metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊5C4, C4⋊Dic15, C2.1D60, C6.4D20, C30.5Q8, C20⋊3Dic3, C12⋊1Dic5, C30.22D4, C10.4D12, C2.2Dic30, C6.5Dic10, C10.5Dic6, C22.5D30, C15⋊8(C4⋊C4), (C2×C20).3S3, (C2×C60).5C2, C5⋊3(C4⋊Dic3), C3⋊2(C4⋊Dic5), (C2×C4).3D15, (C2×C12).3D5, C30.52(C2×C4), (C2×C6).23D10, (C2×C10).23D6, C6.9(C2×Dic5), C2.4(C2×Dic15), (C2×C30).24C22, (C2×Dic15).2C2, C10.16(C2×Dic3), SmallGroup(240,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊5C4
G = < a,b | a60=b4=1, bab-1=a-1 >
Smallest permutation representation of C60⋊5C4
►Regular action 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)
(1 143 61 224)(2 142 62 223)(3 141 63 222)(4 140 64 221)(5 139 65 220)(6 138 66 219)(7 137 67 218)(8 136 68 217)(9 135 69 216)(10 134 70 215)(11 133 71 214)(12 132 72 213)(13 131 73 212)(14 130 74 211)(15 129 75 210)(16 128 76 209)(17 127 77 208)(18 126 78 207)(19 125 79 206)(20 124 80 205)(21 123 81 204)(22 122 82 203)(23 121 83 202)(24 180 84 201)(25 179 85 200)(26 178 86 199)(27 177 87 198)(28 176 88 197)(29 175 89 196)(30 174 90 195)(31 173 91 194)(32 172 92 193)(33 171 93 192)(34 170 94 191)(35 169 95 190)(36 168 96 189)(37 167 97 188)(38 166 98 187)(39 165 99 186)(40 164 100 185)(41 163 101 184)(42 162 102 183)(43 161 103 182)(44 160 104 181)(45 159 105 240)(46 158 106 239)(47 157 107 238)(48 156 108 237)(49 155 109 236)(50 154 110 235)(51 153 111 234)(52 152 112 233)(53 151 113 232)(54 150 114 231)(55 149 115 230)(56 148 116 229)(57 147 117 228)(58 146 118 227)(59 145 119 226)(60 144 120 225)
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), (1,143,61,224)(2,142,62,223)(3,141,63,222)(4,140,64,221)(5,139,65,220)(6,138,66,219)(7,137,67,218)(8,136,68,217)(9,135,69,216)(10,134,70,215)(11,133,71,214)(12,132,72,213)(13,131,73,212)(14,130,74,211)(15,129,75,210)(16,128,76,209)(17,127,77,208)(18,126,78,207)(19,125,79,206)(20,124,80,205)(21,123,81,204)(22,122,82,203)(23,121,83,202)(24,180,84,201)(25,179,85,200)(26,178,86,199)(27,177,87,198)(28,176,88,197)(29,175,89,196)(30,174,90,195)(31,173,91,194)(32,172,92,193)(33,171,93,192)(34,170,94,191)(35,169,95,190)(36,168,96,189)(37,167,97,188)(38,166,98,187)(39,165,99,186)(40,164,100,185)(41,163,101,184)(42,162,102,183)(43,161,103,182)(44,160,104,181)(45,159,105,240)(46,158,106,239)(47,157,107,238)(48,156,108,237)(49,155,109,236)(50,154,110,235)(51,153,111,234)(52,152,112,233)(53,151,113,232)(54,150,114,231)(55,149,115,230)(56,148,116,229)(57,147,117,228)(58,146,118,227)(59,145,119,226)(60,144,120,225)>;
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), (1,143,61,224)(2,142,62,223)(3,141,63,222)(4,140,64,221)(5,139,65,220)(6,138,66,219)(7,137,67,218)(8,136,68,217)(9,135,69,216)(10,134,70,215)(11,133,71,214)(12,132,72,213)(13,131,73,212)(14,130,74,211)(15,129,75,210)(16,128,76,209)(17,127,77,208)(18,126,78,207)(19,125,79,206)(20,124,80,205)(21,123,81,204)(22,122,82,203)(23,121,83,202)(24,180,84,201)(25,179,85,200)(26,178,86,199)(27,177,87,198)(28,176,88,197)(29,175,89,196)(30,174,90,195)(31,173,91,194)(32,172,92,193)(33,171,93,192)(34,170,94,191)(35,169,95,190)(36,168,96,189)(37,167,97,188)(38,166,98,187)(39,165,99,186)(40,164,100,185)(41,163,101,184)(42,162,102,183)(43,161,103,182)(44,160,104,181)(45,159,105,240)(46,158,106,239)(47,157,107,238)(48,156,108,237)(49,155,109,236)(50,154,110,235)(51,153,111,234)(52,152,112,233)(53,151,113,232)(54,150,114,231)(55,149,115,230)(56,148,116,229)(57,147,117,228)(58,146,118,227)(59,145,119,226)(60,144,120,225) );
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)], [(1,143,61,224),(2,142,62,223),(3,141,63,222),(4,140,64,221),(5,139,65,220),(6,138,66,219),(7,137,67,218),(8,136,68,217),(9,135,69,216),(10,134,70,215),(11,133,71,214),(12,132,72,213),(13,131,73,212),(14,130,74,211),(15,129,75,210),(16,128,76,209),(17,127,77,208),(18,126,78,207),(19,125,79,206),(20,124,80,205),(21,123,81,204),(22,122,82,203),(23,121,83,202),(24,180,84,201),(25,179,85,200),(26,178,86,199),(27,177,87,198),(28,176,88,197),(29,175,89,196),(30,174,90,195),(31,173,91,194),(32,172,92,193),(33,171,93,192),(34,170,94,191),(35,169,95,190),(36,168,96,189),(37,167,97,188),(38,166,98,187),(39,165,99,186),(40,164,100,185),(41,163,101,184),(42,162,102,183),(43,161,103,182),(44,160,104,181),(45,159,105,240),(46,158,106,239),(47,157,107,238),(48,156,108,237),(49,155,109,236),(50,154,110,235),(51,153,111,234),(52,152,112,233),(53,151,113,232),(54,150,114,231),(55,149,115,230),(56,148,116,229),(57,147,117,228),(58,146,118,227),(59,145,119,226),(60,144,120,225)]])
C60⋊5C4 is a maximal subgroup of
C6.D40 C10.D24 C6.Dic20 C10.Dic12 C60.7Q8 C60.Q8 C60.8Q8 C60.5Q8 C60.1Q8 C60.2Q8 Dic30⋊8C4 C120⋊10C4 C120⋊9C4 D60⋊8C4 D4⋊Dic15 Q8⋊2Dic15 Dic3×Dic10 Dic5×Dic6 Dic5.1Dic6 (S3×C20)⋊5C4 C60⋊5C4⋊C2 Dic3.Dic10 (C4×D5)⋊Dic3 C60.67D4 (C2×C60).C22 C60.46D4 C60.6Q8 C20.Dic6 D5×C4⋊Dic3 D10.17D12 Dic5×D12 D6⋊2Dic10 D10⋊2Dic6 Dic3×D20 S3×C4⋊Dic5 C60⋊D4 C60⋊4D4 D6.9D20 C60⋊Q8 C20⋊4Dic6 C4×Dic30 C60⋊8Q8 C60.24Q8 C4×D60 C22⋊2Dic30 C23.8D30 D30.28D4 C22.D60 C4⋊Dic30 Dic15.3Q8 C4.Dic30 C4⋊C4×D15 C4⋊C4⋊7D15 D30⋊6Q8 C4⋊C4⋊D15 C60.205D4 C23.26D30 C60⋊29D4 D4×Dic15 C60⋊2D4 Q8×Dic15 D30⋊7Q8
C60⋊5C4 is a maximal quotient of
C60⋊5C8 C120⋊10C4 C120⋊9C4 C4.18D60 C30.29C42
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | Dic3 | D6 | Dic5 | D10 | Dic6 | D12 | D15 | Dic10 | D20 | Dic15 | D30 | Dic30 | D60 |
| kernel | C60⋊5C4 | C2×Dic15 | C2×C60 | C60 | C2×C20 | C30 | C30 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 |
Matrix representation of C60⋊5C4 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 55 | 28 |
| 0 | 33 | 39 |
| 50 | 0 | 0 |
| 0 | 45 | 8 |
| 0 | 52 | 16 |
G:=sub<GL(3,GF(61))| [1,0,0,0,55,33,0,28,39],[50,0,0,0,45,52,0,8,16] >;
C60⋊5C4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_5C_4 % in TeX
G:=Group("C60:5C4"); // GroupNames label
G:=SmallGroup(240,74);
// by ID
G=gap.SmallGroup(240,74);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,964,6917]);
// Polycyclic
G:=Group<a,b|a^60=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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