metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.3Q8, Dic3⋊Dic5, C30.18D4, C6.12D20, C6.3Dic10, C10.3Dic6, C15⋊6(C4⋊C4), C3⋊1(C4⋊Dic5), C2.3(C15⋊Q8), C10.21(C4×S3), C30.34(C2×C4), C5⋊3(Dic3⋊C4), (C5×Dic3)⋊4C4, (C2×C6).11D10, (C2×C10).11D6, C2.5(S3×Dic5), C6.5(C2×Dic5), C2.3(C3⋊D20), C10.6(C3⋊D4), (C2×C30).8C22, (C2×Dic5).3S3, (C6×Dic5).4C2, (C2×Dic3).3D5, C22.10(S3×D5), (C10×Dic3).4C2, (C2×Dic15).7C2, SmallGroup(240,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Dic10
G = < a,b,c | a6=b20=1, c2=a3b10, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6.Dic10
►Regular action on 240 points
Generators in S240
(1 221 147 52 216 170)(2 171 217 53 148 222)(3 223 149 54 218 172)(4 173 219 55 150 224)(5 225 151 56 220 174)(6 175 201 57 152 226)(7 227 153 58 202 176)(8 177 203 59 154 228)(9 229 155 60 204 178)(10 179 205 41 156 230)(11 231 157 42 206 180)(12 161 207 43 158 232)(13 233 159 44 208 162)(14 163 209 45 160 234)(15 235 141 46 210 164)(16 165 211 47 142 236)(17 237 143 48 212 166)(18 167 213 49 144 238)(19 239 145 50 214 168)(20 169 215 51 146 240)(21 74 132 96 181 119)(22 120 182 97 133 75)(23 76 134 98 183 101)(24 102 184 99 135 77)(25 78 136 100 185 103)(26 104 186 81 137 79)(27 80 138 82 187 105)(28 106 188 83 139 61)(29 62 140 84 189 107)(30 108 190 85 121 63)(31 64 122 86 191 109)(32 110 192 87 123 65)(33 66 124 88 193 111)(34 112 194 89 125 67)(35 68 126 90 195 113)(36 114 196 91 127 69)(37 70 128 92 197 115)(38 116 198 93 129 71)(39 72 130 94 199 117)(40 118 200 95 131 73)
(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)
(1 132 42 109)(2 131 43 108)(3 130 44 107)(4 129 45 106)(5 128 46 105)(6 127 47 104)(7 126 48 103)(8 125 49 102)(9 124 50 101)(10 123 51 120)(11 122 52 119)(12 121 53 118)(13 140 54 117)(14 139 55 116)(15 138 56 115)(16 137 57 114)(17 136 58 113)(18 135 59 112)(19 134 60 111)(20 133 41 110)(21 231 86 216)(22 230 87 215)(23 229 88 214)(24 228 89 213)(25 227 90 212)(26 226 91 211)(27 225 92 210)(28 224 93 209)(29 223 94 208)(30 222 95 207)(31 221 96 206)(32 240 97 205)(33 239 98 204)(34 238 99 203)(35 237 100 202)(36 236 81 201)(37 235 82 220)(38 234 83 219)(39 233 84 218)(40 232 85 217)(61 150 198 163)(62 149 199 162)(63 148 200 161)(64 147 181 180)(65 146 182 179)(66 145 183 178)(67 144 184 177)(68 143 185 176)(69 142 186 175)(70 141 187 174)(71 160 188 173)(72 159 189 172)(73 158 190 171)(74 157 191 170)(75 156 192 169)(76 155 193 168)(77 154 194 167)(78 153 195 166)(79 152 196 165)(80 151 197 164)
G:=sub<Sym(240)| (1,221,147,52,216,170)(2,171,217,53,148,222)(3,223,149,54,218,172)(4,173,219,55,150,224)(5,225,151,56,220,174)(6,175,201,57,152,226)(7,227,153,58,202,176)(8,177,203,59,154,228)(9,229,155,60,204,178)(10,179,205,41,156,230)(11,231,157,42,206,180)(12,161,207,43,158,232)(13,233,159,44,208,162)(14,163,209,45,160,234)(15,235,141,46,210,164)(16,165,211,47,142,236)(17,237,143,48,212,166)(18,167,213,49,144,238)(19,239,145,50,214,168)(20,169,215,51,146,240)(21,74,132,96,181,119)(22,120,182,97,133,75)(23,76,134,98,183,101)(24,102,184,99,135,77)(25,78,136,100,185,103)(26,104,186,81,137,79)(27,80,138,82,187,105)(28,106,188,83,139,61)(29,62,140,84,189,107)(30,108,190,85,121,63)(31,64,122,86,191,109)(32,110,192,87,123,65)(33,66,124,88,193,111)(34,112,194,89,125,67)(35,68,126,90,195,113)(36,114,196,91,127,69)(37,70,128,92,197,115)(38,116,198,93,129,71)(39,72,130,94,199,117)(40,118,200,95,131,73), (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), (1,132,42,109)(2,131,43,108)(3,130,44,107)(4,129,45,106)(5,128,46,105)(6,127,47,104)(7,126,48,103)(8,125,49,102)(9,124,50,101)(10,123,51,120)(11,122,52,119)(12,121,53,118)(13,140,54,117)(14,139,55,116)(15,138,56,115)(16,137,57,114)(17,136,58,113)(18,135,59,112)(19,134,60,111)(20,133,41,110)(21,231,86,216)(22,230,87,215)(23,229,88,214)(24,228,89,213)(25,227,90,212)(26,226,91,211)(27,225,92,210)(28,224,93,209)(29,223,94,208)(30,222,95,207)(31,221,96,206)(32,240,97,205)(33,239,98,204)(34,238,99,203)(35,237,100,202)(36,236,81,201)(37,235,82,220)(38,234,83,219)(39,233,84,218)(40,232,85,217)(61,150,198,163)(62,149,199,162)(63,148,200,161)(64,147,181,180)(65,146,182,179)(66,145,183,178)(67,144,184,177)(68,143,185,176)(69,142,186,175)(70,141,187,174)(71,160,188,173)(72,159,189,172)(73,158,190,171)(74,157,191,170)(75,156,192,169)(76,155,193,168)(77,154,194,167)(78,153,195,166)(79,152,196,165)(80,151,197,164)>;
G:=Group( (1,221,147,52,216,170)(2,171,217,53,148,222)(3,223,149,54,218,172)(4,173,219,55,150,224)(5,225,151,56,220,174)(6,175,201,57,152,226)(7,227,153,58,202,176)(8,177,203,59,154,228)(9,229,155,60,204,178)(10,179,205,41,156,230)(11,231,157,42,206,180)(12,161,207,43,158,232)(13,233,159,44,208,162)(14,163,209,45,160,234)(15,235,141,46,210,164)(16,165,211,47,142,236)(17,237,143,48,212,166)(18,167,213,49,144,238)(19,239,145,50,214,168)(20,169,215,51,146,240)(21,74,132,96,181,119)(22,120,182,97,133,75)(23,76,134,98,183,101)(24,102,184,99,135,77)(25,78,136,100,185,103)(26,104,186,81,137,79)(27,80,138,82,187,105)(28,106,188,83,139,61)(29,62,140,84,189,107)(30,108,190,85,121,63)(31,64,122,86,191,109)(32,110,192,87,123,65)(33,66,124,88,193,111)(34,112,194,89,125,67)(35,68,126,90,195,113)(36,114,196,91,127,69)(37,70,128,92,197,115)(38,116,198,93,129,71)(39,72,130,94,199,117)(40,118,200,95,131,73), (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), (1,132,42,109)(2,131,43,108)(3,130,44,107)(4,129,45,106)(5,128,46,105)(6,127,47,104)(7,126,48,103)(8,125,49,102)(9,124,50,101)(10,123,51,120)(11,122,52,119)(12,121,53,118)(13,140,54,117)(14,139,55,116)(15,138,56,115)(16,137,57,114)(17,136,58,113)(18,135,59,112)(19,134,60,111)(20,133,41,110)(21,231,86,216)(22,230,87,215)(23,229,88,214)(24,228,89,213)(25,227,90,212)(26,226,91,211)(27,225,92,210)(28,224,93,209)(29,223,94,208)(30,222,95,207)(31,221,96,206)(32,240,97,205)(33,239,98,204)(34,238,99,203)(35,237,100,202)(36,236,81,201)(37,235,82,220)(38,234,83,219)(39,233,84,218)(40,232,85,217)(61,150,198,163)(62,149,199,162)(63,148,200,161)(64,147,181,180)(65,146,182,179)(66,145,183,178)(67,144,184,177)(68,143,185,176)(69,142,186,175)(70,141,187,174)(71,160,188,173)(72,159,189,172)(73,158,190,171)(74,157,191,170)(75,156,192,169)(76,155,193,168)(77,154,194,167)(78,153,195,166)(79,152,196,165)(80,151,197,164) );
G=PermutationGroup([[(1,221,147,52,216,170),(2,171,217,53,148,222),(3,223,149,54,218,172),(4,173,219,55,150,224),(5,225,151,56,220,174),(6,175,201,57,152,226),(7,227,153,58,202,176),(8,177,203,59,154,228),(9,229,155,60,204,178),(10,179,205,41,156,230),(11,231,157,42,206,180),(12,161,207,43,158,232),(13,233,159,44,208,162),(14,163,209,45,160,234),(15,235,141,46,210,164),(16,165,211,47,142,236),(17,237,143,48,212,166),(18,167,213,49,144,238),(19,239,145,50,214,168),(20,169,215,51,146,240),(21,74,132,96,181,119),(22,120,182,97,133,75),(23,76,134,98,183,101),(24,102,184,99,135,77),(25,78,136,100,185,103),(26,104,186,81,137,79),(27,80,138,82,187,105),(28,106,188,83,139,61),(29,62,140,84,189,107),(30,108,190,85,121,63),(31,64,122,86,191,109),(32,110,192,87,123,65),(33,66,124,88,193,111),(34,112,194,89,125,67),(35,68,126,90,195,113),(36,114,196,91,127,69),(37,70,128,92,197,115),(38,116,198,93,129,71),(39,72,130,94,199,117),(40,118,200,95,131,73)], [(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)], [(1,132,42,109),(2,131,43,108),(3,130,44,107),(4,129,45,106),(5,128,46,105),(6,127,47,104),(7,126,48,103),(8,125,49,102),(9,124,50,101),(10,123,51,120),(11,122,52,119),(12,121,53,118),(13,140,54,117),(14,139,55,116),(15,138,56,115),(16,137,57,114),(17,136,58,113),(18,135,59,112),(19,134,60,111),(20,133,41,110),(21,231,86,216),(22,230,87,215),(23,229,88,214),(24,228,89,213),(25,227,90,212),(26,226,91,211),(27,225,92,210),(28,224,93,209),(29,223,94,208),(30,222,95,207),(31,221,96,206),(32,240,97,205),(33,239,98,204),(34,238,99,203),(35,237,100,202),(36,236,81,201),(37,235,82,220),(38,234,83,219),(39,233,84,218),(40,232,85,217),(61,150,198,163),(62,149,199,162),(63,148,200,161),(64,147,181,180),(65,146,182,179),(66,145,183,178),(67,144,184,177),(68,143,185,176),(69,142,186,175),(70,141,187,174),(71,160,188,173),(72,159,189,172),(73,158,190,171),(74,157,191,170),(75,156,192,169),(76,155,193,168),(77,154,194,167),(78,153,195,166),(79,152,196,165),(80,151,197,164)]])
C6.Dic10 is a maximal subgroup of
Dic3⋊5Dic10 Dic15⋊1Q8 Dic3⋊Dic10 Dic5×Dic6 Dic5.2Dic6 Dic15.Q8 C4⋊Dic3⋊D5 Dic3.Dic10 Dic15⋊7Q8 Dic3.2Dic10 D6⋊Dic10 D30.35D4 C60.68D4 (C2×C12).D10 (C4×Dic15)⋊C2 D6⋊Dic5.C2 (S3×C20)⋊7C4 C5⋊(C42⋊3S3) Dic5.7Dic6 Dic3.3Dic10 Dic15.4Q8 C12.Dic10 (C4×Dic5)⋊S3 C20.Dic6 C60.48D4 D30⋊10Q8 D5×Dic3⋊C4 D10.19(C4×S3) Dic3⋊4D20 D30.C2⋊C4 D30.Q8 D6⋊1Dic10 D10⋊1Dic6 D10⋊2Dic6 D30⋊3Q8 S3×C4⋊Dic5 D6.D20 D30⋊4Q8 D6⋊3Dic10 C4×C3⋊D20 D6⋊C4⋊D5 D6⋊D20 D6.9D20 C4×C15⋊Q8 C20⋊4Dic6 C23.26(S3×D5) C23.13(S3×D5) C23.14(S3×D5) C6.(C2×D20) C6.D4⋊D5 C23.17(S3×D5) (C6×D5)⋊D4 Dic5×C3⋊D4 D30⋊7D4 Dic15⋊17D4 D30.16D4 (C2×C6)⋊D20 (C2×C30)⋊Q8 (C2×C10)⋊8Dic6 Dic15.48D4
C6.Dic10 is a maximal quotient of
C60.15Q8 C60.Q8 C60.5Q8 C12.59D20 C30.24C42
42 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 | 20A | ··· | 20H | 30A | ··· | 30F |
| 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 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 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 | C4 | S3 | D4 | Q8 | D5 | D6 | Dic5 | D10 | Dic6 | C4×S3 | C3⋊D4 | Dic10 | D20 | S3×D5 | S3×Dic5 | C3⋊D20 | C15⋊Q8 |
| kernel | C6.Dic10 | C6×Dic5 | C10×Dic3 | C2×Dic15 | C5×Dic3 | C2×Dic5 | C30 | C30 | C2×Dic3 | C2×C10 | Dic3 | C2×C6 | C10 | C10 | C10 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C6.Dic10 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 1 |
| 25 | 29 | 0 | 0 |
| 59 | 27 | 0 | 0 |
| 0 | 0 | 41 | 12 |
| 0 | 0 | 53 | 20 |
| 34 | 30 | 0 | 0 |
| 57 | 27 | 0 | 0 |
| 0 | 0 | 52 | 18 |
| 0 | 0 | 43 | 9 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,1],[25,59,0,0,29,27,0,0,0,0,41,53,0,0,12,20],[34,57,0,0,30,27,0,0,0,0,52,43,0,0,18,9] >;
C6.Dic10 in GAP, Magma, Sage, TeX
C_6.{\rm Dic}_{10} % in TeX
G:=Group("C6.Dic10"); // GroupNames label
G:=SmallGroup(240,31);
// by ID
G=gap.SmallGroup(240,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^6=b^20=1,c^2=a^3*b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export