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G = C174C16order 272 = 24·17

The semidirect product of C17 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C174C16, C68.5C4, C34.3C8, C8.2D17, C136.2C2, C4.2Dic17, C2.(C173C8), SmallGroup(272,1)

Series: Derived Chief Lower central Upper central

C1C17 — C174C16
C1C17C34C68C136 — C174C16
C17 — C174C16
C1C8

Generators and relations for C174C16
 G = < a,b | a17=b16=1, bab-1=a-1 >

17C16

Smallest permutation representation of C174C16
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)
(1 256 130 188 67 222 100 154 20 239 116 171 45 205 85 137)(2 272 131 204 68 238 101 170 21 255 117 187 46 221 69 153)(3 271 132 203 52 237 102 169 22 254 118 186 47 220 70 152)(4 270 133 202 53 236 86 168 23 253 119 185 48 219 71 151)(5 269 134 201 54 235 87 167 24 252 103 184 49 218 72 150)(6 268 135 200 55 234 88 166 25 251 104 183 50 217 73 149)(7 267 136 199 56 233 89 165 26 250 105 182 51 216 74 148)(8 266 120 198 57 232 90 164 27 249 106 181 35 215 75 147)(9 265 121 197 58 231 91 163 28 248 107 180 36 214 76 146)(10 264 122 196 59 230 92 162 29 247 108 179 37 213 77 145)(11 263 123 195 60 229 93 161 30 246 109 178 38 212 78 144)(12 262 124 194 61 228 94 160 31 245 110 177 39 211 79 143)(13 261 125 193 62 227 95 159 32 244 111 176 40 210 80 142)(14 260 126 192 63 226 96 158 33 243 112 175 41 209 81 141)(15 259 127 191 64 225 97 157 34 242 113 174 42 208 82 140)(16 258 128 190 65 224 98 156 18 241 114 173 43 207 83 139)(17 257 129 189 66 223 99 155 19 240 115 172 44 206 84 138)

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

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

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

80 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H17A···17H34A···34H68A···68P136A···136AF
order1244888816···1617···1734···3468···68136···136
size1111111117···172···22···22···22···2

80 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D17Dic17C173C8C174C16
kernelC174C16C136C68C34C17C8C4C2C1
# reps11248881632

Matrix representation of C174C16 in GL2(𝔽1361) generated by

01
13601205
,
934198
121427
G:=sub<GL(2,GF(1361))| [0,1360,1,1205],[934,121,198,427] >;

C174C16 in GAP, Magma, Sage, TeX

C_{17}\rtimes_4C_{16}
% in TeX

G:=Group("C17:4C16");
// GroupNames label

G:=SmallGroup(272,1);
// by ID

G=gap.SmallGroup(272,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,6404]);
// Polycyclic

G:=Group<a,b|a^17=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C174C16 in TeX

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