C2^2xDic19 - GroupNames

G = C₂²×Dic₁₉ order 304 = 2⁴·19

Direct product of C₂² and Dic₁₉

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂²×Dic₁₉, C₃₈.₉C₂³, C₂³.₂D₁₉, C₂².₁₁D₃₈, C₃₈⋊₂(C₂×C₄), (C₂×C₃₈)⋊₃C₄, C₁₉⋊₂(C₂²×C₄), (C₂²×C₃₈).₃C₂, C₂.₂(C₂²×D₁₉), (C₂×C₃₈).₁₂C₂², SmallGroup(304,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₉ — C₂²×Dic₁₉

Chief series

C₁ — C₁₉ — C₃₈ — Dic₁₉ — C₂×Dic₁₉ — C₂²×Dic₁₉

Lower central

C₁₉ — C₂²×Dic₁₉

Upper central

C₁ — C₂³

Generators and relations for C₂²×Dic₁₉
G = < a,b,c,d | a²=b²=c³⁸=1, d²=c¹⁹, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 252 in 54 conjugacy classes, 43 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₁₉, C₃₈, C₃₈, Dic₁₉, C₂×C₃₈, C₂×Dic₁₉, C₂²×C₃₈, C₂²×Dic₁₉
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, D₁₉, Dic₁₉, D₃₈, C₂×Dic₁₉, C₂²×D₁₉, C₂²×Dic₁₉

Smallest permutation representation of C₂²×Dic₁₉
►Regular action on 304 points

Generators in S₃₀₄

(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(153 229)(154 230)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)(161 237)(162 238)(163 239)(164 240)(165 241)(166 242)(167 243)(168 244)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 253)(178 254)(179 255)(180 256)(181 257)(182 258)(183 259)(184 260)(185 261)(186 262)(187 263)(188 264)(189 265)(190 266)(191 286)(192 287)(193 288)(194 289)(195 290)(196 291)(197 292)(198 293)(199 294)(200 295)(201 296)(202 297)(203 298)(204 299)(205 300)(206 301)(207 302)(208 303)(209 304)(210 267)(211 268)(212 269)(213 270)(214 271)(215 272)(216 273)(217 274)(218 275)(219 276)(220 277)(221 278)(222 279)(223 280)(224 281)(225 282)(226 283)(227 284)(228 285)

(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 211)(174 212)(175 213)(176 214)(177 215)(178 216)(179 217)(180 218)(181 219)(182 220)(183 221)(184 222)(185 223)(186 224)(187 225)(188 226)(189 227)(190 228)(229 286)(230 287)(231 288)(232 289)(233 290)(234 291)(235 292)(236 293)(237 294)(238 295)(239 296)(240 297)(241 298)(242 299)(243 300)(244 301)(245 302)(246 303)(247 304)(248 267)(249 268)(250 269)(251 270)(252 271)(253 272)(254 273)(255 274)(256 275)(257 276)(258 277)(259 278)(260 279)(261 280)(262 281)(263 282)(264 283)(265 284)(266 285)

(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)(267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)

(1 210 20 191)(2 209 21 228)(3 208 22 227)(4 207 23 226)(5 206 24 225)(6 205 25 224)(7 204 26 223)(8 203 27 222)(9 202 28 221)(10 201 29 220)(11 200 30 219)(12 199 31 218)(13 198 32 217)(14 197 33 216)(15 196 34 215)(16 195 35 214)(17 194 36 213)(18 193 37 212)(19 192 38 211)(39 185 58 166)(40 184 59 165)(41 183 60 164)(42 182 61 163)(43 181 62 162)(44 180 63 161)(45 179 64 160)(46 178 65 159)(47 177 66 158)(48 176 67 157)(49 175 68 156)(50 174 69 155)(51 173 70 154)(52 172 71 153)(53 171 72 190)(54 170 73 189)(55 169 74 188)(56 168 75 187)(57 167 76 186)(77 284 96 303)(78 283 97 302)(79 282 98 301)(80 281 99 300)(81 280 100 299)(82 279 101 298)(83 278 102 297)(84 277 103 296)(85 276 104 295)(86 275 105 294)(87 274 106 293)(88 273 107 292)(89 272 108 291)(90 271 109 290)(91 270 110 289)(92 269 111 288)(93 268 112 287)(94 267 113 286)(95 304 114 285)(115 247 134 266)(116 246 135 265)(117 245 136 264)(118 244 137 263)(119 243 138 262)(120 242 139 261)(121 241 140 260)(122 240 141 259)(123 239 142 258)(124 238 143 257)(125 237 144 256)(126 236 145 255)(127 235 146 254)(128 234 147 253)(129 233 148 252)(130 232 149 251)(131 231 150 250)(132 230 151 249)(133 229 152 248)

G:=sub<Sym(304)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,241)(166,242)(167,243)(168,244)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,286)(192,287)(193,288)(194,289)(195,290)(196,291)(197,292)(198,293)(199,294)(200,295)(201,296)(202,297)(203,298)(204,299)(205,300)(206,301)(207,302)(208,303)(209,304)(210,267)(211,268)(212,269)(213,270)(214,271)(215,272)(216,273)(217,274)(218,275)(219,276)(220,277)(221,278)(222,279)(223,280)(224,281)(225,282)(226,283)(227,284)(228,285), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,225)(188,226)(189,227)(190,228)(229,286)(230,287)(231,288)(232,289)(233,290)(234,291)(235,292)(236,293)(237,294)(238,295)(239,296)(240,297)(241,298)(242,299)(243,300)(244,301)(245,302)(246,303)(247,304)(248,267)(249,268)(250,269)(251,270)(252,271)(253,272)(254,273)(255,274)(256,275)(257,276)(258,277)(259,278)(260,279)(261,280)(262,281)(263,282)(264,283)(265,284)(266,285), (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)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,210,20,191)(2,209,21,228)(3,208,22,227)(4,207,23,226)(5,206,24,225)(6,205,25,224)(7,204,26,223)(8,203,27,222)(9,202,28,221)(10,201,29,220)(11,200,30,219)(12,199,31,218)(13,198,32,217)(14,197,33,216)(15,196,34,215)(16,195,35,214)(17,194,36,213)(18,193,37,212)(19,192,38,211)(39,185,58,166)(40,184,59,165)(41,183,60,164)(42,182,61,163)(43,181,62,162)(44,180,63,161)(45,179,64,160)(46,178,65,159)(47,177,66,158)(48,176,67,157)(49,175,68,156)(50,174,69,155)(51,173,70,154)(52,172,71,153)(53,171,72,190)(54,170,73,189)(55,169,74,188)(56,168,75,187)(57,167,76,186)(77,284,96,303)(78,283,97,302)(79,282,98,301)(80,281,99,300)(81,280,100,299)(82,279,101,298)(83,278,102,297)(84,277,103,296)(85,276,104,295)(86,275,105,294)(87,274,106,293)(88,273,107,292)(89,272,108,291)(90,271,109,290)(91,270,110,289)(92,269,111,288)(93,268,112,287)(94,267,113,286)(95,304,114,285)(115,247,134,266)(116,246,135,265)(117,245,136,264)(118,244,137,263)(119,243,138,262)(120,242,139,261)(121,241,140,260)(122,240,141,259)(123,239,142,258)(124,238,143,257)(125,237,144,256)(126,236,145,255)(127,235,146,254)(128,234,147,253)(129,233,148,252)(130,232,149,251)(131,231,150,250)(132,230,151,249)(133,229,152,248)>;

G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,241)(166,242)(167,243)(168,244)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,286)(192,287)(193,288)(194,289)(195,290)(196,291)(197,292)(198,293)(199,294)(200,295)(201,296)(202,297)(203,298)(204,299)(205,300)(206,301)(207,302)(208,303)(209,304)(210,267)(211,268)(212,269)(213,270)(214,271)(215,272)(216,273)(217,274)(218,275)(219,276)(220,277)(221,278)(222,279)(223,280)(224,281)(225,282)(226,283)(227,284)(228,285), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,225)(188,226)(189,227)(190,228)(229,286)(230,287)(231,288)(232,289)(233,290)(234,291)(235,292)(236,293)(237,294)(238,295)(239,296)(240,297)(241,298)(242,299)(243,300)(244,301)(245,302)(246,303)(247,304)(248,267)(249,268)(250,269)(251,270)(252,271)(253,272)(254,273)(255,274)(256,275)(257,276)(258,277)(259,278)(260,279)(261,280)(262,281)(263,282)(264,283)(265,284)(266,285), (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)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,210,20,191)(2,209,21,228)(3,208,22,227)(4,207,23,226)(5,206,24,225)(6,205,25,224)(7,204,26,223)(8,203,27,222)(9,202,28,221)(10,201,29,220)(11,200,30,219)(12,199,31,218)(13,198,32,217)(14,197,33,216)(15,196,34,215)(16,195,35,214)(17,194,36,213)(18,193,37,212)(19,192,38,211)(39,185,58,166)(40,184,59,165)(41,183,60,164)(42,182,61,163)(43,181,62,162)(44,180,63,161)(45,179,64,160)(46,178,65,159)(47,177,66,158)(48,176,67,157)(49,175,68,156)(50,174,69,155)(51,173,70,154)(52,172,71,153)(53,171,72,190)(54,170,73,189)(55,169,74,188)(56,168,75,187)(57,167,76,186)(77,284,96,303)(78,283,97,302)(79,282,98,301)(80,281,99,300)(81,280,100,299)(82,279,101,298)(83,278,102,297)(84,277,103,296)(85,276,104,295)(86,275,105,294)(87,274,106,293)(88,273,107,292)(89,272,108,291)(90,271,109,290)(91,270,110,289)(92,269,111,288)(93,268,112,287)(94,267,113,286)(95,304,114,285)(115,247,134,266)(116,246,135,265)(117,245,136,264)(118,244,137,263)(119,243,138,262)(120,242,139,261)(121,241,140,260)(122,240,141,259)(123,239,142,258)(124,238,143,257)(125,237,144,256)(126,236,145,255)(127,235,146,254)(128,234,147,253)(129,233,148,252)(130,232,149,251)(131,231,150,250)(132,230,151,249)(133,229,152,248) );

G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,109),(17,110),(18,111),(19,112),(20,113),(21,114),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,151),(52,152),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(153,229),(154,230),(155,231),(156,232),(157,233),(158,234),(159,235),(160,236),(161,237),(162,238),(163,239),(164,240),(165,241),(166,242),(167,243),(168,244),(169,245),(170,246),(171,247),(172,248),(173,249),(174,250),(175,251),(176,252),(177,253),(178,254),(179,255),(180,256),(181,257),(182,258),(183,259),(184,260),(185,261),(186,262),(187,263),(188,264),(189,265),(190,266),(191,286),(192,287),(193,288),(194,289),(195,290),(196,291),(197,292),(198,293),(199,294),(200,295),(201,296),(202,297),(203,298),(204,299),(205,300),(206,301),(207,302),(208,303),(209,304),(210,267),(211,268),(212,269),(213,270),(214,271),(215,272),(216,273),(217,274),(218,275),(219,276),(220,277),(221,278),(222,279),(223,280),(224,281),(225,282),(226,283),(227,284),(228,285)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(77,135),(78,136),(79,137),(80,138),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(153,191),(154,192),(155,193),(156,194),(157,195),(158,196),(159,197),(160,198),(161,199),(162,200),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206),(169,207),(170,208),(171,209),(172,210),(173,211),(174,212),(175,213),(176,214),(177,215),(178,216),(179,217),(180,218),(181,219),(182,220),(183,221),(184,222),(185,223),(186,224),(187,225),(188,226),(189,227),(190,228),(229,286),(230,287),(231,288),(232,289),(233,290),(234,291),(235,292),(236,293),(237,294),(238,295),(239,296),(240,297),(241,298),(242,299),(243,300),(244,301),(245,302),(246,303),(247,304),(248,267),(249,268),(250,269),(251,270),(252,271),(253,272),(254,273),(255,274),(256,275),(257,276),(258,277),(259,278),(260,279),(261,280),(262,281),(263,282),(264,283),(265,284),(266,285)], [(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),(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)], [(1,210,20,191),(2,209,21,228),(3,208,22,227),(4,207,23,226),(5,206,24,225),(6,205,25,224),(7,204,26,223),(8,203,27,222),(9,202,28,221),(10,201,29,220),(11,200,30,219),(12,199,31,218),(13,198,32,217),(14,197,33,216),(15,196,34,215),(16,195,35,214),(17,194,36,213),(18,193,37,212),(19,192,38,211),(39,185,58,166),(40,184,59,165),(41,183,60,164),(42,182,61,163),(43,181,62,162),(44,180,63,161),(45,179,64,160),(46,178,65,159),(47,177,66,158),(48,176,67,157),(49,175,68,156),(50,174,69,155),(51,173,70,154),(52,172,71,153),(53,171,72,190),(54,170,73,189),(55,169,74,188),(56,168,75,187),(57,167,76,186),(77,284,96,303),(78,283,97,302),(79,282,98,301),(80,281,99,300),(81,280,100,299),(82,279,101,298),(83,278,102,297),(84,277,103,296),(85,276,104,295),(86,275,105,294),(87,274,106,293),(88,273,107,292),(89,272,108,291),(90,271,109,290),(91,270,110,289),(92,269,111,288),(93,268,112,287),(94,267,113,286),(95,304,114,285),(115,247,134,266),(116,246,135,265),(117,245,136,264),(118,244,137,263),(119,243,138,262),(120,242,139,261),(121,241,140,260),(122,240,141,259),(123,239,142,258),(124,238,143,257),(125,237,144,256),(126,236,145,255),(127,235,146,254),(128,234,147,253),(129,233,148,252),(130,232,149,251),(131,231,150,250),(132,230,151,249),(133,229,152,248)]])

88 conjugacy classes

class	1	2A	···	2G	4A	···	4H	19A	···	19I	38A	···	38BK
order	1	2	···	2	4	···	4	19	···	19	38	···	38
size	1	1	···	1	19	···	19	2	···	2	2	···	2

88 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+	+		+	-	+
image	C₁	C₂	C₂	C₄	D₁₉	Dic₁₉	D₃₈
kernel	C₂²×Dic₁₉	C₂×Dic₁₉	C₂²×C₃₈	C₂×C₃₈	C₂³	C₂²	C₂²
# reps	1	6	1	8	9	36	27

Matrix representation of C₂²×Dic₁₉ ►in GL₅(𝔽₂₂₉)

228	0	0	0	0
0	1	0	0	0
0	0	228	0	0
0	0	0	1	0
0	0	0	0	1

228	0	0	0	0
0	228	0	0	0
0	0	228	0	0
0	0	0	1	0
0	0	0	0	1

228	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	228	97

122	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	97	228

G:=sub<GL(5,GF(229))| [228,0,0,0,0,0,1,0,0,0,0,0,228,0,0,0,0,0,1,0,0,0,0,0,1],[228,0,0,0,0,0,228,0,0,0,0,0,228,0,0,0,0,0,1,0,0,0,0,0,1],[228,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,228,0,0,0,1,97],[122,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,97,0,0,0,0,228] >;

C₂²×Dic₁₉ in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{19}

% in TeX

G:=Group("C2^2xDic19");

// GroupNames label

G:=SmallGroup(304,35);

// by ID

G=gap.SmallGroup(304,35);

# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,40,7204]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^38=1,d^2=c^19,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

G = C22×Dic19 order 304 = 24·19

Direct product of C22 and Dic19

G = C₂²×Dic₁₉ order 304 = 2⁴·19

Direct product of C₂² and Dic₁₉