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G = C22×C5⋊Q16order 320 = 26·5

Direct product of C22 and C5⋊Q16

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

Aliases: C22×C5⋊Q16, C20.32C24, Dic10.28C23, C103(C2×Q16), (C2×C10)⋊9Q16, C53(C22×Q16), (C2×C20).212D4, C20.256(C2×D4), C4.32(C23×D5), (C22×Q8).7D5, C52C8.29C23, (C2×Q8).185D10, (C5×Q8).21C23, Q8.21(C22×D5), (C2×C20).549C23, (C22×C10).211D4, C10.151(C22×D4), (C22×C4).382D10, C23.107(C5⋊D4), (Q8×C10).230C22, (C22×C20).281C22, (C22×Dic10).19C2, (C2×Dic10).314C22, (Q8×C2×C10).6C2, C4.26(C2×C5⋊D4), (C2×C10).586(C2×D4), C2.24(C22×C5⋊D4), (C2×C4).155(C5⋊D4), (C22×C52C8).14C2, (C2×C4).630(C22×D5), C22.114(C2×C5⋊D4), (C2×C52C8).295C22, SmallGroup(320,1481)

Series: Derived Chief Lower central Upper central

C1C20 — C22×C5⋊Q16
C1C5C10C20Dic10C2×Dic10C22×Dic10 — C22×C5⋊Q16
C5C10C20 — C22×C5⋊Q16
C1C23C22×C4C22×Q8

Generators and relations for C22×C5⋊Q16
 G = < a,b,c,d,e | a2=b2=c5=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 702 in 258 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C5, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×4], Q8 [×16], C23, C10, C10 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×12], Dic5 [×4], C20, C20 [×3], C20 [×4], C2×C10 [×7], C22×C8, C2×Q16 [×12], C22×Q8, C22×Q8, C52C8 [×4], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22×Q16, C2×C52C8 [×6], C5⋊Q16 [×16], C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C22×C52C8, C2×C5⋊Q16 [×12], C22×Dic10, Q8×C2×C10, C22×C5⋊Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C22×Q16, C5⋊Q16 [×4], C2×C5⋊D4 [×6], C23×D5, C2×C5⋊Q16 [×6], C22×C5⋊D4, C22×C5⋊Q16

Smallest permutation representation of C22×C5⋊Q16
Regular action on 320 points
Generators in S320
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 183)(18 184)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(41 151)(42 152)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 153)(57 249)(58 250)(59 251)(60 252)(61 253)(62 254)(63 255)(64 256)(73 198)(74 199)(75 200)(76 193)(77 194)(78 195)(79 196)(80 197)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)(89 133)(90 134)(91 135)(92 136)(93 129)(94 130)(95 131)(96 132)(97 141)(98 142)(99 143)(100 144)(101 137)(102 138)(103 139)(104 140)(113 222)(114 223)(115 224)(116 217)(117 218)(118 219)(119 220)(120 221)(161 205)(162 206)(163 207)(164 208)(165 201)(166 202)(167 203)(168 204)(169 213)(170 214)(171 215)(172 216)(173 209)(174 210)(175 211)(176 212)(185 242)(186 243)(187 244)(188 245)(189 246)(190 247)(191 248)(192 241)(225 294)(226 295)(227 296)(228 289)(229 290)(230 291)(231 292)(232 293)(233 287)(234 288)(235 281)(236 282)(237 283)(238 284)(239 285)(240 286)(257 301)(258 302)(259 303)(260 304)(261 297)(262 298)(263 299)(264 300)(265 309)(266 310)(267 311)(268 312)(269 305)(270 306)(271 307)(272 308)(273 317)(274 318)(275 319)(276 320)(277 313)(278 314)(279 315)(280 316)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 81)(16 82)(17 197)(18 198)(19 199)(20 200)(21 193)(22 194)(23 195)(24 196)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 65)(48 66)(57 173)(58 174)(59 175)(60 176)(61 169)(62 170)(63 171)(64 172)(73 184)(74 177)(75 178)(76 179)(77 180)(78 181)(79 182)(80 183)(89 223)(90 224)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 228)(98 229)(99 230)(100 231)(101 232)(102 225)(103 226)(104 227)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 132)(114 133)(115 134)(116 135)(117 136)(118 129)(119 130)(120 131)(137 293)(138 294)(139 295)(140 296)(141 289)(142 290)(143 291)(144 292)(161 188)(162 189)(163 190)(164 191)(165 192)(166 185)(167 186)(168 187)(201 241)(202 242)(203 243)(204 244)(205 245)(206 246)(207 247)(208 248)(209 249)(210 250)(211 251)(212 252)(213 253)(214 254)(215 255)(216 256)(233 310)(234 311)(235 312)(236 305)(237 306)(238 307)(239 308)(240 309)(257 275)(258 276)(259 277)(260 278)(261 279)(262 280)(263 273)(264 274)(265 286)(266 287)(267 288)(268 281)(269 282)(270 283)(271 284)(272 285)(297 315)(298 316)(299 317)(300 318)(301 319)(302 320)(303 313)(304 314)
(1 110 75 287 291)(2 292 288 76 111)(3 112 77 281 293)(4 294 282 78 105)(5 106 79 283 295)(6 296 284 80 107)(7 108 73 285 289)(8 290 286 74 109)(9 37 225 236 195)(10 196 237 226 38)(11 39 227 238 197)(12 198 239 228 40)(13 33 229 240 199)(14 200 233 230 34)(15 35 231 234 193)(16 194 235 232 36)(17 85 159 104 307)(18 308 97 160 86)(19 87 153 98 309)(20 310 99 154 88)(21 81 155 100 311)(22 312 101 156 82)(23 83 157 102 305)(24 306 103 158 84)(25 313 206 214 224)(26 217 215 207 314)(27 315 208 216 218)(28 219 209 201 316)(29 317 202 210 220)(30 221 211 203 318)(31 319 204 212 222)(32 223 213 205 320)(41 257 187 60 132)(42 133 61 188 258)(43 259 189 62 134)(44 135 63 190 260)(45 261 191 64 136)(46 129 57 192 262)(47 263 185 58 130)(48 131 59 186 264)(49 124 178 266 143)(50 144 267 179 125)(51 126 180 268 137)(52 138 269 181 127)(53 128 182 270 139)(54 140 271 183 121)(55 122 184 272 141)(56 142 265 177 123)(65 273 166 174 119)(66 120 175 167 274)(67 275 168 176 113)(68 114 169 161 276)(69 277 162 170 115)(70 116 171 163 278)(71 279 164 172 117)(72 118 173 165 280)(89 253 245 302 152)(90 145 303 246 254)(91 255 247 304 146)(92 147 297 248 256)(93 249 241 298 148)(94 149 299 242 250)(95 251 243 300 150)(96 151 301 244 252)
(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)(305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320)
(1 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 44 13 48)(10 43 14 47)(11 42 15 46)(12 41 16 45)(17 276 21 280)(18 275 22 279)(19 274 23 278)(20 273 24 277)(25 124 29 128)(26 123 30 127)(27 122 31 126)(28 121 32 125)(33 131 37 135)(34 130 38 134)(35 129 39 133)(36 136 40 132)(49 220 53 224)(50 219 54 223)(51 218 55 222)(52 217 56 221)(57 227 61 231)(58 226 62 230)(59 225 63 229)(60 232 64 228)(65 84 69 88)(66 83 70 87)(67 82 71 86)(68 81 72 85)(73 301 77 297)(74 300 78 304)(75 299 79 303)(76 298 80 302)(97 176 101 172)(98 175 102 171)(99 174 103 170)(100 173 104 169)(105 146 109 150)(106 145 110 149)(107 152 111 148)(108 151 112 147)(113 156 117 160)(114 155 118 159)(115 154 119 158)(116 153 120 157)(137 216 141 212)(138 215 142 211)(139 214 143 210)(140 213 144 209)(161 311 165 307)(162 310 166 306)(163 309 167 305)(164 308 168 312)(177 318 181 314)(178 317 182 313)(179 316 183 320)(180 315 184 319)(185 237 189 233)(186 236 190 240)(187 235 191 239)(188 234 192 238)(193 262 197 258)(194 261 198 257)(195 260 199 264)(196 259 200 263)(201 271 205 267)(202 270 206 266)(203 269 207 265)(204 268 208 272)(241 284 245 288)(242 283 246 287)(243 282 247 286)(244 281 248 285)(249 296 253 292)(250 295 254 291)(251 294 255 290)(252 293 256 289)

G:=sub<Sym(320)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,183)(18,184)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(41,151)(42,152)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,153)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(73,198)(74,199)(75,200)(76,193)(77,194)(78,195)(79,196)(80,197)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(113,222)(114,223)(115,224)(116,217)(117,218)(118,219)(119,220)(120,221)(161,205)(162,206)(163,207)(164,208)(165,201)(166,202)(167,203)(168,204)(169,213)(170,214)(171,215)(172,216)(173,209)(174,210)(175,211)(176,212)(185,242)(186,243)(187,244)(188,245)(189,246)(190,247)(191,248)(192,241)(225,294)(226,295)(227,296)(228,289)(229,290)(230,291)(231,292)(232,293)(233,287)(234,288)(235,281)(236,282)(237,283)(238,284)(239,285)(240,286)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308)(273,317)(274,318)(275,319)(276,320)(277,313)(278,314)(279,315)(280,316), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,197)(18,198)(19,199)(20,200)(21,193)(22,194)(23,195)(24,196)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,228)(98,229)(99,230)(100,231)(101,232)(102,225)(103,226)(104,227)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,132)(114,133)(115,134)(116,135)(117,136)(118,129)(119,130)(120,131)(137,293)(138,294)(139,295)(140,296)(141,289)(142,290)(143,291)(144,292)(161,188)(162,189)(163,190)(164,191)(165,192)(166,185)(167,186)(168,187)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(233,310)(234,311)(235,312)(236,305)(237,306)(238,307)(239,308)(240,309)(257,275)(258,276)(259,277)(260,278)(261,279)(262,280)(263,273)(264,274)(265,286)(266,287)(267,288)(268,281)(269,282)(270,283)(271,284)(272,285)(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,313)(304,314), (1,110,75,287,291)(2,292,288,76,111)(3,112,77,281,293)(4,294,282,78,105)(5,106,79,283,295)(6,296,284,80,107)(7,108,73,285,289)(8,290,286,74,109)(9,37,225,236,195)(10,196,237,226,38)(11,39,227,238,197)(12,198,239,228,40)(13,33,229,240,199)(14,200,233,230,34)(15,35,231,234,193)(16,194,235,232,36)(17,85,159,104,307)(18,308,97,160,86)(19,87,153,98,309)(20,310,99,154,88)(21,81,155,100,311)(22,312,101,156,82)(23,83,157,102,305)(24,306,103,158,84)(25,313,206,214,224)(26,217,215,207,314)(27,315,208,216,218)(28,219,209,201,316)(29,317,202,210,220)(30,221,211,203,318)(31,319,204,212,222)(32,223,213,205,320)(41,257,187,60,132)(42,133,61,188,258)(43,259,189,62,134)(44,135,63,190,260)(45,261,191,64,136)(46,129,57,192,262)(47,263,185,58,130)(48,131,59,186,264)(49,124,178,266,143)(50,144,267,179,125)(51,126,180,268,137)(52,138,269,181,127)(53,128,182,270,139)(54,140,271,183,121)(55,122,184,272,141)(56,142,265,177,123)(65,273,166,174,119)(66,120,175,167,274)(67,275,168,176,113)(68,114,169,161,276)(69,277,162,170,115)(70,116,171,163,278)(71,279,164,172,117)(72,118,173,165,280)(89,253,245,302,152)(90,145,303,246,254)(91,255,247,304,146)(92,147,297,248,256)(93,249,241,298,148)(94,149,299,242,250)(95,251,243,300,150)(96,151,301,244,252), (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)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,44,13,48)(10,43,14,47)(11,42,15,46)(12,41,16,45)(17,276,21,280)(18,275,22,279)(19,274,23,278)(20,273,24,277)(25,124,29,128)(26,123,30,127)(27,122,31,126)(28,121,32,125)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(49,220,53,224)(50,219,54,223)(51,218,55,222)(52,217,56,221)(57,227,61,231)(58,226,62,230)(59,225,63,229)(60,232,64,228)(65,84,69,88)(66,83,70,87)(67,82,71,86)(68,81,72,85)(73,301,77,297)(74,300,78,304)(75,299,79,303)(76,298,80,302)(97,176,101,172)(98,175,102,171)(99,174,103,170)(100,173,104,169)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147)(113,156,117,160)(114,155,118,159)(115,154,119,158)(116,153,120,157)(137,216,141,212)(138,215,142,211)(139,214,143,210)(140,213,144,209)(161,311,165,307)(162,310,166,306)(163,309,167,305)(164,308,168,312)(177,318,181,314)(178,317,182,313)(179,316,183,320)(180,315,184,319)(185,237,189,233)(186,236,190,240)(187,235,191,239)(188,234,192,238)(193,262,197,258)(194,261,198,257)(195,260,199,264)(196,259,200,263)(201,271,205,267)(202,270,206,266)(203,269,207,265)(204,268,208,272)(241,284,245,288)(242,283,246,287)(243,282,247,286)(244,281,248,285)(249,296,253,292)(250,295,254,291)(251,294,255,290)(252,293,256,289)>;

G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,183)(18,184)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(41,151)(42,152)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,153)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(73,198)(74,199)(75,200)(76,193)(77,194)(78,195)(79,196)(80,197)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(113,222)(114,223)(115,224)(116,217)(117,218)(118,219)(119,220)(120,221)(161,205)(162,206)(163,207)(164,208)(165,201)(166,202)(167,203)(168,204)(169,213)(170,214)(171,215)(172,216)(173,209)(174,210)(175,211)(176,212)(185,242)(186,243)(187,244)(188,245)(189,246)(190,247)(191,248)(192,241)(225,294)(226,295)(227,296)(228,289)(229,290)(230,291)(231,292)(232,293)(233,287)(234,288)(235,281)(236,282)(237,283)(238,284)(239,285)(240,286)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308)(273,317)(274,318)(275,319)(276,320)(277,313)(278,314)(279,315)(280,316), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,197)(18,198)(19,199)(20,200)(21,193)(22,194)(23,195)(24,196)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,228)(98,229)(99,230)(100,231)(101,232)(102,225)(103,226)(104,227)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,132)(114,133)(115,134)(116,135)(117,136)(118,129)(119,130)(120,131)(137,293)(138,294)(139,295)(140,296)(141,289)(142,290)(143,291)(144,292)(161,188)(162,189)(163,190)(164,191)(165,192)(166,185)(167,186)(168,187)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(233,310)(234,311)(235,312)(236,305)(237,306)(238,307)(239,308)(240,309)(257,275)(258,276)(259,277)(260,278)(261,279)(262,280)(263,273)(264,274)(265,286)(266,287)(267,288)(268,281)(269,282)(270,283)(271,284)(272,285)(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,313)(304,314), (1,110,75,287,291)(2,292,288,76,111)(3,112,77,281,293)(4,294,282,78,105)(5,106,79,283,295)(6,296,284,80,107)(7,108,73,285,289)(8,290,286,74,109)(9,37,225,236,195)(10,196,237,226,38)(11,39,227,238,197)(12,198,239,228,40)(13,33,229,240,199)(14,200,233,230,34)(15,35,231,234,193)(16,194,235,232,36)(17,85,159,104,307)(18,308,97,160,86)(19,87,153,98,309)(20,310,99,154,88)(21,81,155,100,311)(22,312,101,156,82)(23,83,157,102,305)(24,306,103,158,84)(25,313,206,214,224)(26,217,215,207,314)(27,315,208,216,218)(28,219,209,201,316)(29,317,202,210,220)(30,221,211,203,318)(31,319,204,212,222)(32,223,213,205,320)(41,257,187,60,132)(42,133,61,188,258)(43,259,189,62,134)(44,135,63,190,260)(45,261,191,64,136)(46,129,57,192,262)(47,263,185,58,130)(48,131,59,186,264)(49,124,178,266,143)(50,144,267,179,125)(51,126,180,268,137)(52,138,269,181,127)(53,128,182,270,139)(54,140,271,183,121)(55,122,184,272,141)(56,142,265,177,123)(65,273,166,174,119)(66,120,175,167,274)(67,275,168,176,113)(68,114,169,161,276)(69,277,162,170,115)(70,116,171,163,278)(71,279,164,172,117)(72,118,173,165,280)(89,253,245,302,152)(90,145,303,246,254)(91,255,247,304,146)(92,147,297,248,256)(93,249,241,298,148)(94,149,299,242,250)(95,251,243,300,150)(96,151,301,244,252), (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)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,44,13,48)(10,43,14,47)(11,42,15,46)(12,41,16,45)(17,276,21,280)(18,275,22,279)(19,274,23,278)(20,273,24,277)(25,124,29,128)(26,123,30,127)(27,122,31,126)(28,121,32,125)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(49,220,53,224)(50,219,54,223)(51,218,55,222)(52,217,56,221)(57,227,61,231)(58,226,62,230)(59,225,63,229)(60,232,64,228)(65,84,69,88)(66,83,70,87)(67,82,71,86)(68,81,72,85)(73,301,77,297)(74,300,78,304)(75,299,79,303)(76,298,80,302)(97,176,101,172)(98,175,102,171)(99,174,103,170)(100,173,104,169)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147)(113,156,117,160)(114,155,118,159)(115,154,119,158)(116,153,120,157)(137,216,141,212)(138,215,142,211)(139,214,143,210)(140,213,144,209)(161,311,165,307)(162,310,166,306)(163,309,167,305)(164,308,168,312)(177,318,181,314)(178,317,182,313)(179,316,183,320)(180,315,184,319)(185,237,189,233)(186,236,190,240)(187,235,191,239)(188,234,192,238)(193,262,197,258)(194,261,198,257)(195,260,199,264)(196,259,200,263)(201,271,205,267)(202,270,206,266)(203,269,207,265)(204,268,208,272)(241,284,245,288)(242,283,246,287)(243,282,247,286)(244,281,248,285)(249,296,253,292)(250,295,254,291)(251,294,255,290)(252,293,256,289) );

G=PermutationGroup([(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,183),(18,184),(19,177),(20,178),(21,179),(22,180),(23,181),(24,182),(25,69),(26,70),(27,71),(28,72),(29,65),(30,66),(31,67),(32,68),(41,151),(42,152),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,153),(57,249),(58,250),(59,251),(60,252),(61,253),(62,254),(63,255),(64,256),(73,198),(74,199),(75,200),(76,193),(77,194),(78,195),(79,196),(80,197),(81,125),(82,126),(83,127),(84,128),(85,121),(86,122),(87,123),(88,124),(89,133),(90,134),(91,135),(92,136),(93,129),(94,130),(95,131),(96,132),(97,141),(98,142),(99,143),(100,144),(101,137),(102,138),(103,139),(104,140),(113,222),(114,223),(115,224),(116,217),(117,218),(118,219),(119,220),(120,221),(161,205),(162,206),(163,207),(164,208),(165,201),(166,202),(167,203),(168,204),(169,213),(170,214),(171,215),(172,216),(173,209),(174,210),(175,211),(176,212),(185,242),(186,243),(187,244),(188,245),(189,246),(190,247),(191,248),(192,241),(225,294),(226,295),(227,296),(228,289),(229,290),(230,291),(231,292),(232,293),(233,287),(234,288),(235,281),(236,282),(237,283),(238,284),(239,285),(240,286),(257,301),(258,302),(259,303),(260,304),(261,297),(262,298),(263,299),(264,300),(265,309),(266,310),(267,311),(268,312),(269,305),(270,306),(271,307),(272,308),(273,317),(274,318),(275,319),(276,320),(277,313),(278,314),(279,315),(280,316)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,81),(16,82),(17,197),(18,198),(19,199),(20,200),(21,193),(22,194),(23,195),(24,196),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,65),(48,66),(57,173),(58,174),(59,175),(60,176),(61,169),(62,170),(63,171),(64,172),(73,184),(74,177),(75,178),(76,179),(77,180),(78,181),(79,182),(80,183),(89,223),(90,224),(91,217),(92,218),(93,219),(94,220),(95,221),(96,222),(97,228),(98,229),(99,230),(100,231),(101,232),(102,225),(103,226),(104,227),(105,127),(106,128),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,132),(114,133),(115,134),(116,135),(117,136),(118,129),(119,130),(120,131),(137,293),(138,294),(139,295),(140,296),(141,289),(142,290),(143,291),(144,292),(161,188),(162,189),(163,190),(164,191),(165,192),(166,185),(167,186),(168,187),(201,241),(202,242),(203,243),(204,244),(205,245),(206,246),(207,247),(208,248),(209,249),(210,250),(211,251),(212,252),(213,253),(214,254),(215,255),(216,256),(233,310),(234,311),(235,312),(236,305),(237,306),(238,307),(239,308),(240,309),(257,275),(258,276),(259,277),(260,278),(261,279),(262,280),(263,273),(264,274),(265,286),(266,287),(267,288),(268,281),(269,282),(270,283),(271,284),(272,285),(297,315),(298,316),(299,317),(300,318),(301,319),(302,320),(303,313),(304,314)], [(1,110,75,287,291),(2,292,288,76,111),(3,112,77,281,293),(4,294,282,78,105),(5,106,79,283,295),(6,296,284,80,107),(7,108,73,285,289),(8,290,286,74,109),(9,37,225,236,195),(10,196,237,226,38),(11,39,227,238,197),(12,198,239,228,40),(13,33,229,240,199),(14,200,233,230,34),(15,35,231,234,193),(16,194,235,232,36),(17,85,159,104,307),(18,308,97,160,86),(19,87,153,98,309),(20,310,99,154,88),(21,81,155,100,311),(22,312,101,156,82),(23,83,157,102,305),(24,306,103,158,84),(25,313,206,214,224),(26,217,215,207,314),(27,315,208,216,218),(28,219,209,201,316),(29,317,202,210,220),(30,221,211,203,318),(31,319,204,212,222),(32,223,213,205,320),(41,257,187,60,132),(42,133,61,188,258),(43,259,189,62,134),(44,135,63,190,260),(45,261,191,64,136),(46,129,57,192,262),(47,263,185,58,130),(48,131,59,186,264),(49,124,178,266,143),(50,144,267,179,125),(51,126,180,268,137),(52,138,269,181,127),(53,128,182,270,139),(54,140,271,183,121),(55,122,184,272,141),(56,142,265,177,123),(65,273,166,174,119),(66,120,175,167,274),(67,275,168,176,113),(68,114,169,161,276),(69,277,162,170,115),(70,116,171,163,278),(71,279,164,172,117),(72,118,173,165,280),(89,253,245,302,152),(90,145,303,246,254),(91,255,247,304,146),(92,147,297,248,256),(93,249,241,298,148),(94,149,299,242,250),(95,251,243,300,150),(96,151,301,244,252)], [(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),(305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320)], [(1,94,5,90),(2,93,6,89),(3,92,7,96),(4,91,8,95),(9,44,13,48),(10,43,14,47),(11,42,15,46),(12,41,16,45),(17,276,21,280),(18,275,22,279),(19,274,23,278),(20,273,24,277),(25,124,29,128),(26,123,30,127),(27,122,31,126),(28,121,32,125),(33,131,37,135),(34,130,38,134),(35,129,39,133),(36,136,40,132),(49,220,53,224),(50,219,54,223),(51,218,55,222),(52,217,56,221),(57,227,61,231),(58,226,62,230),(59,225,63,229),(60,232,64,228),(65,84,69,88),(66,83,70,87),(67,82,71,86),(68,81,72,85),(73,301,77,297),(74,300,78,304),(75,299,79,303),(76,298,80,302),(97,176,101,172),(98,175,102,171),(99,174,103,170),(100,173,104,169),(105,146,109,150),(106,145,110,149),(107,152,111,148),(108,151,112,147),(113,156,117,160),(114,155,118,159),(115,154,119,158),(116,153,120,157),(137,216,141,212),(138,215,142,211),(139,214,143,210),(140,213,144,209),(161,311,165,307),(162,310,166,306),(163,309,167,305),(164,308,168,312),(177,318,181,314),(178,317,182,313),(179,316,183,320),(180,315,184,319),(185,237,189,233),(186,236,190,240),(187,235,191,239),(188,234,192,238),(193,262,197,258),(194,261,198,257),(195,260,199,264),(196,259,200,263),(201,271,205,267),(202,270,206,266),(203,269,207,265),(204,268,208,272),(241,284,245,288),(242,283,246,287),(243,282,247,286),(244,281,248,285),(249,296,253,292),(250,295,254,291),(251,294,255,290),(252,293,256,289)])

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10N20A···20X
order12···2444444444444558···810···1020···20
size11···122224444202020202210···102···24···4

68 irreducible representations

dim11111222222224
type++++++++-++-
imageC1C2C2C2C2D4D4D5Q16D10D10C5⋊D4C5⋊D4C5⋊Q16
kernelC22×C5⋊Q16C22×C52C8C2×C5⋊Q16C22×Dic10Q8×C2×C10C2×C20C22×C10C22×Q8C2×C10C22×C4C2×Q8C2×C4C23C22
# reps11121131282121248

Matrix representation of C22×C5⋊Q16 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
0040000
0004000
0000400
0000040
,
0400000
160000
0064000
001000
000010
000001
,
0400000
4000000
0040600
000100
00001229
00001212
,
4000000
0400000
001000
000100
00001140
00004030

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,6,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,40,0,0,0,0,40,30] >;

C22×C5⋊Q16 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C2^2xC5:Q16");
// GroupNames label

G:=SmallGroup(320,1481);
// by ID

G=gap.SmallGroup(320,1481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,136,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽