direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×Dic10, C30⋊3Q8, C12.55D10, C60.62C22, C30.38C23, C10⋊(C3×Q8), C5⋊1(C6×Q8), C15⋊6(C2×Q8), (C2×C20).4C6, C4.11(C6×D5), C20.11(C2×C6), (C2×C60).11C2, (C2×C12).10D5, (C2×C6).36D10, C22.8(C6×D5), C10.1(C22×C6), (C6×Dic5).8C2, Dic5.1(C2×C6), (C2×Dic5).3C6, C6.38(C22×D5), (C2×C30).37C22, (C3×Dic5).17C22, C2.3(D5×C2×C6), (C2×C4).4(C3×D5), (C2×C10).8(C2×C6), SmallGroup(240,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Dic10
G = < a,b,c | a6=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 164 in 76 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, C2×Q8, Dic5, C20, C2×C10, C2×C12, C2×C12, C3×Q8, C30, C30, Dic10, C2×Dic5, C2×C20, C6×Q8, C3×Dic5, C60, C2×C30, C2×Dic10, C3×Dic10, C6×Dic5, C2×C60, C6×Dic10
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, D10, C3×Q8, C22×C6, C3×D5, Dic10, C22×D5, C6×Q8, C6×D5, C2×Dic10, C3×Dic10, D5×C2×C6, C6×Dic10
►Regular action on 240 points
Generators in S240
(1 67 218 108 129 59)(2 68 219 109 130 60)(3 69 220 110 131 41)(4 70 201 111 132 42)(5 71 202 112 133 43)(6 72 203 113 134 44)(7 73 204 114 135 45)(8 74 205 115 136 46)(9 75 206 116 137 47)(10 76 207 117 138 48)(11 77 208 118 139 49)(12 78 209 119 140 50)(13 79 210 120 121 51)(14 80 211 101 122 52)(15 61 212 102 123 53)(16 62 213 103 124 54)(17 63 214 104 125 55)(18 64 215 105 126 56)(19 65 216 106 127 57)(20 66 217 107 128 58)(21 172 143 184 224 99)(22 173 144 185 225 100)(23 174 145 186 226 81)(24 175 146 187 227 82)(25 176 147 188 228 83)(26 177 148 189 229 84)(27 178 149 190 230 85)(28 179 150 191 231 86)(29 180 151 192 232 87)(30 161 152 193 233 88)(31 162 153 194 234 89)(32 163 154 195 235 90)(33 164 155 196 236 91)(34 165 156 197 237 92)(35 166 157 198 238 93)(36 167 158 199 239 94)(37 168 159 200 240 95)(38 169 160 181 221 96)(39 170 141 182 222 97)(40 171 142 183 223 98)
(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 94 11 84)(2 93 12 83)(3 92 13 82)(4 91 14 81)(5 90 15 100)(6 89 16 99)(7 88 17 98)(8 87 18 97)(9 86 19 96)(10 85 20 95)(21 72 31 62)(22 71 32 61)(23 70 33 80)(24 69 34 79)(25 68 35 78)(26 67 36 77)(27 66 37 76)(28 65 38 75)(29 64 39 74)(30 63 40 73)(41 237 51 227)(42 236 52 226)(43 235 53 225)(44 234 54 224)(45 233 55 223)(46 232 56 222)(47 231 57 221)(48 230 58 240)(49 229 59 239)(50 228 60 238)(101 145 111 155)(102 144 112 154)(103 143 113 153)(104 142 114 152)(105 141 115 151)(106 160 116 150)(107 159 117 149)(108 158 118 148)(109 157 119 147)(110 156 120 146)(121 187 131 197)(122 186 132 196)(123 185 133 195)(124 184 134 194)(125 183 135 193)(126 182 136 192)(127 181 137 191)(128 200 138 190)(129 199 139 189)(130 198 140 188)(161 214 171 204)(162 213 172 203)(163 212 173 202)(164 211 174 201)(165 210 175 220)(166 209 176 219)(167 208 177 218)(168 207 178 217)(169 206 179 216)(170 205 180 215)
G:=sub<Sym(240)| (1,67,218,108,129,59)(2,68,219,109,130,60)(3,69,220,110,131,41)(4,70,201,111,132,42)(5,71,202,112,133,43)(6,72,203,113,134,44)(7,73,204,114,135,45)(8,74,205,115,136,46)(9,75,206,116,137,47)(10,76,207,117,138,48)(11,77,208,118,139,49)(12,78,209,119,140,50)(13,79,210,120,121,51)(14,80,211,101,122,52)(15,61,212,102,123,53)(16,62,213,103,124,54)(17,63,214,104,125,55)(18,64,215,105,126,56)(19,65,216,106,127,57)(20,66,217,107,128,58)(21,172,143,184,224,99)(22,173,144,185,225,100)(23,174,145,186,226,81)(24,175,146,187,227,82)(25,176,147,188,228,83)(26,177,148,189,229,84)(27,178,149,190,230,85)(28,179,150,191,231,86)(29,180,151,192,232,87)(30,161,152,193,233,88)(31,162,153,194,234,89)(32,163,154,195,235,90)(33,164,155,196,236,91)(34,165,156,197,237,92)(35,166,157,198,238,93)(36,167,158,199,239,94)(37,168,159,200,240,95)(38,169,160,181,221,96)(39,170,141,182,222,97)(40,171,142,183,223,98), (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), (1,94,11,84)(2,93,12,83)(3,92,13,82)(4,91,14,81)(5,90,15,100)(6,89,16,99)(7,88,17,98)(8,87,18,97)(9,86,19,96)(10,85,20,95)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(41,237,51,227)(42,236,52,226)(43,235,53,225)(44,234,54,224)(45,233,55,223)(46,232,56,222)(47,231,57,221)(48,230,58,240)(49,229,59,239)(50,228,60,238)(101,145,111,155)(102,144,112,154)(103,143,113,153)(104,142,114,152)(105,141,115,151)(106,160,116,150)(107,159,117,149)(108,158,118,148)(109,157,119,147)(110,156,120,146)(121,187,131,197)(122,186,132,196)(123,185,133,195)(124,184,134,194)(125,183,135,193)(126,182,136,192)(127,181,137,191)(128,200,138,190)(129,199,139,189)(130,198,140,188)(161,214,171,204)(162,213,172,203)(163,212,173,202)(164,211,174,201)(165,210,175,220)(166,209,176,219)(167,208,177,218)(168,207,178,217)(169,206,179,216)(170,205,180,215)>;
G:=Group( (1,67,218,108,129,59)(2,68,219,109,130,60)(3,69,220,110,131,41)(4,70,201,111,132,42)(5,71,202,112,133,43)(6,72,203,113,134,44)(7,73,204,114,135,45)(8,74,205,115,136,46)(9,75,206,116,137,47)(10,76,207,117,138,48)(11,77,208,118,139,49)(12,78,209,119,140,50)(13,79,210,120,121,51)(14,80,211,101,122,52)(15,61,212,102,123,53)(16,62,213,103,124,54)(17,63,214,104,125,55)(18,64,215,105,126,56)(19,65,216,106,127,57)(20,66,217,107,128,58)(21,172,143,184,224,99)(22,173,144,185,225,100)(23,174,145,186,226,81)(24,175,146,187,227,82)(25,176,147,188,228,83)(26,177,148,189,229,84)(27,178,149,190,230,85)(28,179,150,191,231,86)(29,180,151,192,232,87)(30,161,152,193,233,88)(31,162,153,194,234,89)(32,163,154,195,235,90)(33,164,155,196,236,91)(34,165,156,197,237,92)(35,166,157,198,238,93)(36,167,158,199,239,94)(37,168,159,200,240,95)(38,169,160,181,221,96)(39,170,141,182,222,97)(40,171,142,183,223,98), (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), (1,94,11,84)(2,93,12,83)(3,92,13,82)(4,91,14,81)(5,90,15,100)(6,89,16,99)(7,88,17,98)(8,87,18,97)(9,86,19,96)(10,85,20,95)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(41,237,51,227)(42,236,52,226)(43,235,53,225)(44,234,54,224)(45,233,55,223)(46,232,56,222)(47,231,57,221)(48,230,58,240)(49,229,59,239)(50,228,60,238)(101,145,111,155)(102,144,112,154)(103,143,113,153)(104,142,114,152)(105,141,115,151)(106,160,116,150)(107,159,117,149)(108,158,118,148)(109,157,119,147)(110,156,120,146)(121,187,131,197)(122,186,132,196)(123,185,133,195)(124,184,134,194)(125,183,135,193)(126,182,136,192)(127,181,137,191)(128,200,138,190)(129,199,139,189)(130,198,140,188)(161,214,171,204)(162,213,172,203)(163,212,173,202)(164,211,174,201)(165,210,175,220)(166,209,176,219)(167,208,177,218)(168,207,178,217)(169,206,179,216)(170,205,180,215) );
G=PermutationGroup([[(1,67,218,108,129,59),(2,68,219,109,130,60),(3,69,220,110,131,41),(4,70,201,111,132,42),(5,71,202,112,133,43),(6,72,203,113,134,44),(7,73,204,114,135,45),(8,74,205,115,136,46),(9,75,206,116,137,47),(10,76,207,117,138,48),(11,77,208,118,139,49),(12,78,209,119,140,50),(13,79,210,120,121,51),(14,80,211,101,122,52),(15,61,212,102,123,53),(16,62,213,103,124,54),(17,63,214,104,125,55),(18,64,215,105,126,56),(19,65,216,106,127,57),(20,66,217,107,128,58),(21,172,143,184,224,99),(22,173,144,185,225,100),(23,174,145,186,226,81),(24,175,146,187,227,82),(25,176,147,188,228,83),(26,177,148,189,229,84),(27,178,149,190,230,85),(28,179,150,191,231,86),(29,180,151,192,232,87),(30,161,152,193,233,88),(31,162,153,194,234,89),(32,163,154,195,235,90),(33,164,155,196,236,91),(34,165,156,197,237,92),(35,166,157,198,238,93),(36,167,158,199,239,94),(37,168,159,200,240,95),(38,169,160,181,221,96),(39,170,141,182,222,97),(40,171,142,183,223,98)], [(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)], [(1,94,11,84),(2,93,12,83),(3,92,13,82),(4,91,14,81),(5,90,15,100),(6,89,16,99),(7,88,17,98),(8,87,18,97),(9,86,19,96),(10,85,20,95),(21,72,31,62),(22,71,32,61),(23,70,33,80),(24,69,34,79),(25,68,35,78),(26,67,36,77),(27,66,37,76),(28,65,38,75),(29,64,39,74),(30,63,40,73),(41,237,51,227),(42,236,52,226),(43,235,53,225),(44,234,54,224),(45,233,55,223),(46,232,56,222),(47,231,57,221),(48,230,58,240),(49,229,59,239),(50,228,60,238),(101,145,111,155),(102,144,112,154),(103,143,113,153),(104,142,114,152),(105,141,115,151),(106,160,116,150),(107,159,117,149),(108,158,118,148),(109,157,119,147),(110,156,120,146),(121,187,131,197),(122,186,132,196),(123,185,133,195),(124,184,134,194),(125,183,135,193),(126,182,136,192),(127,181,137,191),(128,200,138,190),(129,199,139,189),(130,198,140,188),(161,214,171,204),(162,213,172,203),(163,212,173,202),(164,211,174,201),(165,210,175,220),(166,209,176,219),(167,208,177,218),(168,207,178,217),(169,206,179,216),(170,205,180,215)]])
C6×Dic10 is a maximal subgroup of
C12.6D20 C30.Q16 C6.Dic20 (C2×C60).C4 D12.37D10 C12.D20 (C2×C20).D6 Dic15⋊1Q8 Dic15⋊6Q8 C60.46D4 C60.89D4 C60.47D4 Dic15⋊8Q8 C60.48D4 D30⋊10Q8 D6⋊1Dic10 D30⋊Q8 C30.C24 C6×Q8×D5
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8 | D5 | D10 | D10 | C3×Q8 | C3×D5 | Dic10 | C6×D5 | C6×D5 | C3×Dic10 |
| kernel | C6×Dic10 | C3×Dic10 | C6×Dic5 | C2×C60 | C2×Dic10 | Dic10 | C2×Dic5 | C2×C20 | C30 | C2×C12 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C6×Dic10 ►in GL4(𝔽61) generated by
| 48 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 34 | 32 |
| 0 | 0 | 2 | 36 |
| 59 | 39 | 0 | 0 |
| 14 | 2 | 0 | 0 |
| 0 | 0 | 9 | 29 |
| 0 | 0 | 14 | 52 |
G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,13,0,0,0,0,13],[0,60,0,0,1,43,0,0,0,0,34,2,0,0,32,36],[59,14,0,0,39,2,0,0,0,0,9,14,0,0,29,52] >;
C6×Dic10 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{10} % in TeX
G:=Group("C6xDic10"); // GroupNames label
G:=SmallGroup(240,155);
// by ID
G=gap.SmallGroup(240,155);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,144,506,122,6917]);
// Polycyclic
G:=Group<a,b,c|a^6=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations