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G = Q16×C17order 272 = 24·17

Direct product of C17 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: Q16×C17, C8.C34, Q8.C34, C136.3C2, C34.16D4, C68.19C22, C4.3(C2×C34), C2.5(D4×C17), (Q8×C17).2C2, SmallGroup(272,27)

Series: Derived Chief Lower central Upper central

C1C4 — Q16×C17
C1C2C4C68Q8×C17 — Q16×C17
C1C2C4 — Q16×C17
C1C34C68 — Q16×C17

Generators and relations for Q16×C17
 G = < a,b,c | a17=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C68
2C68

Smallest permutation representation of Q16×C17
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 40 32 85 179 63 260 190)(2 41 33 69 180 64 261 191)(3 42 34 70 181 65 262 192)(4 43 18 71 182 66 263 193)(5 44 19 72 183 67 264 194)(6 45 20 73 184 68 265 195)(7 46 21 74 185 52 266 196)(8 47 22 75 186 53 267 197)(9 48 23 76 187 54 268 198)(10 49 24 77 171 55 269 199)(11 50 25 78 172 56 270 200)(12 51 26 79 173 57 271 201)(13 35 27 80 174 58 272 202)(14 36 28 81 175 59 256 203)(15 37 29 82 176 60 257 204)(16 38 30 83 177 61 258 188)(17 39 31 84 178 62 259 189)(86 118 231 246 150 207 127 169)(87 119 232 247 151 208 128 170)(88 103 233 248 152 209 129 154)(89 104 234 249 153 210 130 155)(90 105 235 250 137 211 131 156)(91 106 236 251 138 212 132 157)(92 107 237 252 139 213 133 158)(93 108 238 253 140 214 134 159)(94 109 222 254 141 215 135 160)(95 110 223 255 142 216 136 161)(96 111 224 239 143 217 120 162)(97 112 225 240 144 218 121 163)(98 113 226 241 145 219 122 164)(99 114 227 242 146 220 123 165)(100 115 228 243 147 221 124 166)(101 116 229 244 148 205 125 167)(102 117 230 245 149 206 126 168)
(1 227 179 123)(2 228 180 124)(3 229 181 125)(4 230 182 126)(5 231 183 127)(6 232 184 128)(7 233 185 129)(8 234 186 130)(9 235 187 131)(10 236 171 132)(11 237 172 133)(12 238 173 134)(13 222 174 135)(14 223 175 136)(15 224 176 120)(16 225 177 121)(17 226 178 122)(18 102 263 149)(19 86 264 150)(20 87 265 151)(21 88 266 152)(22 89 267 153)(23 90 268 137)(24 91 269 138)(25 92 270 139)(26 93 271 140)(27 94 272 141)(28 95 256 142)(29 96 257 143)(30 97 258 144)(31 98 259 145)(32 99 260 146)(33 100 261 147)(34 101 262 148)(35 109 58 215)(36 110 59 216)(37 111 60 217)(38 112 61 218)(39 113 62 219)(40 114 63 220)(41 115 64 221)(42 116 65 205)(43 117 66 206)(44 118 67 207)(45 119 68 208)(46 103 52 209)(47 104 53 210)(48 105 54 211)(49 106 55 212)(50 107 56 213)(51 108 57 214)(69 166 191 243)(70 167 192 244)(71 168 193 245)(72 169 194 246)(73 170 195 247)(74 154 196 248)(75 155 197 249)(76 156 198 250)(77 157 199 251)(78 158 200 252)(79 159 201 253)(80 160 202 254)(81 161 203 255)(82 162 204 239)(83 163 188 240)(84 164 189 241)(85 165 190 242)

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), (1,40,32,85,179,63,260,190)(2,41,33,69,180,64,261,191)(3,42,34,70,181,65,262,192)(4,43,18,71,182,66,263,193)(5,44,19,72,183,67,264,194)(6,45,20,73,184,68,265,195)(7,46,21,74,185,52,266,196)(8,47,22,75,186,53,267,197)(9,48,23,76,187,54,268,198)(10,49,24,77,171,55,269,199)(11,50,25,78,172,56,270,200)(12,51,26,79,173,57,271,201)(13,35,27,80,174,58,272,202)(14,36,28,81,175,59,256,203)(15,37,29,82,176,60,257,204)(16,38,30,83,177,61,258,188)(17,39,31,84,178,62,259,189)(86,118,231,246,150,207,127,169)(87,119,232,247,151,208,128,170)(88,103,233,248,152,209,129,154)(89,104,234,249,153,210,130,155)(90,105,235,250,137,211,131,156)(91,106,236,251,138,212,132,157)(92,107,237,252,139,213,133,158)(93,108,238,253,140,214,134,159)(94,109,222,254,141,215,135,160)(95,110,223,255,142,216,136,161)(96,111,224,239,143,217,120,162)(97,112,225,240,144,218,121,163)(98,113,226,241,145,219,122,164)(99,114,227,242,146,220,123,165)(100,115,228,243,147,221,124,166)(101,116,229,244,148,205,125,167)(102,117,230,245,149,206,126,168), (1,227,179,123)(2,228,180,124)(3,229,181,125)(4,230,182,126)(5,231,183,127)(6,232,184,128)(7,233,185,129)(8,234,186,130)(9,235,187,131)(10,236,171,132)(11,237,172,133)(12,238,173,134)(13,222,174,135)(14,223,175,136)(15,224,176,120)(16,225,177,121)(17,226,178,122)(18,102,263,149)(19,86,264,150)(20,87,265,151)(21,88,266,152)(22,89,267,153)(23,90,268,137)(24,91,269,138)(25,92,270,139)(26,93,271,140)(27,94,272,141)(28,95,256,142)(29,96,257,143)(30,97,258,144)(31,98,259,145)(32,99,260,146)(33,100,261,147)(34,101,262,148)(35,109,58,215)(36,110,59,216)(37,111,60,217)(38,112,61,218)(39,113,62,219)(40,114,63,220)(41,115,64,221)(42,116,65,205)(43,117,66,206)(44,118,67,207)(45,119,68,208)(46,103,52,209)(47,104,53,210)(48,105,54,211)(49,106,55,212)(50,107,56,213)(51,108,57,214)(69,166,191,243)(70,167,192,244)(71,168,193,245)(72,169,194,246)(73,170,195,247)(74,154,196,248)(75,155,197,249)(76,156,198,250)(77,157,199,251)(78,158,200,252)(79,159,201,253)(80,160,202,254)(81,161,203,255)(82,162,204,239)(83,163,188,240)(84,164,189,241)(85,165,190,242)>;

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), (1,40,32,85,179,63,260,190)(2,41,33,69,180,64,261,191)(3,42,34,70,181,65,262,192)(4,43,18,71,182,66,263,193)(5,44,19,72,183,67,264,194)(6,45,20,73,184,68,265,195)(7,46,21,74,185,52,266,196)(8,47,22,75,186,53,267,197)(9,48,23,76,187,54,268,198)(10,49,24,77,171,55,269,199)(11,50,25,78,172,56,270,200)(12,51,26,79,173,57,271,201)(13,35,27,80,174,58,272,202)(14,36,28,81,175,59,256,203)(15,37,29,82,176,60,257,204)(16,38,30,83,177,61,258,188)(17,39,31,84,178,62,259,189)(86,118,231,246,150,207,127,169)(87,119,232,247,151,208,128,170)(88,103,233,248,152,209,129,154)(89,104,234,249,153,210,130,155)(90,105,235,250,137,211,131,156)(91,106,236,251,138,212,132,157)(92,107,237,252,139,213,133,158)(93,108,238,253,140,214,134,159)(94,109,222,254,141,215,135,160)(95,110,223,255,142,216,136,161)(96,111,224,239,143,217,120,162)(97,112,225,240,144,218,121,163)(98,113,226,241,145,219,122,164)(99,114,227,242,146,220,123,165)(100,115,228,243,147,221,124,166)(101,116,229,244,148,205,125,167)(102,117,230,245,149,206,126,168), (1,227,179,123)(2,228,180,124)(3,229,181,125)(4,230,182,126)(5,231,183,127)(6,232,184,128)(7,233,185,129)(8,234,186,130)(9,235,187,131)(10,236,171,132)(11,237,172,133)(12,238,173,134)(13,222,174,135)(14,223,175,136)(15,224,176,120)(16,225,177,121)(17,226,178,122)(18,102,263,149)(19,86,264,150)(20,87,265,151)(21,88,266,152)(22,89,267,153)(23,90,268,137)(24,91,269,138)(25,92,270,139)(26,93,271,140)(27,94,272,141)(28,95,256,142)(29,96,257,143)(30,97,258,144)(31,98,259,145)(32,99,260,146)(33,100,261,147)(34,101,262,148)(35,109,58,215)(36,110,59,216)(37,111,60,217)(38,112,61,218)(39,113,62,219)(40,114,63,220)(41,115,64,221)(42,116,65,205)(43,117,66,206)(44,118,67,207)(45,119,68,208)(46,103,52,209)(47,104,53,210)(48,105,54,211)(49,106,55,212)(50,107,56,213)(51,108,57,214)(69,166,191,243)(70,167,192,244)(71,168,193,245)(72,169,194,246)(73,170,195,247)(74,154,196,248)(75,155,197,249)(76,156,198,250)(77,157,199,251)(78,158,200,252)(79,159,201,253)(80,160,202,254)(81,161,203,255)(82,162,204,239)(83,163,188,240)(84,164,189,241)(85,165,190,242) );

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)], [(1,40,32,85,179,63,260,190),(2,41,33,69,180,64,261,191),(3,42,34,70,181,65,262,192),(4,43,18,71,182,66,263,193),(5,44,19,72,183,67,264,194),(6,45,20,73,184,68,265,195),(7,46,21,74,185,52,266,196),(8,47,22,75,186,53,267,197),(9,48,23,76,187,54,268,198),(10,49,24,77,171,55,269,199),(11,50,25,78,172,56,270,200),(12,51,26,79,173,57,271,201),(13,35,27,80,174,58,272,202),(14,36,28,81,175,59,256,203),(15,37,29,82,176,60,257,204),(16,38,30,83,177,61,258,188),(17,39,31,84,178,62,259,189),(86,118,231,246,150,207,127,169),(87,119,232,247,151,208,128,170),(88,103,233,248,152,209,129,154),(89,104,234,249,153,210,130,155),(90,105,235,250,137,211,131,156),(91,106,236,251,138,212,132,157),(92,107,237,252,139,213,133,158),(93,108,238,253,140,214,134,159),(94,109,222,254,141,215,135,160),(95,110,223,255,142,216,136,161),(96,111,224,239,143,217,120,162),(97,112,225,240,144,218,121,163),(98,113,226,241,145,219,122,164),(99,114,227,242,146,220,123,165),(100,115,228,243,147,221,124,166),(101,116,229,244,148,205,125,167),(102,117,230,245,149,206,126,168)], [(1,227,179,123),(2,228,180,124),(3,229,181,125),(4,230,182,126),(5,231,183,127),(6,232,184,128),(7,233,185,129),(8,234,186,130),(9,235,187,131),(10,236,171,132),(11,237,172,133),(12,238,173,134),(13,222,174,135),(14,223,175,136),(15,224,176,120),(16,225,177,121),(17,226,178,122),(18,102,263,149),(19,86,264,150),(20,87,265,151),(21,88,266,152),(22,89,267,153),(23,90,268,137),(24,91,269,138),(25,92,270,139),(26,93,271,140),(27,94,272,141),(28,95,256,142),(29,96,257,143),(30,97,258,144),(31,98,259,145),(32,99,260,146),(33,100,261,147),(34,101,262,148),(35,109,58,215),(36,110,59,216),(37,111,60,217),(38,112,61,218),(39,113,62,219),(40,114,63,220),(41,115,64,221),(42,116,65,205),(43,117,66,206),(44,118,67,207),(45,119,68,208),(46,103,52,209),(47,104,53,210),(48,105,54,211),(49,106,55,212),(50,107,56,213),(51,108,57,214),(69,166,191,243),(70,167,192,244),(71,168,193,245),(72,169,194,246),(73,170,195,247),(74,154,196,248),(75,155,197,249),(76,156,198,250),(77,157,199,251),(78,158,200,252),(79,159,201,253),(80,160,202,254),(81,161,203,255),(82,162,204,239),(83,163,188,240),(84,164,189,241),(85,165,190,242)])

119 conjugacy classes

class 1  2 4A4B4C8A8B17A···17P34A···34P68A···68P68Q···68AV136A···136AF
order124448817···1734···3468···6868···68136···136
size11244221···11···12···24···42···2

119 irreducible representations

dim1111112222
type++++-
imageC1C2C2C17C34C34D4Q16D4×C17Q16×C17
kernelQ16×C17C136Q8×C17Q16C8Q8C34C17C2C1
# reps112161632121632

Matrix representation of Q16×C17 in GL3(𝔽137) generated by

11900
010
001
,
100
0031
05331
,
100
0918
09546
G:=sub<GL(3,GF(137))| [119,0,0,0,1,0,0,0,1],[1,0,0,0,0,53,0,31,31],[1,0,0,0,91,95,0,8,46] >;

Q16×C17 in GAP, Magma, Sage, TeX

Q_{16}\times C_{17}
% in TeX

G:=Group("Q16xC17");
// GroupNames label

G:=SmallGroup(272,27);
// by ID

G=gap.SmallGroup(272,27);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,680,701,686,4083,2048,58]);
// Polycyclic

G:=Group<a,b,c|a^17=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q16×C17 in TeX

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