direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×C3⋊C8, C6⋊C40, C30⋊5C8, C12.3C20, C60.15C4, C20.58D6, C20.9Dic3, C60.75C22, C3⋊2(C2×C40), C15⋊15(C2×C8), C6.5(C2×C20), (C2×C6).2C20, C4.14(S3×C10), (C2×C30).10C4, C30.58(C2×C4), (C2×C20).11S3, (C2×C12).6C10, (C2×C60).18C2, C4.3(C5×Dic3), C12.14(C2×C10), C2.1(C10×Dic3), (C2×C10).6Dic3, C10.18(C2×Dic3), C22.2(C5×Dic3), (C2×C4).5(C5×S3), SmallGroup(240,54)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C10×C3⋊C8 |
Generators and relations for C10×C3⋊C8
G = < a,b,c | a10=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C10×C3⋊C8
►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)
(1 195 189)(2 196 190)(3 197 181)(4 198 182)(5 199 183)(6 200 184)(7 191 185)(8 192 186)(9 193 187)(10 194 188)(11 168 49)(12 169 50)(13 170 41)(14 161 42)(15 162 43)(16 163 44)(17 164 45)(18 165 46)(19 166 47)(20 167 48)(21 35 239)(22 36 240)(23 37 231)(24 38 232)(25 39 233)(26 40 234)(27 31 235)(28 32 236)(29 33 237)(30 34 238)(51 136 212)(52 137 213)(53 138 214)(54 139 215)(55 140 216)(56 131 217)(57 132 218)(58 133 219)(59 134 220)(60 135 211)(61 105 99)(62 106 100)(63 107 91)(64 108 92)(65 109 93)(66 110 94)(67 101 95)(68 102 96)(69 103 97)(70 104 98)(71 81 148)(72 82 149)(73 83 150)(74 84 141)(75 85 142)(76 86 143)(77 87 144)(78 88 145)(79 89 146)(80 90 147)(111 156 205)(112 157 206)(113 158 207)(114 159 208)(115 160 209)(116 151 210)(117 152 201)(118 153 202)(119 154 203)(120 155 204)(121 225 179)(122 226 180)(123 227 171)(124 228 172)(125 229 173)(126 230 174)(127 221 175)(128 222 176)(129 223 177)(130 224 178)
(1 73 175 45 217 233 118 91)(2 74 176 46 218 234 119 92)(3 75 177 47 219 235 120 93)(4 76 178 48 220 236 111 94)(5 77 179 49 211 237 112 95)(6 78 180 50 212 238 113 96)(7 79 171 41 213 239 114 97)(8 80 172 42 214 240 115 98)(9 71 173 43 215 231 116 99)(10 72 174 44 216 232 117 100)(11 135 29 206 67 183 87 225)(12 136 30 207 68 184 88 226)(13 137 21 208 69 185 89 227)(14 138 22 209 70 186 90 228)(15 139 23 210 61 187 81 229)(16 140 24 201 62 188 82 230)(17 131 25 202 63 189 83 221)(18 132 26 203 64 190 84 222)(19 133 27 204 65 181 85 223)(20 134 28 205 66 182 86 224)(31 155 109 197 142 129 166 58)(32 156 110 198 143 130 167 59)(33 157 101 199 144 121 168 60)(34 158 102 200 145 122 169 51)(35 159 103 191 146 123 170 52)(36 160 104 192 147 124 161 53)(37 151 105 193 148 125 162 54)(38 152 106 194 149 126 163 55)(39 153 107 195 150 127 164 56)(40 154 108 196 141 128 165 57)
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), (1,195,189)(2,196,190)(3,197,181)(4,198,182)(5,199,183)(6,200,184)(7,191,185)(8,192,186)(9,193,187)(10,194,188)(11,168,49)(12,169,50)(13,170,41)(14,161,42)(15,162,43)(16,163,44)(17,164,45)(18,165,46)(19,166,47)(20,167,48)(21,35,239)(22,36,240)(23,37,231)(24,38,232)(25,39,233)(26,40,234)(27,31,235)(28,32,236)(29,33,237)(30,34,238)(51,136,212)(52,137,213)(53,138,214)(54,139,215)(55,140,216)(56,131,217)(57,132,218)(58,133,219)(59,134,220)(60,135,211)(61,105,99)(62,106,100)(63,107,91)(64,108,92)(65,109,93)(66,110,94)(67,101,95)(68,102,96)(69,103,97)(70,104,98)(71,81,148)(72,82,149)(73,83,150)(74,84,141)(75,85,142)(76,86,143)(77,87,144)(78,88,145)(79,89,146)(80,90,147)(111,156,205)(112,157,206)(113,158,207)(114,159,208)(115,160,209)(116,151,210)(117,152,201)(118,153,202)(119,154,203)(120,155,204)(121,225,179)(122,226,180)(123,227,171)(124,228,172)(125,229,173)(126,230,174)(127,221,175)(128,222,176)(129,223,177)(130,224,178), (1,73,175,45,217,233,118,91)(2,74,176,46,218,234,119,92)(3,75,177,47,219,235,120,93)(4,76,178,48,220,236,111,94)(5,77,179,49,211,237,112,95)(6,78,180,50,212,238,113,96)(7,79,171,41,213,239,114,97)(8,80,172,42,214,240,115,98)(9,71,173,43,215,231,116,99)(10,72,174,44,216,232,117,100)(11,135,29,206,67,183,87,225)(12,136,30,207,68,184,88,226)(13,137,21,208,69,185,89,227)(14,138,22,209,70,186,90,228)(15,139,23,210,61,187,81,229)(16,140,24,201,62,188,82,230)(17,131,25,202,63,189,83,221)(18,132,26,203,64,190,84,222)(19,133,27,204,65,181,85,223)(20,134,28,205,66,182,86,224)(31,155,109,197,142,129,166,58)(32,156,110,198,143,130,167,59)(33,157,101,199,144,121,168,60)(34,158,102,200,145,122,169,51)(35,159,103,191,146,123,170,52)(36,160,104,192,147,124,161,53)(37,151,105,193,148,125,162,54)(38,152,106,194,149,126,163,55)(39,153,107,195,150,127,164,56)(40,154,108,196,141,128,165,57)>;
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), (1,195,189)(2,196,190)(3,197,181)(4,198,182)(5,199,183)(6,200,184)(7,191,185)(8,192,186)(9,193,187)(10,194,188)(11,168,49)(12,169,50)(13,170,41)(14,161,42)(15,162,43)(16,163,44)(17,164,45)(18,165,46)(19,166,47)(20,167,48)(21,35,239)(22,36,240)(23,37,231)(24,38,232)(25,39,233)(26,40,234)(27,31,235)(28,32,236)(29,33,237)(30,34,238)(51,136,212)(52,137,213)(53,138,214)(54,139,215)(55,140,216)(56,131,217)(57,132,218)(58,133,219)(59,134,220)(60,135,211)(61,105,99)(62,106,100)(63,107,91)(64,108,92)(65,109,93)(66,110,94)(67,101,95)(68,102,96)(69,103,97)(70,104,98)(71,81,148)(72,82,149)(73,83,150)(74,84,141)(75,85,142)(76,86,143)(77,87,144)(78,88,145)(79,89,146)(80,90,147)(111,156,205)(112,157,206)(113,158,207)(114,159,208)(115,160,209)(116,151,210)(117,152,201)(118,153,202)(119,154,203)(120,155,204)(121,225,179)(122,226,180)(123,227,171)(124,228,172)(125,229,173)(126,230,174)(127,221,175)(128,222,176)(129,223,177)(130,224,178), (1,73,175,45,217,233,118,91)(2,74,176,46,218,234,119,92)(3,75,177,47,219,235,120,93)(4,76,178,48,220,236,111,94)(5,77,179,49,211,237,112,95)(6,78,180,50,212,238,113,96)(7,79,171,41,213,239,114,97)(8,80,172,42,214,240,115,98)(9,71,173,43,215,231,116,99)(10,72,174,44,216,232,117,100)(11,135,29,206,67,183,87,225)(12,136,30,207,68,184,88,226)(13,137,21,208,69,185,89,227)(14,138,22,209,70,186,90,228)(15,139,23,210,61,187,81,229)(16,140,24,201,62,188,82,230)(17,131,25,202,63,189,83,221)(18,132,26,203,64,190,84,222)(19,133,27,204,65,181,85,223)(20,134,28,205,66,182,86,224)(31,155,109,197,142,129,166,58)(32,156,110,198,143,130,167,59)(33,157,101,199,144,121,168,60)(34,158,102,200,145,122,169,51)(35,159,103,191,146,123,170,52)(36,160,104,192,147,124,161,53)(37,151,105,193,148,125,162,54)(38,152,106,194,149,126,163,55)(39,153,107,195,150,127,164,56)(40,154,108,196,141,128,165,57) );
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)], [(1,195,189),(2,196,190),(3,197,181),(4,198,182),(5,199,183),(6,200,184),(7,191,185),(8,192,186),(9,193,187),(10,194,188),(11,168,49),(12,169,50),(13,170,41),(14,161,42),(15,162,43),(16,163,44),(17,164,45),(18,165,46),(19,166,47),(20,167,48),(21,35,239),(22,36,240),(23,37,231),(24,38,232),(25,39,233),(26,40,234),(27,31,235),(28,32,236),(29,33,237),(30,34,238),(51,136,212),(52,137,213),(53,138,214),(54,139,215),(55,140,216),(56,131,217),(57,132,218),(58,133,219),(59,134,220),(60,135,211),(61,105,99),(62,106,100),(63,107,91),(64,108,92),(65,109,93),(66,110,94),(67,101,95),(68,102,96),(69,103,97),(70,104,98),(71,81,148),(72,82,149),(73,83,150),(74,84,141),(75,85,142),(76,86,143),(77,87,144),(78,88,145),(79,89,146),(80,90,147),(111,156,205),(112,157,206),(113,158,207),(114,159,208),(115,160,209),(116,151,210),(117,152,201),(118,153,202),(119,154,203),(120,155,204),(121,225,179),(122,226,180),(123,227,171),(124,228,172),(125,229,173),(126,230,174),(127,221,175),(128,222,176),(129,223,177),(130,224,178)], [(1,73,175,45,217,233,118,91),(2,74,176,46,218,234,119,92),(3,75,177,47,219,235,120,93),(4,76,178,48,220,236,111,94),(5,77,179,49,211,237,112,95),(6,78,180,50,212,238,113,96),(7,79,171,41,213,239,114,97),(8,80,172,42,214,240,115,98),(9,71,173,43,215,231,116,99),(10,72,174,44,216,232,117,100),(11,135,29,206,67,183,87,225),(12,136,30,207,68,184,88,226),(13,137,21,208,69,185,89,227),(14,138,22,209,70,186,90,228),(15,139,23,210,61,187,81,229),(16,140,24,201,62,188,82,230),(17,131,25,202,63,189,83,221),(18,132,26,203,64,190,84,222),(19,133,27,204,65,181,85,223),(20,134,28,205,66,182,86,224),(31,155,109,197,142,129,166,58),(32,156,110,198,143,130,167,59),(33,157,101,199,144,121,168,60),(34,158,102,200,145,122,169,51),(35,159,103,191,146,123,170,52),(36,160,104,192,147,124,161,53),(37,151,105,193,148,125,162,54),(38,152,106,194,149,126,163,55),(39,153,107,195,150,127,164,56),(40,154,108,196,141,128,165,57)]])
C10×C3⋊C8 is a maximal subgroup of
Dic15⋊4C8 C30.21C42 C30.23C42 C60.93D4 D30⋊4C8 C6.D40 D60⋊12C4 C6.Dic20 Dic30⋊12C4 C60.13Q8 C60.14Q8 C60.Q8 C60.5Q8 C12.59D20 Dic3×C40 D20.3Dic3 D60.4C4 D20.31D6 S3×C2×C40
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | ··· | 8H | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30L | 40A | ··· | 40AF | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C5×S3 | C5×Dic3 | S3×C10 | C5×Dic3 | C5×C3⋊C8 |
| kernel | C10×C3⋊C8 | C5×C3⋊C8 | C2×C60 | C60 | C2×C30 | C2×C3⋊C8 | C30 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C6 | C2×C20 | C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of C10×C3⋊C8 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 143 | 0 |
| 0 | 0 | 143 |
| 1 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 1 | 240 |
| 1 | 0 | 0 |
| 0 | 233 | 8 |
| 0 | 0 | 8 |
G:=sub<GL(3,GF(241))| [240,0,0,0,143,0,0,0,143],[1,0,0,0,0,1,0,240,240],[1,0,0,0,233,0,0,8,8] >;
C10×C3⋊C8 in GAP, Magma, Sage, TeX
C_{10}\times C_3\rtimes C_8 % in TeX
G:=Group("C10xC3:C8"); // GroupNames label
G:=SmallGroup(240,54);
// by ID
G=gap.SmallGroup(240,54);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,69,5765]);
// Polycyclic
G:=Group<a,b,c|a^10=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export