metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C68⋊3C4, C4⋊Dic17, C2.1D68, C34.4D4, C34.2Q8, C2.2Dic34, C22.5D34, C17⋊3(C4⋊C4), (C2×C68).3C2, (C2×C4).3D17, C34.15(C2×C4), (C2×C34).5C22, C2.4(C2×Dic17), (C2×Dic17).2C2, SmallGroup(272,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C68⋊3C4
G = < a,b | a68=b4=1, bab-1=a-1 >
Smallest permutation representation of C68⋊3C4
►Regular action on 272 points
Generators in S272
(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)(205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272)
(1 184 261 128)(2 183 262 127)(3 182 263 126)(4 181 264 125)(5 180 265 124)(6 179 266 123)(7 178 267 122)(8 177 268 121)(9 176 269 120)(10 175 270 119)(11 174 271 118)(12 173 272 117)(13 172 205 116)(14 171 206 115)(15 170 207 114)(16 169 208 113)(17 168 209 112)(18 167 210 111)(19 166 211 110)(20 165 212 109)(21 164 213 108)(22 163 214 107)(23 162 215 106)(24 161 216 105)(25 160 217 104)(26 159 218 103)(27 158 219 102)(28 157 220 101)(29 156 221 100)(30 155 222 99)(31 154 223 98)(32 153 224 97)(33 152 225 96)(34 151 226 95)(35 150 227 94)(36 149 228 93)(37 148 229 92)(38 147 230 91)(39 146 231 90)(40 145 232 89)(41 144 233 88)(42 143 234 87)(43 142 235 86)(44 141 236 85)(45 140 237 84)(46 139 238 83)(47 138 239 82)(48 137 240 81)(49 204 241 80)(50 203 242 79)(51 202 243 78)(52 201 244 77)(53 200 245 76)(54 199 246 75)(55 198 247 74)(56 197 248 73)(57 196 249 72)(58 195 250 71)(59 194 251 70)(60 193 252 69)(61 192 253 136)(62 191 254 135)(63 190 255 134)(64 189 256 133)(65 188 257 132)(66 187 258 131)(67 186 259 130)(68 185 260 129)
G:=sub<Sym(272)| (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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272), (1,184,261,128)(2,183,262,127)(3,182,263,126)(4,181,264,125)(5,180,265,124)(6,179,266,123)(7,178,267,122)(8,177,268,121)(9,176,269,120)(10,175,270,119)(11,174,271,118)(12,173,272,117)(13,172,205,116)(14,171,206,115)(15,170,207,114)(16,169,208,113)(17,168,209,112)(18,167,210,111)(19,166,211,110)(20,165,212,109)(21,164,213,108)(22,163,214,107)(23,162,215,106)(24,161,216,105)(25,160,217,104)(26,159,218,103)(27,158,219,102)(28,157,220,101)(29,156,221,100)(30,155,222,99)(31,154,223,98)(32,153,224,97)(33,152,225,96)(34,151,226,95)(35,150,227,94)(36,149,228,93)(37,148,229,92)(38,147,230,91)(39,146,231,90)(40,145,232,89)(41,144,233,88)(42,143,234,87)(43,142,235,86)(44,141,236,85)(45,140,237,84)(46,139,238,83)(47,138,239,82)(48,137,240,81)(49,204,241,80)(50,203,242,79)(51,202,243,78)(52,201,244,77)(53,200,245,76)(54,199,246,75)(55,198,247,74)(56,197,248,73)(57,196,249,72)(58,195,250,71)(59,194,251,70)(60,193,252,69)(61,192,253,136)(62,191,254,135)(63,190,255,134)(64,189,256,133)(65,188,257,132)(66,187,258,131)(67,186,259,130)(68,185,260,129)>;
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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272), (1,184,261,128)(2,183,262,127)(3,182,263,126)(4,181,264,125)(5,180,265,124)(6,179,266,123)(7,178,267,122)(8,177,268,121)(9,176,269,120)(10,175,270,119)(11,174,271,118)(12,173,272,117)(13,172,205,116)(14,171,206,115)(15,170,207,114)(16,169,208,113)(17,168,209,112)(18,167,210,111)(19,166,211,110)(20,165,212,109)(21,164,213,108)(22,163,214,107)(23,162,215,106)(24,161,216,105)(25,160,217,104)(26,159,218,103)(27,158,219,102)(28,157,220,101)(29,156,221,100)(30,155,222,99)(31,154,223,98)(32,153,224,97)(33,152,225,96)(34,151,226,95)(35,150,227,94)(36,149,228,93)(37,148,229,92)(38,147,230,91)(39,146,231,90)(40,145,232,89)(41,144,233,88)(42,143,234,87)(43,142,235,86)(44,141,236,85)(45,140,237,84)(46,139,238,83)(47,138,239,82)(48,137,240,81)(49,204,241,80)(50,203,242,79)(51,202,243,78)(52,201,244,77)(53,200,245,76)(54,199,246,75)(55,198,247,74)(56,197,248,73)(57,196,249,72)(58,195,250,71)(59,194,251,70)(60,193,252,69)(61,192,253,136)(62,191,254,135)(63,190,255,134)(64,189,256,133)(65,188,257,132)(66,187,258,131)(67,186,259,130)(68,185,260,129) );
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),(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)], [(1,184,261,128),(2,183,262,127),(3,182,263,126),(4,181,264,125),(5,180,265,124),(6,179,266,123),(7,178,267,122),(8,177,268,121),(9,176,269,120),(10,175,270,119),(11,174,271,118),(12,173,272,117),(13,172,205,116),(14,171,206,115),(15,170,207,114),(16,169,208,113),(17,168,209,112),(18,167,210,111),(19,166,211,110),(20,165,212,109),(21,164,213,108),(22,163,214,107),(23,162,215,106),(24,161,216,105),(25,160,217,104),(26,159,218,103),(27,158,219,102),(28,157,220,101),(29,156,221,100),(30,155,222,99),(31,154,223,98),(32,153,224,97),(33,152,225,96),(34,151,226,95),(35,150,227,94),(36,149,228,93),(37,148,229,92),(38,147,230,91),(39,146,231,90),(40,145,232,89),(41,144,233,88),(42,143,234,87),(43,142,235,86),(44,141,236,85),(45,140,237,84),(46,139,238,83),(47,138,239,82),(48,137,240,81),(49,204,241,80),(50,203,242,79),(51,202,243,78),(52,201,244,77),(53,200,245,76),(54,199,246,75),(55,198,247,74),(56,197,248,73),(57,196,249,72),(58,195,250,71),(59,194,251,70),(60,193,252,69),(61,192,253,136),(62,191,254,135),(63,190,255,134),(64,189,256,133),(65,188,257,132),(66,187,258,131),(67,186,259,130),(68,185,260,129)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D17 | Dic17 | D34 | Dic34 | D68 |
| kernel | C68⋊3C4 | C2×Dic17 | C2×C68 | C68 | C34 | C34 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 8 | 16 | 8 | 16 | 16 |
Matrix representation of C68⋊3C4 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 100 | 130 |
| 0 | 7 | 5 |
| 100 | 0 | 0 |
| 0 | 93 | 80 |
| 0 | 70 | 44 |
G:=sub<GL(3,GF(137))| [1,0,0,0,100,7,0,130,5],[100,0,0,0,93,70,0,80,44] >;
C68⋊3C4 in GAP, Magma, Sage, TeX
C_{68}\rtimes_3C_4 % in TeX
G:=Group("C68:3C4"); // GroupNames label
G:=SmallGroup(272,13);
// by ID
G=gap.SmallGroup(272,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,46,6404]);
// Polycyclic
G:=Group<a,b|a^68=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export