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G = Q8×C37order 296 = 23·37

Direct product of C37 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C37, C4.C74, C148.3C2, C74.7C22, C2.2(C2×C74), SmallGroup(296,11)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C37
C1C2C74C148 — Q8×C37
C1C2 — Q8×C37
C1C74 — Q8×C37

Generators and relations for Q8×C37
 G = < a,b,c | a37=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C37
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)
(1 124 62 86)(2 125 63 87)(3 126 64 88)(4 127 65 89)(5 128 66 90)(6 129 67 91)(7 130 68 92)(8 131 69 93)(9 132 70 94)(10 133 71 95)(11 134 72 96)(12 135 73 97)(13 136 74 98)(14 137 38 99)(15 138 39 100)(16 139 40 101)(17 140 41 102)(18 141 42 103)(19 142 43 104)(20 143 44 105)(21 144 45 106)(22 145 46 107)(23 146 47 108)(24 147 48 109)(25 148 49 110)(26 112 50 111)(27 113 51 75)(28 114 52 76)(29 115 53 77)(30 116 54 78)(31 117 55 79)(32 118 56 80)(33 119 57 81)(34 120 58 82)(35 121 59 83)(36 122 60 84)(37 123 61 85)(149 216 254 277)(150 217 255 278)(151 218 256 279)(152 219 257 280)(153 220 258 281)(154 221 259 282)(155 222 223 283)(156 186 224 284)(157 187 225 285)(158 188 226 286)(159 189 227 287)(160 190 228 288)(161 191 229 289)(162 192 230 290)(163 193 231 291)(164 194 232 292)(165 195 233 293)(166 196 234 294)(167 197 235 295)(168 198 236 296)(169 199 237 260)(170 200 238 261)(171 201 239 262)(172 202 240 263)(173 203 241 264)(174 204 242 265)(175 205 243 266)(176 206 244 267)(177 207 245 268)(178 208 246 269)(179 209 247 270)(180 210 248 271)(181 211 249 272)(182 212 250 273)(183 213 251 274)(184 214 252 275)(185 215 253 276)
(1 166 62 234)(2 167 63 235)(3 168 64 236)(4 169 65 237)(5 170 66 238)(6 171 67 239)(7 172 68 240)(8 173 69 241)(9 174 70 242)(10 175 71 243)(11 176 72 244)(12 177 73 245)(13 178 74 246)(14 179 38 247)(15 180 39 248)(16 181 40 249)(17 182 41 250)(18 183 42 251)(19 184 43 252)(20 185 44 253)(21 149 45 254)(22 150 46 255)(23 151 47 256)(24 152 48 257)(25 153 49 258)(26 154 50 259)(27 155 51 223)(28 156 52 224)(29 157 53 225)(30 158 54 226)(31 159 55 227)(32 160 56 228)(33 161 57 229)(34 162 58 230)(35 163 59 231)(36 164 60 232)(37 165 61 233)(75 222 113 283)(76 186 114 284)(77 187 115 285)(78 188 116 286)(79 189 117 287)(80 190 118 288)(81 191 119 289)(82 192 120 290)(83 193 121 291)(84 194 122 292)(85 195 123 293)(86 196 124 294)(87 197 125 295)(88 198 126 296)(89 199 127 260)(90 200 128 261)(91 201 129 262)(92 202 130 263)(93 203 131 264)(94 204 132 265)(95 205 133 266)(96 206 134 267)(97 207 135 268)(98 208 136 269)(99 209 137 270)(100 210 138 271)(101 211 139 272)(102 212 140 273)(103 213 141 274)(104 214 142 275)(105 215 143 276)(106 216 144 277)(107 217 145 278)(108 218 146 279)(109 219 147 280)(110 220 148 281)(111 221 112 282)

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

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

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

185 conjugacy classes

class 1  2 4A4B4C37A···37AJ74A···74AJ148A···148DD
order1244437···3774···74148···148
size112221···11···12···2

185 irreducible representations

dim111122
type++-
imageC1C2C37C74Q8Q8×C37
kernelQ8×C37C148Q8C4C37C1
# reps1336108136

Matrix representation of Q8×C37 in GL2(𝔽149) generated by

190
019
,
14147
24135
,
5468
8295
G:=sub<GL(2,GF(149))| [19,0,0,19],[14,24,147,135],[54,82,68,95] >;

Q8×C37 in GAP, Magma, Sage, TeX

Q_8\times C_{37}
% in TeX

G:=Group("Q8xC37");
// GroupNames label

G:=SmallGroup(296,11);
// by ID

G=gap.SmallGroup(296,11);
# by ID

G:=PCGroup([4,-2,-2,-37,-2,592,1201,597]);
// Polycyclic

G:=Group<a,b,c|a^37=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C37 in TeX

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