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G = C372C8order 296 = 23·37

The semidirect product of C37 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C372C8, C74.2C4, C4.2D37, C2.Dic37, C148.2C2, SmallGroup(296,1)

Series: Derived Chief Lower central Upper central

C1C37 — C372C8
C1C37C74C148 — C372C8
C37 — C372C8
C1C4

Generators and relations for C372C8
 G = < a,b | a37=b8=1, bab-1=a-1 >

37C8

Smallest permutation representation of C372C8
Regular action on 296 points
Generators in S296
(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)
(1 260 135 186 57 223 79 149)(2 296 136 222 58 259 80 185)(3 295 137 221 59 258 81 184)(4 294 138 220 60 257 82 183)(5 293 139 219 61 256 83 182)(6 292 140 218 62 255 84 181)(7 291 141 217 63 254 85 180)(8 290 142 216 64 253 86 179)(9 289 143 215 65 252 87 178)(10 288 144 214 66 251 88 177)(11 287 145 213 67 250 89 176)(12 286 146 212 68 249 90 175)(13 285 147 211 69 248 91 174)(14 284 148 210 70 247 92 173)(15 283 112 209 71 246 93 172)(16 282 113 208 72 245 94 171)(17 281 114 207 73 244 95 170)(18 280 115 206 74 243 96 169)(19 279 116 205 38 242 97 168)(20 278 117 204 39 241 98 167)(21 277 118 203 40 240 99 166)(22 276 119 202 41 239 100 165)(23 275 120 201 42 238 101 164)(24 274 121 200 43 237 102 163)(25 273 122 199 44 236 103 162)(26 272 123 198 45 235 104 161)(27 271 124 197 46 234 105 160)(28 270 125 196 47 233 106 159)(29 269 126 195 48 232 107 158)(30 268 127 194 49 231 108 157)(31 267 128 193 50 230 109 156)(32 266 129 192 51 229 110 155)(33 265 130 191 52 228 111 154)(34 264 131 190 53 227 75 153)(35 263 132 189 54 226 76 152)(36 262 133 188 55 225 77 151)(37 261 134 187 56 224 78 150)

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

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

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

80 conjugacy classes

class 1  2 4A4B8A8B8C8D37A···37R74A···74R148A···148AJ
order1244888837···3774···74148···148
size1111373737372···22···22···2

80 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D37Dic37C372C8
kernelC372C8C148C74C37C4C2C1
# reps1124181836

Matrix representation of C372C8 in GL2(𝔽593) generated by

01
592561
,
352426
429241
G:=sub<GL(2,GF(593))| [0,592,1,561],[352,429,426,241] >;

C372C8 in GAP, Magma, Sage, TeX

C_{37}\rtimes_2C_8
% in TeX

G:=Group("C37:2C8");
// GroupNames label

G:=SmallGroup(296,1);
// by ID

G=gap.SmallGroup(296,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,8,21,4611]);
// Polycyclic

G:=Group<a,b|a^37=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C372C8 in TeX

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