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G = C173C16order 272 = 24·17

The semidirect product of C17 and C16 acting via C16/C4=C4

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

Aliases: C173C16, C68.2C4, C34.2C8, C2.(C172C8), C4.2(C17⋊C4), C173C8.2C2, SmallGroup(272,3)

Series: Derived Chief Lower central Upper central

C1C17 — C173C16
C1C17C34C68C173C8 — C173C16
C17 — C173C16
C1C4

Generators and relations for C173C16
 G = < a,b | a17=b16=1, bab-1=a4 >

17C8
17C16

Smallest permutation representation of C173C16
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)
(1 270 120 200 53 235 86 159 20 255 103 179 49 214 69 152)(2 266 136 204 54 231 102 163 21 251 119 183 50 210 85 139)(3 262 135 191 55 227 101 167 22 247 118 187 51 206 84 143)(4 258 134 195 56 223 100 154 23 243 117 174 35 219 83 147)(5 271 133 199 57 236 99 158 24 239 116 178 36 215 82 151)(6 267 132 203 58 232 98 162 25 252 115 182 37 211 81 138)(7 263 131 190 59 228 97 166 26 248 114 186 38 207 80 142)(8 259 130 194 60 224 96 170 27 244 113 173 39 220 79 146)(9 272 129 198 61 237 95 157 28 240 112 177 40 216 78 150)(10 268 128 202 62 233 94 161 29 253 111 181 41 212 77 137)(11 264 127 189 63 229 93 165 30 249 110 185 42 208 76 141)(12 260 126 193 64 225 92 169 31 245 109 172 43 221 75 145)(13 256 125 197 65 238 91 156 32 241 108 176 44 217 74 149)(14 269 124 201 66 234 90 160 33 254 107 180 45 213 73 153)(15 265 123 188 67 230 89 164 34 250 106 184 46 209 72 140)(16 261 122 192 68 226 88 168 18 246 105 171 47 205 71 144)(17 257 121 196 52 222 87 155 19 242 104 175 48 218 70 148)

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

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

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

32 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H17A17B17C17D34A34B34C34D68A···68H
order1244888816···16171717173434343468···68
size11111717171717···17444444444···4

32 irreducible representations

dim11111444
type+++-
imageC1C2C4C8C16C17⋊C4C172C8C173C16
kernelC173C16C173C8C68C34C17C4C2C1
# reps11248448

Matrix representation of C173C16 in GL5(𝔽1361)

10000
01360100
01360010
01360001
0480191342880
,
7870000
01458619741303
0112654502848
08919163611080
051661001801

G:=sub<GL(5,GF(1361))| [1,0,0,0,0,0,1360,1360,1360,480,0,1,0,0,19,0,0,1,0,1342,0,0,0,1,880],[787,0,0,0,0,0,145,1126,891,51,0,861,54,916,66,0,974,502,361,1001,0,1303,848,1080,801] >;

C173C16 in GAP, Magma, Sage, TeX

C_{17}\rtimes_3C_{16}
% in TeX

G:=Group("C17:3C16");
// GroupNames label

G:=SmallGroup(272,3);
// by ID

G=gap.SmallGroup(272,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,5204,3209]);
// Polycyclic

G:=Group<a,b|a^17=b^16=1,b*a*b^-1=a^4>;
// generators/relations

Export

Subgroup lattice of C173C16 in TeX

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