direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C60, C42⋊7C30, C4⋊C4⋊7C30, C4⋊1(C2×C60), C12⋊6(C2×C20), (C4×C60)⋊25C2, C60⋊38(C2×C4), C20⋊9(C2×C12), (C4×C20)⋊17C6, C2.3(D4×C30), (C4×C12)⋊11C10, C22⋊C4⋊6C30, (C22×C4)⋊4C30, (C22×C20)⋊8C6, (C22×C60)⋊8C2, C22⋊2(C2×C60), (C2×D4).7C30, C6.66(D4×C10), C10.66(C6×D4), (C22×C12)⋊4C10, (C6×D4).14C10, (D4×C30).28C2, (D4×C10).14C6, C30.449(C2×D4), C2.4(C22×C60), C6.32(C22×C20), C23.10(C2×C30), C30.275(C4○D4), (C2×C30).453C23, C30.239(C22×C4), C10.45(C22×C12), (C2×C60).576C22, C22.7(C22×C30), (C22×C30).131C22, (C5×C4⋊C4)⋊16C6, (C2×C6)⋊4(C2×C20), (C15×C4⋊C4)⋊34C2, (C3×C4⋊C4)⋊16C10, (C2×C30)⋊32(C2×C4), C6.39(C5×C4○D4), C2.2(C15×C4○D4), (C2×C10)⋊11(C2×C12), (C5×C22⋊C4)⋊14C6, (C2×C4).11(C2×C30), C10.39(C3×C4○D4), (C15×C22⋊C4)⋊30C2, (C3×C22⋊C4)⋊14C10, (C2×C12).79(C2×C10), (C2×C20).136(C2×C6), (C2×C6).73(C22×C10), (C2×C10).73(C22×C6), (C22×C6).26(C2×C10), (C22×C10).34(C2×C6), SmallGroup(480,923)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C60
G = < a,b,c | a60=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 248 in 188 conjugacy classes, 128 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C4×D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C60, C60, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C12, C2×C60, C2×C60, C2×C60, D4×C15, C22×C30, D4×C20, C4×C60, C15×C22⋊C4, C15×C4⋊C4, C22×C60, D4×C30, D4×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C23, C10, C12, C2×C6, C15, C22×C4, C2×D4, C4○D4, C20, C2×C10, C2×C12, C3×D4, C22×C6, C30, C4×D4, C2×C20, C5×D4, C22×C10, C22×C12, C6×D4, C3×C4○D4, C60, C2×C30, C22×C20, D4×C10, C5×C4○D4, D4×C12, C2×C60, D4×C15, C22×C30, D4×C20, C22×C60, D4×C30, C15×C4○D4, D4×C60
►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)
(1 123 83 198)(2 124 84 199)(3 125 85 200)(4 126 86 201)(5 127 87 202)(6 128 88 203)(7 129 89 204)(8 130 90 205)(9 131 91 206)(10 132 92 207)(11 133 93 208)(12 134 94 209)(13 135 95 210)(14 136 96 211)(15 137 97 212)(16 138 98 213)(17 139 99 214)(18 140 100 215)(19 141 101 216)(20 142 102 217)(21 143 103 218)(22 144 104 219)(23 145 105 220)(24 146 106 221)(25 147 107 222)(26 148 108 223)(27 149 109 224)(28 150 110 225)(29 151 111 226)(30 152 112 227)(31 153 113 228)(32 154 114 229)(33 155 115 230)(34 156 116 231)(35 157 117 232)(36 158 118 233)(37 159 119 234)(38 160 120 235)(39 161 61 236)(40 162 62 237)(41 163 63 238)(42 164 64 239)(43 165 65 240)(44 166 66 181)(45 167 67 182)(46 168 68 183)(47 169 69 184)(48 170 70 185)(49 171 71 186)(50 172 72 187)(51 173 73 188)(52 174 74 189)(53 175 75 190)(54 176 76 191)(55 177 77 192)(56 178 78 193)(57 179 79 194)(58 180 80 195)(59 121 81 196)(60 122 82 197)
(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 181)(167 182)(168 183)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)
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), (1,123,83,198)(2,124,84,199)(3,125,85,200)(4,126,86,201)(5,127,87,202)(6,128,88,203)(7,129,89,204)(8,130,90,205)(9,131,91,206)(10,132,92,207)(11,133,93,208)(12,134,94,209)(13,135,95,210)(14,136,96,211)(15,137,97,212)(16,138,98,213)(17,139,99,214)(18,140,100,215)(19,141,101,216)(20,142,102,217)(21,143,103,218)(22,144,104,219)(23,145,105,220)(24,146,106,221)(25,147,107,222)(26,148,108,223)(27,149,109,224)(28,150,110,225)(29,151,111,226)(30,152,112,227)(31,153,113,228)(32,154,114,229)(33,155,115,230)(34,156,116,231)(35,157,117,232)(36,158,118,233)(37,159,119,234)(38,160,120,235)(39,161,61,236)(40,162,62,237)(41,163,63,238)(42,164,64,239)(43,165,65,240)(44,166,66,181)(45,167,67,182)(46,168,68,183)(47,169,69,184)(48,170,70,185)(49,171,71,186)(50,172,72,187)(51,173,73,188)(52,174,74,189)(53,175,75,190)(54,176,76,191)(55,177,77,192)(56,178,78,193)(57,179,79,194)(58,180,80,195)(59,121,81,196)(60,122,82,197), (121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)>;
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), (1,123,83,198)(2,124,84,199)(3,125,85,200)(4,126,86,201)(5,127,87,202)(6,128,88,203)(7,129,89,204)(8,130,90,205)(9,131,91,206)(10,132,92,207)(11,133,93,208)(12,134,94,209)(13,135,95,210)(14,136,96,211)(15,137,97,212)(16,138,98,213)(17,139,99,214)(18,140,100,215)(19,141,101,216)(20,142,102,217)(21,143,103,218)(22,144,104,219)(23,145,105,220)(24,146,106,221)(25,147,107,222)(26,148,108,223)(27,149,109,224)(28,150,110,225)(29,151,111,226)(30,152,112,227)(31,153,113,228)(32,154,114,229)(33,155,115,230)(34,156,116,231)(35,157,117,232)(36,158,118,233)(37,159,119,234)(38,160,120,235)(39,161,61,236)(40,162,62,237)(41,163,63,238)(42,164,64,239)(43,165,65,240)(44,166,66,181)(45,167,67,182)(46,168,68,183)(47,169,69,184)(48,170,70,185)(49,171,71,186)(50,172,72,187)(51,173,73,188)(52,174,74,189)(53,175,75,190)(54,176,76,191)(55,177,77,192)(56,178,78,193)(57,179,79,194)(58,180,80,195)(59,121,81,196)(60,122,82,197), (121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195) );
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)], [(1,123,83,198),(2,124,84,199),(3,125,85,200),(4,126,86,201),(5,127,87,202),(6,128,88,203),(7,129,89,204),(8,130,90,205),(9,131,91,206),(10,132,92,207),(11,133,93,208),(12,134,94,209),(13,135,95,210),(14,136,96,211),(15,137,97,212),(16,138,98,213),(17,139,99,214),(18,140,100,215),(19,141,101,216),(20,142,102,217),(21,143,103,218),(22,144,104,219),(23,145,105,220),(24,146,106,221),(25,147,107,222),(26,148,108,223),(27,149,109,224),(28,150,110,225),(29,151,111,226),(30,152,112,227),(31,153,113,228),(32,154,114,229),(33,155,115,230),(34,156,116,231),(35,157,117,232),(36,158,118,233),(37,159,119,234),(38,160,120,235),(39,161,61,236),(40,162,62,237),(41,163,63,238),(42,164,64,239),(43,165,65,240),(44,166,66,181),(45,167,67,182),(46,168,68,183),(47,169,69,184),(48,170,70,185),(49,171,71,186),(50,172,72,187),(51,173,73,188),(52,174,74,189),(53,175,75,190),(54,176,76,191),(55,177,77,192),(56,178,78,193),(57,179,79,194),(58,180,80,195),(59,121,81,196),(60,122,82,197)], [(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,226),(152,227),(153,228),(154,229),(155,230),(156,231),(157,232),(158,233),(159,234),(160,235),(161,236),(162,237),(163,238),(164,239),(165,240),(166,181),(167,182),(168,183),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12H | 12I | ··· | 12X | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20AV | 30A | ··· | 30X | 30Y | ··· | 30BD | 60A | ··· | 60AF | 60AG | ··· | 60CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C30 | C30 | C60 | D4 | C4○D4 | C3×D4 | C5×D4 | C3×C4○D4 | C5×C4○D4 | D4×C15 | C15×C4○D4 |
| kernel | D4×C60 | C4×C60 | C15×C22⋊C4 | C15×C4⋊C4 | C22×C60 | D4×C30 | D4×C20 | D4×C15 | D4×C12 | C4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C5×D4 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C60 | C30 | C20 | C12 | C10 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 8 | 4 | 16 | 8 | 32 | 8 | 16 | 8 | 16 | 8 | 64 | 2 | 2 | 4 | 8 | 4 | 8 | 16 | 16 |
Matrix representation of D4×C60 ►in GL4(𝔽61) generated by
| 27 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [27,0,0,0,0,50,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60] >;
D4×C60 in GAP, Magma, Sage, TeX
D_4\times C_{60} % in TeX
G:=Group("D4xC60"); // GroupNames label
G:=SmallGroup(480,923);
// by ID
G=gap.SmallGroup(480,923);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,1276]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations