119)(53,203,137,76,120)(54,204,138,77,109)(55,193,139,78,110)(56,194,140,79,111)(57,195,141,80,112)(58,196,142,81,113)(59,197,143,82,114)(60,198,144,83,115)(61,93,214,191,180)(62,94,215,192,169)(63,95,216,181,170)(64,96,205,182,171)(65,85,206,183,172)(66,86,207,184,173)(67,87,208,185,174)(68,88,209,186,175)(69,89,210,187,176)(70,90,211,188,177)(71,91,212,189,178)(72,92,213,190,179), (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), (1,169,159,138)(2,176,160,133)(3,171,161,140)(4,178,162,135)(5,173,163,142)(6,180,164,137)(7,175,165,144)(8,170,166,139)(9,177,167,134)(10,172,168,141)(11,179,157,136)(12,174,158,143)(13,115,148,88)(14,110,149,95)(15,117,150,90)(16,112,151,85)(17,119,152,92)(18,114,153,87)(19,109,154,94)(20,116,155,89)(21,111,156,96)(22,118,145,91)(23,113,146,86)(24,120,147,93)(25,84,41,69)(26,79,42,64)(27,74,43,71)(28,81,44,66)(29,76,45,61)(30,83,46,68)(31,78,47,63)(32,73,48,70)(33,80,37,65)(34,75,38,72)(35,82,39,67)(36,77,40,62)(49,107,210,231)(50,102,211,238)(51,97,212,233)(52,104,213,240)(53,99,214,235)(54,106,215,230)(55,101,216,237)(56,108,205,232)(57,103,206,239)(58,98,207,234)(59,105,208,229)(60,100,209,236)(121,202,220,190)(122,197,221,185)(123,204,222,192)(124,199,223,187)(125,194,224,182)(126,201,225,189)(127,196,226,184)(128,203,227,191)(129,198,228,186)(130,193,217,181)(131,200,218,188)(132,195,219,183), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,99)(50,98)(51,97)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,156)(120,155)(121,182)(122,181)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(157,171)(158,170)(159,169)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(193,221)(194,220)(195,219)(196,218)(197,217)(198,228)(199,227)(200,226)(201,225)(202,224)(203,223)(204,222)(205,240)(206,239)(207,238)(208,237)(209,236)(210,235)(211,234)(212,233)(213,232)(214,231)(215,230)(216,229)>;
G:=Group( (1,40,154,106,222)(2,41,155,107,223)(3,42,156,108,224)(4,43,145,97,225)(5,44,146,98,226)(6,45,147,99,227)(7,46,148,100,228)(8,47,149,101,217)(9,48,150,102,218)(10,37,151,103,219)(11,38,152,104,220)(12,39,153,105,221)(13,236,129,165,30)(14,237,130,166,31)(15,238,131,167,32)(16,239,132,168,33)(17,240,121,157,34)(18,229,122,158,35)(19,230,123,159,36)(20,231,124,160,25)(21,232,125,161,26)(22,233,126,162,27)(23,234,127,163,28)(24,235,128,164,29)(49,199,133,84,116)(50,200,134,73,117)(51,201,135,74,118)(52,202,136,75,119)(53,203,137,76,120)(54,204,138,77,109)(55,193,139,78,110)(56,194,140,79,111)(57,195,141,80,112)(58,196,142,81,113)(59,197,143,82,114)(60,198,144,83,115)(61,93,214,191,180)(62,94,215,192,169)(63,95,216,181,170)(64,96,205,182,171)(65,85,206,183,172)(66,86,207,184,173)(67,87,208,185,174)(68,88,209,186,175)(69,89,210,187,176)(70,90,211,188,177)(71,91,212,189,178)(72,92,213,190,179), (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), (1,169,159,138)(2,176,160,133)(3,171,161,140)(4,178,162,135)(5,173,163,142)(6,180,164,137)(7,175,165,144)(8,170,166,139)(9,177,167,134)(10,172,168,141)(11,179,157,136)(12,174,158,143)(13,115,148,88)(14,110,149,95)(15,117,150,90)(16,112,151,85)(17,119,152,92)(18,114,153,87)(19,109,154,94)(20,116,155,89)(21,111,156,96)(22,118,145,91)(23,113,146,86)(24,120,147,93)(25,84,41,69)(26,79,42,64)(27,74,43,71)(28,81,44,66)(29,76,45,61)(30,83,46,68)(31,78,47,63)(32,73,48,70)(33,80,37,65)(34,75,38,72)(35,82,39,67)(36,77,40,62)(49,107,210,231)(50,102,211,238)(51,97,212,233)(52,104,213,240)(53,99,214,235)(54,106,215,230)(55,101,216,237)(56,108,205,232)(57,103,206,239)(58,98,207,234)(59,105,208,229)(60,100,209,236)(121,202,220,190)(122,197,221,185)(123,204,222,192)(124,199,223,187)(125,194,224,182)(126,201,225,189)(127,196,226,184)(128,203,227,191)(129,198,228,186)(130,193,217,181)(131,200,218,188)(132,195,219,183), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,99)(50,98)(51,97)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,156)(120,155)(121,182)(122,181)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(157,171)(158,170)(159,169)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(193,221)(194,220)(195,219)(196,218)(197,217)(198,228)(199,227)(200,226)(201,225)(202,224)(203,223)(204,222)(205,240)(206,239)(207,238)(208,237)(209,236)(210,235)(211,234)(212,233)(213,232)(214,231)(215,230)(216,229) );
G=PermutationGroup([[(1,40,154,106,222),(2,41,155,107,223),(3,42,156,108,224),(4,43,145,97,225),(5,44,146,98,226),(6,45,147,99,227),(7,46,148,100,228),(8,47,149,101,217),(9,48,150,102,218),(10,37,151,103,219),(11,38,152,104,220),(12,39,153,105,221),(13,236,129,165,30),(14,237,130,166,31),(15,238,131,167,32),(16,239,132,168,33),(17,240,121,157,34),(18,229,122,158,35),(19,230,123,159,36),(20,231,124,160,25),(21,232,125,161,26),(22,233,126,162,27),(23,234,127,163,28),(24,235,128,164,29),(49,199,133,84,116),(50,200,134,73,117),(51,201,135,74,118),(52,202,136,75,119),(53,203,137,76,120),(54,204,138,77,109),(55,193,139,78,110),(56,194,140,79,111),(57,195,141,80,112),(58,196,142,81,113),(59,197,143,82,114),(60,198,144,83,115),(61,93,214,191,180),(62,94,215,192,169),(63,95,216,181,170),(64,96,205,182,171),(65,85,206,183,172),(66,86,207,184,173),(67,87,208,185,174),(68,88,209,186,175),(69,89,210,187,176),(70,90,211,188,177),(71,91,212,189,178),(72,92,213,190,179)], [(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)], [(1,169,159,138),(2,176,160,133),(3,171,161,140),(4,178,162,135),(5,173,163,142),(6,180,164,137),(7,175,165,144),(8,170,166,139),(9,177,167,134),(10,172,168,141),(11,179,157,136),(12,174,158,143),(13,115,148,88),(14,110,149,95),(15,117,150,90),(16,112,151,85),(17,119,152,92),(18,114,153,87),(19,109,154,94),(20,116,155,89),(21,111,156,96),(22,118,145,91),(23,113,146,86),(24,120,147,93),(25,84,41,69),(26,79,42,64),(27,74,43,71),(28,81,44,66),(29,76,45,61),(30,83,46,68),(31,78,47,63),(32,73,48,70),(33,80,37,65),(34,75,38,72),(35,82,39,67),(36,77,40,62),(49,107,210,231),(50,102,211,238),(51,97,212,233),(52,104,213,240),(53,99,214,235),(54,106,215,230),(55,101,216,237),(56,108,205,232),(57,103,206,239),(58,98,207,234),(59,105,208,229),(60,100,209,236),(121,202,220,190),(122,197,221,185),(123,204,222,192),(124,199,223,187),(125,194,224,182),(126,201,225,189),(127,196,226,184),(128,203,227,191),(129,198,228,186),(130,193,217,181),(131,200,218,188),(132,195,219,183)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,88),(14,87),(15,86),(16,85),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,61),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,84),(46,83),(47,82),(48,81),(49,99),(50,98),(51,97),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,100),(109,154),(110,153),(111,152),(112,151),(113,150),(114,149),(115,148),(116,147),(117,146),(118,145),(119,156),(120,155),(121,182),(122,181),(123,192),(124,191),(125,190),(126,189),(127,188),(128,187),(129,186),(130,185),(131,184),(132,183),(157,171),(158,170),(159,169),(160,180),(161,179),(162,178),(163,177),(164,176),(165,175),(166,174),(167,173),(168,172),(193,221),(194,220),(195,219),(196,218),(197,217),(198,228),(199,227),(200,226),(201,225),(202,224),(203,223),(204,222),(205,240),(206,239),(207,238),(208,237),(209,236),(210,235),(211,234),(212,233),(213,232),(214,231),(215,230),(216,229)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 12 | ··· | 12 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | C4○D4 | D12 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | C5×C4○D4 | C5×D12 | S3×D4 | Q8⋊3S3 | C5×S3×D4 | C5×Q8⋊3S3 |
| kernel | C5×C12⋊D4 | C5×D6⋊C4 | C15×C4⋊C4 | S3×C2×C20 | C10×D12 | C12⋊D4 | D6⋊C4 | C3×C4⋊C4 | S3×C2×C4 | C2×D12 | C5×C4⋊C4 | C60 | S3×C10 | C2×C20 | C30 | C20 | C4⋊C4 | C12 | D6 | C2×C4 | C6 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 8 | 4 | 4 | 12 | 1 | 2 | 2 | 3 | 2 | 4 | 4 | 8 | 8 | 12 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C12⋊D4 ►in GL6(𝔽61)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 53 | 59 | 0 | 0 | 0 | 0 |
| 2 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 53 | 59 | 0 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[53,2,0,0,0,0,59,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[53,1,0,0,0,0,59,8,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C5×C12⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}\rtimes D_4 % in TeX
G:=Group("C5xC12:D4"); // GroupNames label
G:=SmallGroup(480,774);
// by ID
G=gap.SmallGroup(480,774);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations