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## G = Dic3×C23order 276 = 22·3·23

### Direct product of C23 and Dic3

Aliases: Dic3×C23, C3⋊C92, C693C4, C6.C46, C46.2S3, C138.3C2, C2.(S3×C23), SmallGroup(276,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C23
 Chief series C1 — C3 — C6 — C138 — Dic3×C23
 Lower central C3 — Dic3×C23
 Upper central C1 — C46

Generators and relations for Dic3×C23
G = < a,b,c | a23=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C23
Regular action on 276 points
Generators in S276
(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)
(1 25 109 155 65 71)(2 26 110 156 66 72)(3 27 111 157 67 73)(4 28 112 158 68 74)(5 29 113 159 69 75)(6 30 114 160 47 76)(7 31 115 161 48 77)(8 32 93 139 49 78)(9 33 94 140 50 79)(10 34 95 141 51 80)(11 35 96 142 52 81)(12 36 97 143 53 82)(13 37 98 144 54 83)(14 38 99 145 55 84)(15 39 100 146 56 85)(16 40 101 147 57 86)(17 41 102 148 58 87)(18 42 103 149 59 88)(19 43 104 150 60 89)(20 44 105 151 61 90)(21 45 106 152 62 91)(22 46 107 153 63 92)(23 24 108 154 64 70)(116 208 236 266 173 188)(117 209 237 267 174 189)(118 210 238 268 175 190)(119 211 239 269 176 191)(120 212 240 270 177 192)(121 213 241 271 178 193)(122 214 242 272 179 194)(123 215 243 273 180 195)(124 216 244 274 181 196)(125 217 245 275 182 197)(126 218 246 276 183 198)(127 219 247 254 184 199)(128 220 248 255 162 200)(129 221 249 256 163 201)(130 222 250 257 164 202)(131 223 251 258 165 203)(132 224 252 259 166 204)(133 225 253 260 167 205)(134 226 231 261 168 206)(135 227 232 262 169 207)(136 228 233 263 170 185)(137 229 234 264 171 186)(138 230 235 265 172 187)
(1 117 155 267)(2 118 156 268)(3 119 157 269)(4 120 158 270)(5 121 159 271)(6 122 160 272)(7 123 161 273)(8 124 139 274)(9 125 140 275)(10 126 141 276)(11 127 142 254)(12 128 143 255)(13 129 144 256)(14 130 145 257)(15 131 146 258)(16 132 147 259)(17 133 148 260)(18 134 149 261)(19 135 150 262)(20 136 151 263)(21 137 152 264)(22 138 153 265)(23 116 154 266)(24 188 64 236)(25 189 65 237)(26 190 66 238)(27 191 67 239)(28 192 68 240)(29 193 69 241)(30 194 47 242)(31 195 48 243)(32 196 49 244)(33 197 50 245)(34 198 51 246)(35 199 52 247)(36 200 53 248)(37 201 54 249)(38 202 55 250)(39 203 56 251)(40 204 57 252)(41 205 58 253)(42 206 59 231)(43 207 60 232)(44 185 61 233)(45 186 62 234)(46 187 63 235)(70 208 108 173)(71 209 109 174)(72 210 110 175)(73 211 111 176)(74 212 112 177)(75 213 113 178)(76 214 114 179)(77 215 115 180)(78 216 93 181)(79 217 94 182)(80 218 95 183)(81 219 96 184)(82 220 97 162)(83 221 98 163)(84 222 99 164)(85 223 100 165)(86 224 101 166)(87 225 102 167)(88 226 103 168)(89 227 104 169)(90 228 105 170)(91 229 106 171)(92 230 107 172)

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

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

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

138 conjugacy classes

 class 1 2 3 4A 4B 6 23A ··· 23V 46A ··· 46V 69A ··· 69V 92A ··· 92AR 138A ··· 138V order 1 2 3 4 4 6 23 ··· 23 46 ··· 46 69 ··· 69 92 ··· 92 138 ··· 138 size 1 1 2 3 3 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C23 C46 C92 S3 Dic3 S3×C23 Dic3×C23 kernel Dic3×C23 C138 C69 Dic3 C6 C3 C46 C23 C2 C1 # reps 1 1 2 22 22 44 1 1 22 22

Matrix representation of Dic3×C23 in GL2(𝔽47) generated by

 37 0 0 37
,
 17 15 10 31
,
 0 5 28 0
G:=sub<GL(2,GF(47))| [37,0,0,37],[17,10,15,31],[0,28,5,0] >;

Dic3×C23 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{23}
% in TeX

G:=Group("Dic3xC23");
// GroupNames label

G:=SmallGroup(276,1);
// by ID

G=gap.SmallGroup(276,1);
# by ID

G:=PCGroup([4,-2,-23,-2,-3,184,2947]);
// Polycyclic

G:=Group<a,b,c|a^23=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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