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G = C5×C55order 275 = 52·11

Abelian group of type [5,55]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C55, SmallGroup(275,4)

Series: Derived Chief Lower central Upper central

C1 — C5×C55
C1C11C55 — C5×C55
C1 — C5×C55
C1 — C5×C55

Generators and relations for C5×C55
 G = < a,b | a5=b55=1, ab=ba >


Smallest permutation representation of C5×C55
Regular action on 275 points
Generators in S275
(1 242 205 163 67)(2 243 206 164 68)(3 244 207 165 69)(4 245 208 111 70)(5 246 209 112 71)(6 247 210 113 72)(7 248 211 114 73)(8 249 212 115 74)(9 250 213 116 75)(10 251 214 117 76)(11 252 215 118 77)(12 253 216 119 78)(13 254 217 120 79)(14 255 218 121 80)(15 256 219 122 81)(16 257 220 123 82)(17 258 166 124 83)(18 259 167 125 84)(19 260 168 126 85)(20 261 169 127 86)(21 262 170 128 87)(22 263 171 129 88)(23 264 172 130 89)(24 265 173 131 90)(25 266 174 132 91)(26 267 175 133 92)(27 268 176 134 93)(28 269 177 135 94)(29 270 178 136 95)(30 271 179 137 96)(31 272 180 138 97)(32 273 181 139 98)(33 274 182 140 99)(34 275 183 141 100)(35 221 184 142 101)(36 222 185 143 102)(37 223 186 144 103)(38 224 187 145 104)(39 225 188 146 105)(40 226 189 147 106)(41 227 190 148 107)(42 228 191 149 108)(43 229 192 150 109)(44 230 193 151 110)(45 231 194 152 56)(46 232 195 153 57)(47 233 196 154 58)(48 234 197 155 59)(49 235 198 156 60)(50 236 199 157 61)(51 237 200 158 62)(52 238 201 159 63)(53 239 202 160 64)(54 240 203 161 65)(55 241 204 162 66)
(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)| (1,242,205,163,67)(2,243,206,164,68)(3,244,207,165,69)(4,245,208,111,70)(5,246,209,112,71)(6,247,210,113,72)(7,248,211,114,73)(8,249,212,115,74)(9,250,213,116,75)(10,251,214,117,76)(11,252,215,118,77)(12,253,216,119,78)(13,254,217,120,79)(14,255,218,121,80)(15,256,219,122,81)(16,257,220,123,82)(17,258,166,124,83)(18,259,167,125,84)(19,260,168,126,85)(20,261,169,127,86)(21,262,170,128,87)(22,263,171,129,88)(23,264,172,130,89)(24,265,173,131,90)(25,266,174,132,91)(26,267,175,133,92)(27,268,176,134,93)(28,269,177,135,94)(29,270,178,136,95)(30,271,179,137,96)(31,272,180,138,97)(32,273,181,139,98)(33,274,182,140,99)(34,275,183,141,100)(35,221,184,142,101)(36,222,185,143,102)(37,223,186,144,103)(38,224,187,145,104)(39,225,188,146,105)(40,226,189,147,106)(41,227,190,148,107)(42,228,191,149,108)(43,229,192,150,109)(44,230,193,151,110)(45,231,194,152,56)(46,232,195,153,57)(47,233,196,154,58)(48,234,197,155,59)(49,235,198,156,60)(50,236,199,157,61)(51,237,200,158,62)(52,238,201,159,63)(53,239,202,160,64)(54,240,203,161,65)(55,241,204,162,66), (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( (1,242,205,163,67)(2,243,206,164,68)(3,244,207,165,69)(4,245,208,111,70)(5,246,209,112,71)(6,247,210,113,72)(7,248,211,114,73)(8,249,212,115,74)(9,250,213,116,75)(10,251,214,117,76)(11,252,215,118,77)(12,253,216,119,78)(13,254,217,120,79)(14,255,218,121,80)(15,256,219,122,81)(16,257,220,123,82)(17,258,166,124,83)(18,259,167,125,84)(19,260,168,126,85)(20,261,169,127,86)(21,262,170,128,87)(22,263,171,129,88)(23,264,172,130,89)(24,265,173,131,90)(25,266,174,132,91)(26,267,175,133,92)(27,268,176,134,93)(28,269,177,135,94)(29,270,178,136,95)(30,271,179,137,96)(31,272,180,138,97)(32,273,181,139,98)(33,274,182,140,99)(34,275,183,141,100)(35,221,184,142,101)(36,222,185,143,102)(37,223,186,144,103)(38,224,187,145,104)(39,225,188,146,105)(40,226,189,147,106)(41,227,190,148,107)(42,228,191,149,108)(43,229,192,150,109)(44,230,193,151,110)(45,231,194,152,56)(46,232,195,153,57)(47,233,196,154,58)(48,234,197,155,59)(49,235,198,156,60)(50,236,199,157,61)(51,237,200,158,62)(52,238,201,159,63)(53,239,202,160,64)(54,240,203,161,65)(55,241,204,162,66), (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([(1,242,205,163,67),(2,243,206,164,68),(3,244,207,165,69),(4,245,208,111,70),(5,246,209,112,71),(6,247,210,113,72),(7,248,211,114,73),(8,249,212,115,74),(9,250,213,116,75),(10,251,214,117,76),(11,252,215,118,77),(12,253,216,119,78),(13,254,217,120,79),(14,255,218,121,80),(15,256,219,122,81),(16,257,220,123,82),(17,258,166,124,83),(18,259,167,125,84),(19,260,168,126,85),(20,261,169,127,86),(21,262,170,128,87),(22,263,171,129,88),(23,264,172,130,89),(24,265,173,131,90),(25,266,174,132,91),(26,267,175,133,92),(27,268,176,134,93),(28,269,177,135,94),(29,270,178,136,95),(30,271,179,137,96),(31,272,180,138,97),(32,273,181,139,98),(33,274,182,140,99),(34,275,183,141,100),(35,221,184,142,101),(36,222,185,143,102),(37,223,186,144,103),(38,224,187,145,104),(39,225,188,146,105),(40,226,189,147,106),(41,227,190,148,107),(42,228,191,149,108),(43,229,192,150,109),(44,230,193,151,110),(45,231,194,152,56),(46,232,195,153,57),(47,233,196,154,58),(48,234,197,155,59),(49,235,198,156,60),(50,236,199,157,61),(51,237,200,158,62),(52,238,201,159,63),(53,239,202,160,64),(54,240,203,161,65),(55,241,204,162,66)], [(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)])

275 conjugacy classes

class 1 5A···5X11A···11J55A···55IF
order15···511···1155···55
size11···11···11···1

275 irreducible representations

dim1111
type+
imageC1C5C11C55
kernelC5×C55C55C52C5
# reps12410240

Matrix representation of C5×C55 in GL2(𝔽331) generated by

1500
0124
,
1050
0144
G:=sub<GL(2,GF(331))| [150,0,0,124],[105,0,0,144] >;

C5×C55 in GAP, Magma, Sage, TeX

C_5\times C_{55}
% in TeX

G:=Group("C5xC55");
// GroupNames label

G:=SmallGroup(275,4);
// by ID

G=gap.SmallGroup(275,4);
# by ID

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

G:=Group<a,b|a^5=b^55=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C55 in TeX

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