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G = C11⋊C25order 275 = 52·11

The semidirect product of C11 and C25 acting via C25/C5=C5

metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C11⋊C25, C55.C5, C5.(C11⋊C5), SmallGroup(275,1)

Series: Derived Chief Lower central Upper central

C1C11 — C11⋊C25
C1C11C55 — C11⋊C25
C11 — C11⋊C25
C1C5

Generators and relations for C11⋊C25
 G = < a,b | a11=b25=1, bab-1=a3 >

11C25

Smallest permutation representation of C11⋊C25
Regular action on 275 points
Generators in S275
(1 102 161 130 33 269 51 89 185 219 248)(2 131 52 220 103 34 90 249 162 270 186)(3 221 91 271 132 104 250 187 53 35 163)(4 272 226 36 222 133 188 164 92 105 54)(5 37 189 106 273 223 165 55 227 134 93)(6 107 166 135 38 274 56 94 190 224 228)(7 136 57 225 108 39 95 229 167 275 191)(8 201 96 251 137 109 230 192 58 40 168)(9 252 231 41 202 138 193 169 97 110 59)(10 42 194 111 253 203 170 60 232 139 98)(11 112 171 140 43 254 61 99 195 204 233)(12 141 62 205 113 44 100 234 172 255 196)(13 206 76 256 142 114 235 197 63 45 173)(14 257 236 46 207 143 198 174 77 115 64)(15 47 199 116 258 208 175 65 237 144 78)(16 117 151 145 48 259 66 79 200 209 238)(17 146 67 210 118 49 80 239 152 260 176)(18 211 81 261 147 119 240 177 68 50 153)(19 262 241 26 212 148 178 154 82 120 69)(20 27 179 121 263 213 155 70 242 149 83)(21 122 156 150 28 264 71 84 180 214 243)(22 126 72 215 123 29 85 244 157 265 181)(23 216 86 266 127 124 245 182 73 30 158)(24 267 246 31 217 128 183 159 87 125 74)(25 32 184 101 268 218 160 75 247 129 88)
(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)

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

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

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

35 conjugacy classes

class 1 5A5B5C5D11A11B25A···25T55A···55H
order15555111125···2555···55
size111115511···115···5

35 irreducible representations

dim11155
type+
imageC1C5C25C11⋊C5C11⋊C25
kernelC11⋊C25C55C11C5C1
# reps142028

Matrix representation of C11⋊C25 in GL5(𝔽3301)

33001000
33000100
33000010
33000001
320110132993203100
,
317430413311502232
3279119831942877413
26032787401387169
18191402139311091199
162722395343721081

G:=sub<GL(5,GF(3301))| [3300,3300,3300,3300,3201,1,0,0,0,101,0,1,0,0,3299,0,0,1,0,3203,0,0,0,1,100],[3174,3279,2603,1819,1627,304,1198,2787,1402,2239,133,3194,40,1393,534,1150,2877,1387,1109,372,2232,413,169,1199,1081] >;

C11⋊C25 in GAP, Magma, Sage, TeX

C_{11}\rtimes C_{25}
% in TeX

G:=Group("C11:C25");
// GroupNames label

G:=SmallGroup(275,1);
// by ID

G=gap.SmallGroup(275,1);
# by ID

G:=PCGroup([3,-5,-5,-11,15,902]);
// Polycyclic

G:=Group<a,b|a^11=b^25=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C11⋊C25 in TeX

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