Copied to
clipboard

## G = C2×C17⋊2C8order 272 = 24·17

### Direct product of C2 and C17⋊2C8

Aliases: C2×C172C8, C342C8, Dic17.3C4, Dic17.6C22, C173(C2×C8), C34.5(C2×C4), (C2×C34).1C4, C22.2(C17⋊C4), (C2×Dic17).4C2, C2.3(C2×C17⋊C4), SmallGroup(272,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C2×C17⋊2C8
 Chief series C1 — C17 — C34 — Dic17 — C17⋊2C8 — C2×C17⋊2C8
 Lower central C17 — C2×C17⋊2C8
 Upper central C1 — C22

Generators and relations for C2×C172C8
G = < a,b,c | a2=b17=c8=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C2×C172C8
Regular action on 272 points
Generators in S272
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(137 229)(138 230)(139 231)(140 232)(141 233)(142 234)(143 235)(144 236)(145 237)(146 238)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 205)(163 206)(164 207)(165 208)(166 209)(167 210)(168 211)(169 212)(170 213)(171 271)(172 272)(173 256)(174 257)(175 258)(176 259)(177 260)(178 261)(179 262)(180 263)(181 264)(182 265)(183 266)(184 267)(185 268)(186 269)(187 270)(188 244)(189 245)(190 246)(191 247)(192 248)(193 249)(194 250)(195 251)(196 252)(197 253)(198 254)(199 255)(200 239)(201 240)(202 241)(203 242)(204 243)
(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)
(1 191 60 154 18 177 50 149)(2 204 59 158 19 173 49 153)(3 200 58 162 20 186 48 140)(4 196 57 166 21 182 47 144)(5 192 56 170 22 178 46 148)(6 188 55 157 23 174 45 152)(7 201 54 161 24 187 44 139)(8 197 53 165 25 183 43 143)(9 193 52 169 26 179 42 147)(10 189 68 156 27 175 41 151)(11 202 67 160 28 171 40 138)(12 198 66 164 29 184 39 142)(13 194 65 168 30 180 38 146)(14 190 64 155 31 176 37 150)(15 203 63 159 32 172 36 137)(16 199 62 163 33 185 35 141)(17 195 61 167 34 181 51 145)(69 260 124 224 86 247 117 214)(70 256 123 228 87 243 116 218)(71 269 122 232 88 239 115 205)(72 265 121 236 89 252 114 209)(73 261 120 223 90 248 113 213)(74 257 136 227 91 244 112 217)(75 270 135 231 92 240 111 221)(76 266 134 235 93 253 110 208)(77 262 133 222 94 249 109 212)(78 258 132 226 95 245 108 216)(79 271 131 230 96 241 107 220)(80 267 130 234 97 254 106 207)(81 263 129 238 98 250 105 211)(82 259 128 225 99 246 104 215)(83 272 127 229 100 242 103 219)(84 268 126 233 101 255 119 206)(85 264 125 237 102 251 118 210)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A ··· 8H 17A 17B 17C 17D 34A ··· 34L order 1 2 2 2 4 4 4 4 8 ··· 8 17 17 17 17 34 ··· 34 size 1 1 1 1 17 17 17 17 17 ··· 17 4 4 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C8 C17⋊C4 C17⋊2C8 C2×C17⋊C4 kernel C2×C17⋊2C8 C17⋊2C8 C2×Dic17 Dic17 C2×C34 C34 C22 C2 C2 # reps 1 2 1 2 2 8 4 8 4

Matrix representation of C2×C172C8 in GL5(𝔽137)

 1 0 0 0 0 0 136 0 0 0 0 0 136 0 0 0 0 0 136 0 0 0 0 0 136
,
 1 0 0 0 0 0 63 27 63 136 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 10 0 0 0 0 0 3 119 2 129 0 74 72 18 28 0 90 15 135 64 0 112 77 51 64

G:=sub<GL(5,GF(137))| [1,0,0,0,0,0,136,0,0,0,0,0,136,0,0,0,0,0,136,0,0,0,0,0,136],[1,0,0,0,0,0,63,1,0,0,0,27,0,1,0,0,63,0,0,1,0,136,0,0,0],[10,0,0,0,0,0,3,74,90,112,0,119,72,15,77,0,2,18,135,51,0,129,28,64,64] >;

C2×C172C8 in GAP, Magma, Sage, TeX

C_2\times C_{17}\rtimes_2C_8
% in TeX

G:=Group("C2xC17:2C8");
// GroupNames label

G:=SmallGroup(272,33);
// by ID

G=gap.SmallGroup(272,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,20,42,5204,1614]);
// Polycyclic

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

Export

׿
×
𝔽