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## G = C22×Dic17order 272 = 24·17

### Direct product of C22 and Dic17

Aliases: C22×Dic17, C34.9C23, C23.2D17, C22.11D34, (C2×C34)⋊5C4, C343(C2×C4), C173(C22×C4), (C22×C34).3C2, C2.2(C22×D17), (C2×C34).12C22, SmallGroup(272,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C22×Dic17
 Chief series C1 — C17 — C34 — Dic17 — C2×Dic17 — C22×Dic17
 Lower central C17 — C22×Dic17
 Upper central C1 — C23

Generators and relations for C22×Dic17
G = < a,b,c,d | a2=b2=c34=1, d2=c17, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 230 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C17, C34, C34, Dic17, C2×C34, C2×Dic17, C22×C34, C22×Dic17
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D17, Dic17, D34, C2×Dic17, C22×D17, C22×Dic17

Smallest permutation representation of C22×Dic17
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)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(137 205)(138 206)(139 207)(140 208)(141 209)(142 210)(143 211)(144 212)(145 213)(146 214)(147 215)(148 216)(149 217)(150 218)(151 219)(152 220)(153 221)(154 222)(155 223)(156 224)(157 225)(158 226)(159 227)(160 228)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)(169 237)(170 238)(171 256)(172 257)(173 258)(174 259)(175 260)(176 261)(177 262)(178 263)(179 264)(180 265)(181 266)(182 267)(183 268)(184 269)(185 270)(186 271)(187 272)(188 239)(189 240)(190 241)(191 242)(192 243)(193 244)(194 245)(195 246)(196 247)(197 248)(198 249)(199 250)(200 251)(201 252)(202 253)(203 254)(204 255)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(137 188)(138 189)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 171)(155 172)(156 173)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)(163 180)(164 181)(165 182)(166 183)(167 184)(168 185)(169 186)(170 187)(205 239)(206 240)(207 241)(208 242)(209 243)(210 244)(211 245)(212 246)(213 247)(214 248)(215 249)(216 250)(217 251)(218 252)(219 253)(220 254)(221 255)(222 256)(223 257)(224 258)(225 259)(226 260)(227 261)(228 262)(229 263)(230 264)(231 265)(232 266)(233 267)(234 268)(235 269)(236 270)(237 271)(238 272)
(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)
(1 171 18 188)(2 204 19 187)(3 203 20 186)(4 202 21 185)(5 201 22 184)(6 200 23 183)(7 199 24 182)(8 198 25 181)(9 197 26 180)(10 196 27 179)(11 195 28 178)(12 194 29 177)(13 193 30 176)(14 192 31 175)(15 191 32 174)(16 190 33 173)(17 189 34 172)(35 168 52 151)(36 167 53 150)(37 166 54 149)(38 165 55 148)(39 164 56 147)(40 163 57 146)(41 162 58 145)(42 161 59 144)(43 160 60 143)(44 159 61 142)(45 158 62 141)(46 157 63 140)(47 156 64 139)(48 155 65 138)(49 154 66 137)(50 153 67 170)(51 152 68 169)(69 239 86 256)(70 272 87 255)(71 271 88 254)(72 270 89 253)(73 269 90 252)(74 268 91 251)(75 267 92 250)(76 266 93 249)(77 265 94 248)(78 264 95 247)(79 263 96 246)(80 262 97 245)(81 261 98 244)(82 260 99 243)(83 259 100 242)(84 258 101 241)(85 257 102 240)(103 223 120 206)(104 222 121 205)(105 221 122 238)(106 220 123 237)(107 219 124 236)(108 218 125 235)(109 217 126 234)(110 216 127 233)(111 215 128 232)(112 214 129 231)(113 213 130 230)(114 212 131 229)(115 211 132 228)(116 210 133 227)(117 209 134 226)(118 208 135 225)(119 207 136 224)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 17A ··· 17H 34A ··· 34BD order 1 2 ··· 2 4 ··· 4 17 ··· 17 34 ··· 34 size 1 1 ··· 1 17 ··· 17 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D17 Dic17 D34 kernel C22×Dic17 C2×Dic17 C22×C34 C2×C34 C23 C22 C22 # reps 1 6 1 8 8 32 24

Matrix representation of C22×Dic17 in GL4(𝔽137) generated by

 1 0 0 0 0 136 0 0 0 0 1 0 0 0 0 1
,
 136 0 0 0 0 136 0 0 0 0 136 0 0 0 0 136
,
 1 0 0 0 0 1 0 0 0 0 1 136 0 0 78 60
,
 136 0 0 0 0 1 0 0 0 0 7 61 0 0 98 130
G:=sub<GL(4,GF(137))| [1,0,0,0,0,136,0,0,0,0,1,0,0,0,0,1],[136,0,0,0,0,136,0,0,0,0,136,0,0,0,0,136],[1,0,0,0,0,1,0,0,0,0,1,78,0,0,136,60],[136,0,0,0,0,1,0,0,0,0,7,98,0,0,61,130] >;

C22×Dic17 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{17}
% in TeX

G:=Group("C2^2xDic17");
// GroupNames label

G:=SmallGroup(272,44);
// by ID

G=gap.SmallGroup(272,44);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,40,6404]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^34=1,d^2=c^17,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽