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G = C10.(C4⋊Q8)  order 320 = 26·5

1st non-split extension by C10 of C4⋊Q8 acting via C4⋊Q8/C42=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).3Q8, (C2×C4).18D20, (C2×C20).29D4, C10.1(C4⋊Q8), C2.6(C4⋊D20), (C2×Dic5).19D4, C2.5(C202Q8), (C2×C4).10Dic10, C22.79(C2×D20), (C22×C4).69D10, C22.154(D4×D5), C10.6(C22⋊Q8), C10.33(C4⋊D4), C2.7(C4.Dic10), (C22×C20).45C22, C10.14(C42.C2), C22.43(C2×Dic10), C2.C42.16D5, C23.357(C22×D5), C22.86(D42D5), (C22×C10).286C23, C52(C23.81C23), C22.42(Q82D5), C10.7(C22.D4), C2.7(C22.D20), C10.10C42.26C2, (C22×Dic5).11C22, C2.11(Dic5.14D4), (C2×C10).95(C2×D4), (C2×C10).22(C2×Q8), (C2×C4⋊Dic5).11C2, (C2×C10).181(C4○D4), (C2×C10.D4).22C2, (C5×C2.C42).12C2, SmallGroup(320,288)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C10.(C4⋊Q8)
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — C10.(C4⋊Q8)
C5C22×C10 — C10.(C4⋊Q8)
C1C23C2.C42

Generators and relations for C10.(C4⋊Q8)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a5b, dbd-1=b-1, dcd-1=a5c-1 >

Subgroups: 502 in 150 conjugacy classes, 65 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22 [×3], C22 [×4], C5, C2×C4 [×6], C2×C4 [×19], C23, C10 [×3], C10 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×5], C20 [×6], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×2], C2×C4⋊C4 [×4], C2×Dic5 [×2], C2×Dic5 [×11], C2×C20 [×6], C2×C20 [×6], C22×C10, C23.81C23, C10.D4 [×2], C4⋊Dic5 [×6], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42 [×2], C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2×C4⋊Dic5 [×2], C10.(C4⋊Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D5, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D10 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, Dic10 [×4], D20 [×2], C22×D5, C23.81C23, C2×Dic10 [×2], C2×D20, D4×D5, D42D5 [×2], Q82D5, C202Q8, Dic5.14D4 [×2], C22.D20, C4.Dic10 [×2], C4⋊D20, C10.(C4⋊Q8)

Smallest permutation representation of C10.(C4⋊Q8)
Regular action on 320 points
Generators in S320
(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)
(1 133 33 143)(2 134 34 144)(3 135 35 145)(4 136 36 146)(5 137 37 147)(6 138 38 148)(7 139 39 149)(8 140 40 150)(9 131 31 141)(10 132 32 142)(11 121 21 151)(12 122 22 152)(13 123 23 153)(14 124 24 154)(15 125 25 155)(16 126 26 156)(17 127 27 157)(18 128 28 158)(19 129 29 159)(20 130 30 160)(41 111 71 81)(42 112 72 82)(43 113 73 83)(44 114 74 84)(45 115 75 85)(46 116 76 86)(47 117 77 87)(48 118 78 88)(49 119 79 89)(50 120 80 90)(51 101 61 91)(52 102 62 92)(53 103 63 93)(54 104 64 94)(55 105 65 95)(56 106 66 96)(57 107 67 97)(58 108 68 98)(59 109 69 99)(60 110 70 100)(161 301 191 291)(162 302 192 292)(163 303 193 293)(164 304 194 294)(165 305 195 295)(166 306 196 296)(167 307 197 297)(168 308 198 298)(169 309 199 299)(170 310 200 300)(171 311 181 281)(172 312 182 282)(173 313 183 283)(174 314 184 284)(175 315 185 285)(176 316 186 286)(177 317 187 287)(178 318 188 288)(179 319 189 289)(180 320 190 290)(201 241 231 271)(202 242 232 272)(203 243 233 273)(204 244 234 274)(205 245 235 275)(206 246 236 276)(207 247 237 277)(208 248 238 278)(209 249 239 279)(210 250 240 280)(211 251 221 261)(212 252 222 262)(213 253 223 263)(214 254 224 264)(215 255 225 265)(216 256 226 266)(217 257 227 267)(218 258 228 268)(219 259 229 269)(220 260 230 270)
(1 98 18 83)(2 99 19 84)(3 100 20 85)(4 91 11 86)(5 92 12 87)(6 93 13 88)(7 94 14 89)(8 95 15 90)(9 96 16 81)(10 97 17 82)(21 116 36 101)(22 117 37 102)(23 118 38 103)(24 119 39 104)(25 120 40 105)(26 111 31 106)(27 112 32 107)(28 113 33 108)(29 114 34 109)(30 115 35 110)(41 136 56 121)(42 137 57 122)(43 138 58 123)(44 139 59 124)(45 140 60 125)(46 131 51 126)(47 132 52 127)(48 133 53 128)(49 134 54 129)(50 135 55 130)(61 156 76 141)(62 157 77 142)(63 158 78 143)(64 159 79 144)(65 160 80 145)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)(161 246 176 251)(162 247 177 252)(163 248 178 253)(164 249 179 254)(165 250 180 255)(166 241 171 256)(167 242 172 257)(168 243 173 258)(169 244 174 259)(170 245 175 260)(181 266 196 271)(182 267 197 272)(183 268 198 273)(184 269 199 274)(185 270 200 275)(186 261 191 276)(187 262 192 277)(188 263 193 278)(189 264 194 279)(190 265 195 280)(201 286 216 291)(202 287 217 292)(203 288 218 293)(204 289 219 294)(205 290 220 295)(206 281 211 296)(207 282 212 297)(208 283 213 298)(209 284 214 299)(210 285 215 300)(221 306 236 311)(222 307 237 312)(223 308 238 313)(224 309 239 314)(225 310 240 315)(226 301 231 316)(227 302 232 317)(228 303 233 318)(229 304 234 319)(230 305 235 320)
(1 177 13 167)(2 176 14 166)(3 175 15 165)(4 174 16 164)(5 173 17 163)(6 172 18 162)(7 171 19 161)(8 180 20 170)(9 179 11 169)(10 178 12 168)(21 199 31 189)(22 198 32 188)(23 197 33 187)(24 196 34 186)(25 195 35 185)(26 194 36 184)(27 193 37 183)(28 192 38 182)(29 191 39 181)(30 200 40 190)(41 219 51 209)(42 218 52 208)(43 217 53 207)(44 216 54 206)(45 215 55 205)(46 214 56 204)(47 213 57 203)(48 212 58 202)(49 211 59 201)(50 220 60 210)(61 239 71 229)(62 238 72 228)(63 237 73 227)(64 236 74 226)(65 235 75 225)(66 234 76 224)(67 233 77 223)(68 232 78 222)(69 231 79 221)(70 240 80 230)(81 259 91 249)(82 258 92 248)(83 257 93 247)(84 256 94 246)(85 255 95 245)(86 254 96 244)(87 253 97 243)(88 252 98 242)(89 251 99 241)(90 260 100 250)(101 279 111 269)(102 278 112 268)(103 277 113 267)(104 276 114 266)(105 275 115 265)(106 274 116 264)(107 273 117 263)(108 272 118 262)(109 271 119 261)(110 280 120 270)(121 299 131 289)(122 298 132 288)(123 297 133 287)(124 296 134 286)(125 295 135 285)(126 294 136 284)(127 293 137 283)(128 292 138 282)(129 291 139 281)(130 300 140 290)(141 319 151 309)(142 318 152 308)(143 317 153 307)(144 316 154 306)(145 315 155 305)(146 314 156 304)(147 313 157 303)(148 312 158 302)(149 311 159 301)(150 320 160 310)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim1111122222222444
type+++++++-++-++-+
imageC1C2C2C2C2D4D4Q8D5C4○D4D10Dic10D20D4×D5D42D5Q82D5
kernelC10.(C4⋊Q8)C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×Dic5C2×C20C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps12113224266168242

Matrix representation of C10.(C4⋊Q8) in GL6(𝔽41)

770000
34400000
00353500
0064000
0000400
0000040
,
30320000
9110000
0021300
00283900
00003821
0000213
,
4000000
0400000
0021300
00283900
0000213
0000320
,
3150000
35380000
0003200
0032000
00002527
00002716

G:=sub<GL(6,GF(41))| [7,34,0,0,0,0,7,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[30,9,0,0,0,0,32,11,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,38,21,0,0,0,0,21,3],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,21,3,0,0,0,0,3,20],[3,35,0,0,0,0,15,38,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,25,27,0,0,0,0,27,16] >;

C10.(C4⋊Q8) in GAP, Magma, Sage, TeX

C_{10}.(C_4\rtimes Q_8)
% in TeX

G:=Group("C10.(C4:Q8)");
// GroupNames label

G:=SmallGroup(320,288);
// by ID

G=gap.SmallGroup(320,288);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^5*b,d*b*d^-1=b^-1,d*c*d^-1=a^5*c^-1>;
// generators/relations

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