C10.C4wrC2 - GroupNames

G = C₁₀.C₄≀C₂ order 320 = 2⁶·5

2^nd non-split extension by C₁₀ of C₄≀C₂ acting via C₄≀C₂/D₄=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₀.₂C₄≀C₂, C₄⋊C₄.₁F₅, C₄⋊Dic₅.₄C₄, C₂.₅(D₄⋊F₅), C₂.₅(Q₈⋊₂F₅), (C₂×Dic₅).₉₄D₄, C₅⋊(C₄².₂C₂²), C₄.Dic₁₀.₁C₂, (C₄×Dic₅).₁C₂², C₁₀.₁(C₄.₁₀D₄), C₁₀.C₄².₁C₂, C₂².₅₇(C₂²⋊F₅), C₂.₄(Dic₅.D₄), (C₅×C₄⋊C₄).₁C₄, (C₂×C₂₀).₇(C₂×C₄), (C₂×C₄).₁₀(C₂×F₅), (C₂×C₁₀).₂₀(C₂²⋊C₄), SmallGroup(320,208)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₁₀.C₄≀C₂

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — C₁₀.C₄² — C₁₀.C₄≀C₂

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — C₁₀.C₄≀C₂

Upper central

C₁ — C₂² — C₂×C₄ — C₄⋊C₄

Generators and relations for C₁₀.C₄≀C₂
G = < a,b,c,d | a¹⁰=b⁴=1, c²=d⁴=a⁵, ab=ba, ac=ca, dad^-1=a³, cbc^-1=b^-1, dbd^-1=a⁵b, dcd^-1=b^-1c >

Subgroups: 234 in 60 conjugacy classes, 22 normal (all characteristic)
C₁, C₂, C₄, C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₁₀, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, Dic₅, C₂₀, C₂×C₁₀, C₈⋊C₄, C₄².C₂, C₅⋊C₈, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₄².₂C₂², C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, C₄⋊Dic₅, C₅×C₄⋊C₄, C₂×C₅⋊C₈, C₁₀.C₄², C₄.Dic₁₀, C₁₀.C₄≀C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, F₅, C₄.₁₀D₄, C₄≀C₂, C₂×F₅, C₄².₂C₂², C₂²⋊F₅, Dic₅.D₄, D₄⋊F₅, Q₈⋊₂F₅, C₁₀.C₄≀C₂

Character table of C₁₀.C₄≀C₂

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	5	8A	8B	8C	8D	8E	8F	8G	8H	10A	10B	10C	20A	20B	20C	20D	20E	20F
size	1	1	1	1	4	8	10	10	10	10	40	4	20	20	20	20	20	20	20	20	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	1	1	1	-1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	-1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	-1	1	1	1	1	-1	1	-1	1	1	1	1	-1	-1	-1	1	1	1	-1	1	1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₅	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-i	-i	-i	i	i	i	i	-i	1	1	1	-1	1	1	-1	-1	-1	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	-i	i	i	-i	-i	i	i	-i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₇	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	i	-i	-i	i	i	-i	-i	i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₈	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	i	i	i	-i	-i	-i	-i	i	1	1	1	-1	1	1	-1	-1	-1	linear of order 4
ρ₉	2	2	2	2	-2	0	2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	-2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	0	-2	-2	2	2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	-2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	0	0	-2i	2i	0	2	-1-i	0	0	0	0	-1+i	1-i	1+i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₂	2	-2	2	-2	0	0	0	0	-2i	2i	0	2	1+i	0	0	0	0	1-i	-1+i	-1-i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₃	2	2	-2	-2	0	0	2i	-2i	0	0	0	2	0	1+i	-1-i	1-i	-1+i	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	2	-2	-2	0	0	-2i	2i	0	0	0	2	0	-1+i	1-i	-1-i	1+i	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	2	-2	-2	0	0	2i	-2i	0	0	0	2	0	-1-i	1+i	-1+i	1-i	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₆	2	-2	2	-2	0	0	0	0	2i	-2i	0	2	-1+i	0	0	0	0	-1-i	1+i	1-i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₇	2	-2	2	-2	0	0	0	0	2i	-2i	0	2	1-i	0	0	0	0	1+i	-1-i	-1+i	2	-2	-2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₈	2	2	-2	-2	0	0	-2i	2i	0	0	0	2	0	1-i	-1+i	1+i	-1-i	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₉	4	4	4	4	4	4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₂₀	4	4	4	4	4	-4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	1	-1	-1	1	1	1	orthogonal lifted from C₂×F₅
ρ₂₁	4	4	4	4	-4	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	√5	1	1	√5	-√5	-√5	orthogonal lifted from C₂²⋊F₅
ρ₂₂	4	4	4	4	-4	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-√5	1	1	-√5	√5	√5	orthogonal lifted from C₂²⋊F₅
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	4	0	0	0	0	0	0	0	0	-4	4	-4	0	0	0	0	0	0	symplectic lifted from C₄.₁₀D₄, Schur index 2
ρ₂₄	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	-√5	√5	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₅	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	√5	-√5	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₆	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	√5	-√5	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₇	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	-√5	√5	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₈	8	8	-8	-8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₂F₅
ρ₂₉	8	-8	8	-8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	symplectic lifted from D₄⋊F₅, Schur index 2

Smallest permutation representation of C₁₀.C₄≀C₂
►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 41 30 32)(2 42 21 33)(3 43 22 34)(4 44 23 35)(5 45 24 36)(6 46 25 37)(7 47 26 38)(8 48 27 39)(9 49 28 40)(10 50 29 31)(11 309 311 296)(12 310 312 297)(13 301 313 298)(14 302 314 299)(15 303 315 300)(16 304 316 291)(17 305 317 292)(18 306 318 293)(19 307 319 294)(20 308 320 295)(51 87 68 71)(52 88 69 72)(53 89 70 73)(54 90 61 74)(55 81 62 75)(56 82 63 76)(57 83 64 77)(58 84 65 78)(59 85 66 79)(60 86 67 80)(91 111 109 130)(92 112 110 121)(93 113 101 122)(94 114 102 123)(95 115 103 124)(96 116 104 125)(97 117 105 126)(98 118 106 127)(99 119 107 128)(100 120 108 129)(131 151 148 169)(132 152 149 170)(133 153 150 161)(134 154 141 162)(135 155 142 163)(136 156 143 164)(137 157 144 165)(138 158 145 166)(139 159 146 167)(140 160 147 168)(171 196 183 205)(172 197 184 206)(173 198 185 207)(174 199 186 208)(175 200 187 209)(176 191 188 210)(177 192 189 201)(178 193 190 202)(179 194 181 203)(180 195 182 204)(211 236 224 242)(212 237 225 243)(213 238 226 244)(214 239 227 245)(215 240 228 246)(216 231 229 247)(217 232 230 248)(218 233 221 249)(219 234 222 250)(220 235 223 241)(251 285 263 276)(252 286 264 277)(253 287 265 278)(254 288 266 279)(255 289 267 280)(256 290 268 271)(257 281 269 272)(258 282 270 273)(259 283 261 274)(260 284 262 275)

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

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

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

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

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

Matrix representation of C₁₀.C₄≀C₂ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	1	40
0	0	0	0	1	0
0	0	40	0	1	0
0	0	0	40	1	0

13	5	0	0	0	0
7	28	0	0	0	0
0	0	19	3	0	38
0	0	0	22	3	38
0	0	38	3	22	0
0	0	38	0	3	19

39	30	0	0	0	0
34	2	0	0	0	0
0	0	4	34	6	34
0	0	8	38	40	40
0	0	1	1	3	33
0	0	7	35	7	37

28	23	0	0	0	0
19	13	0	0	0	0
0	0	10	13	6	28
0	0	16	0	31	38
0	0	0	10	3	3
0	0	13	16	31	28

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[13,7,0,0,0,0,5,28,0,0,0,0,0,0,19,0,38,38,0,0,3,22,3,0,0,0,0,3,22,3,0,0,38,38,0,19],[39,34,0,0,0,0,30,2,0,0,0,0,0,0,4,8,1,7,0,0,34,38,1,35,0,0,6,40,3,7,0,0,34,40,33,37],[28,19,0,0,0,0,23,13,0,0,0,0,0,0,10,16,0,13,0,0,13,0,10,16,0,0,6,31,3,31,0,0,28,38,3,28] >;

C₁₀.C₄≀C₂ in GAP, Magma, Sage, TeX

C_{10}.C_4\wr C_2

% in TeX

G:=Group("C10.C4wrC2");

// GroupNames label

G:=SmallGroup(320,208);

// by ID

G=gap.SmallGroup(320,208);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,268,1571,570,136,6278,3156]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^2=d^4=a^5,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,d*c*d^-1=b^-1*c>;

// generators/relations

Export

Character table of C₁₀.C₄≀C₂ in TeX

G = C10.C4≀C2 order 320 = 26·5

2nd non-split extension by C10 of C4≀C2 acting via C4≀C2/D4=C4

G = C₁₀.C₄≀C₂ order 320 = 2⁶·5

2^nd non-split extension by C₁₀ of C₄≀C₂ acting via C₄≀C₂/D₄=C₄