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G = C4×C76order 304 = 24·19

Abelian group of type [4,76]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C76, SmallGroup(304,19)

Series: Derived Chief Lower central Upper central

C1 — C4×C76
C1C2C22C2×C38C2×C76 — C4×C76
C1 — C4×C76
C1 — C4×C76

Generators and relations for C4×C76
 G = < a,b | a4=b76=1, ab=ba >


Smallest permutation representation of C4×C76
Regular action on 304 points
Generators in S304
(1 106 278 227)(2 107 279 228)(3 108 280 153)(4 109 281 154)(5 110 282 155)(6 111 283 156)(7 112 284 157)(8 113 285 158)(9 114 286 159)(10 115 287 160)(11 116 288 161)(12 117 289 162)(13 118 290 163)(14 119 291 164)(15 120 292 165)(16 121 293 166)(17 122 294 167)(18 123 295 168)(19 124 296 169)(20 125 297 170)(21 126 298 171)(22 127 299 172)(23 128 300 173)(24 129 301 174)(25 130 302 175)(26 131 303 176)(27 132 304 177)(28 133 229 178)(29 134 230 179)(30 135 231 180)(31 136 232 181)(32 137 233 182)(33 138 234 183)(34 139 235 184)(35 140 236 185)(36 141 237 186)(37 142 238 187)(38 143 239 188)(39 144 240 189)(40 145 241 190)(41 146 242 191)(42 147 243 192)(43 148 244 193)(44 149 245 194)(45 150 246 195)(46 151 247 196)(47 152 248 197)(48 77 249 198)(49 78 250 199)(50 79 251 200)(51 80 252 201)(52 81 253 202)(53 82 254 203)(54 83 255 204)(55 84 256 205)(56 85 257 206)(57 86 258 207)(58 87 259 208)(59 88 260 209)(60 89 261 210)(61 90 262 211)(62 91 263 212)(63 92 264 213)(64 93 265 214)(65 94 266 215)(66 95 267 216)(67 96 268 217)(68 97 269 218)(69 98 270 219)(70 99 271 220)(71 100 272 221)(72 101 273 222)(73 102 274 223)(74 103 275 224)(75 104 276 225)(76 105 277 226)
(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)

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

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

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

304 conjugacy classes

class 1 2A2B2C4A···4L19A···19R38A···38BB76A···76HH
order12224···419···1938···3876···76
size11111···11···11···11···1

304 irreducible representations

dim111111
type++
imageC1C2C4C19C38C76
kernelC4×C76C2×C76C76C42C2×C4C4
# reps13121854216

Matrix representation of C4×C76 in GL2(𝔽229) generated by

10
0107
,
930
0141
G:=sub<GL(2,GF(229))| [1,0,0,107],[93,0,0,141] >;

C4×C76 in GAP, Magma, Sage, TeX

C_4\times C_{76}
% in TeX

G:=Group("C4xC76");
// GroupNames label

G:=SmallGroup(304,19);
// by ID

G=gap.SmallGroup(304,19);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-2,380,766]);
// Polycyclic

G:=Group<a,b|a^4=b^76=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C76 in TeX

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