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G = C22×C68order 272 = 24·17

Abelian group of type [2,2,68]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C68, SmallGroup(272,46)

Series: Derived Chief Lower central Upper central

C1 — C22×C68
C1C2C34C68C2×C68 — C22×C68
C1 — C22×C68
C1 — C22×C68

Generators and relations for C22×C68
 G = < a,b,c | a2=b2=c68=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C17, C34, C34 [×6], C68 [×4], C2×C34 [×7], C2×C68 [×6], C22×C34, C22×C68
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C17, C34 [×7], C68 [×4], C2×C34 [×7], C2×C68 [×6], C22×C34, C22×C68

Smallest permutation representation of C22×C68
Regular action on 272 points
Generators in S272
(1 203)(2 204)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 157)(24 158)(25 159)(26 160)(27 161)(28 162)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 191)(58 192)(59 193)(60 194)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 254)(70 255)(71 256)(72 257)(73 258)(74 259)(75 260)(76 261)(77 262)(78 263)(79 264)(80 265)(81 266)(82 267)(83 268)(84 269)(85 270)(86 271)(87 272)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(101 218)(102 219)(103 220)(104 221)(105 222)(106 223)(107 224)(108 225)(109 226)(110 227)(111 228)(112 229)(113 230)(114 231)(115 232)(116 233)(117 234)(118 235)(119 236)(120 237)(121 238)(122 239)(123 240)(124 241)(125 242)(126 243)(127 244)(128 245)(129 246)(130 247)(131 248)(132 249)(133 250)(134 251)(135 252)(136 253)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 131)(29 132)(30 133)(31 134)(32 135)(33 136)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(137 223)(138 224)(139 225)(140 226)(141 227)(142 228)(143 229)(144 230)(145 231)(146 232)(147 233)(148 234)(149 235)(150 236)(151 237)(152 238)(153 239)(154 240)(155 241)(156 242)(157 243)(158 244)(159 245)(160 246)(161 247)(162 248)(163 249)(164 250)(165 251)(166 252)(167 253)(168 254)(169 255)(170 256)(171 257)(172 258)(173 259)(174 260)(175 261)(176 262)(177 263)(178 264)(179 265)(180 266)(181 267)(182 268)(183 269)(184 270)(185 271)(186 272)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 211)(194 212)(195 213)(196 214)(197 215)(198 216)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)
(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)

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

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

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

272 conjugacy classes

class 1 2A···2G4A···4H17A···17P34A···34DH68A···68DX
order12···24···417···1734···3468···68
size11···11···11···11···11···1

272 irreducible representations

dim11111111
type+++
imageC1C2C2C4C17C34C34C68
kernelC22×C68C2×C68C22×C34C2×C34C22×C4C2×C4C23C22
# reps1618169616128

Matrix representation of C22×C68 in GL3(𝔽137) generated by

13600
01360
001
,
13600
010
001
,
200
0190
0068
G:=sub<GL(3,GF(137))| [136,0,0,0,136,0,0,0,1],[136,0,0,0,1,0,0,0,1],[2,0,0,0,19,0,0,0,68] >;

C22×C68 in GAP, Magma, Sage, TeX

C_2^2\times C_{68}
% in TeX

G:=Group("C2^2xC68");
// GroupNames label

G:=SmallGroup(272,46);
// by ID

G=gap.SmallGroup(272,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-17,-2,680]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^68=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽