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G = C3×Dic23order 276 = 22·3·23

Direct product of C3 and Dic23

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

Aliases: C3×Dic23, C23⋊C12, C692C4, C46.C6, C6.2D23, C138.2C2, C2.(C3×D23), SmallGroup(276,2)

Series: Derived Chief Lower central Upper central

C1C23 — C3×Dic23
C1C23C46C138 — C3×Dic23
C23 — C3×Dic23
C1C6

Generators and relations for C3×Dic23
 G = < a,b,c | a3=b46=1, c2=b23, ab=ba, ac=ca, cbc-1=b-1 >

23C4
23C12

Smallest permutation representation of C3×Dic23
Regular action on 276 points
Generators in S276
(1 122 58)(2 123 59)(3 124 60)(4 125 61)(5 126 62)(6 127 63)(7 128 64)(8 129 65)(9 130 66)(10 131 67)(11 132 68)(12 133 69)(13 134 70)(14 135 71)(15 136 72)(16 137 73)(17 138 74)(18 93 75)(19 94 76)(20 95 77)(21 96 78)(22 97 79)(23 98 80)(24 99 81)(25 100 82)(26 101 83)(27 102 84)(28 103 85)(29 104 86)(30 105 87)(31 106 88)(32 107 89)(33 108 90)(34 109 91)(35 110 92)(36 111 47)(37 112 48)(38 113 49)(39 114 50)(40 115 51)(41 116 52)(42 117 53)(43 118 54)(44 119 55)(45 120 56)(46 121 57)(139 231 185)(140 232 186)(141 233 187)(142 234 188)(143 235 189)(144 236 190)(145 237 191)(146 238 192)(147 239 193)(148 240 194)(149 241 195)(150 242 196)(151 243 197)(152 244 198)(153 245 199)(154 246 200)(155 247 201)(156 248 202)(157 249 203)(158 250 204)(159 251 205)(160 252 206)(161 253 207)(162 254 208)(163 255 209)(164 256 210)(165 257 211)(166 258 212)(167 259 213)(168 260 214)(169 261 215)(170 262 216)(171 263 217)(172 264 218)(173 265 219)(174 266 220)(175 267 221)(176 268 222)(177 269 223)(178 270 224)(179 271 225)(180 272 226)(181 273 227)(182 274 228)(183 275 229)(184 276 230)
(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)
(1 162 24 139)(2 161 25 184)(3 160 26 183)(4 159 27 182)(5 158 28 181)(6 157 29 180)(7 156 30 179)(8 155 31 178)(9 154 32 177)(10 153 33 176)(11 152 34 175)(12 151 35 174)(13 150 36 173)(14 149 37 172)(15 148 38 171)(16 147 39 170)(17 146 40 169)(18 145 41 168)(19 144 42 167)(20 143 43 166)(21 142 44 165)(22 141 45 164)(23 140 46 163)(47 219 70 196)(48 218 71 195)(49 217 72 194)(50 216 73 193)(51 215 74 192)(52 214 75 191)(53 213 76 190)(54 212 77 189)(55 211 78 188)(56 210 79 187)(57 209 80 186)(58 208 81 185)(59 207 82 230)(60 206 83 229)(61 205 84 228)(62 204 85 227)(63 203 86 226)(64 202 87 225)(65 201 88 224)(66 200 89 223)(67 199 90 222)(68 198 91 221)(69 197 92 220)(93 237 116 260)(94 236 117 259)(95 235 118 258)(96 234 119 257)(97 233 120 256)(98 232 121 255)(99 231 122 254)(100 276 123 253)(101 275 124 252)(102 274 125 251)(103 273 126 250)(104 272 127 249)(105 271 128 248)(106 270 129 247)(107 269 130 246)(108 268 131 245)(109 267 132 244)(110 266 133 243)(111 265 134 242)(112 264 135 241)(113 263 136 240)(114 262 137 239)(115 261 138 238)

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

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

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

78 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D23A···23K46A···46K69A···69V138A···138V
order123344661212121223···2346···4669···69138···138
size1111232311232323232···22···22···22···2

78 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D23Dic23C3×D23C3×Dic23
kernelC3×Dic23C138Dic23C69C46C23C6C3C2C1
# reps11222411112222

Matrix representation of C3×Dic23 in GL3(𝔽277) generated by

100
01160
00116
,
27600
001
027665
,
6000
01269
018265
G:=sub<GL(3,GF(277))| [1,0,0,0,116,0,0,0,116],[276,0,0,0,0,276,0,1,65],[60,0,0,0,12,18,0,69,265] >;

C3×Dic23 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{23}
% in TeX

G:=Group("C3xDic23");
// GroupNames label

G:=SmallGroup(276,2);
// by ID

G=gap.SmallGroup(276,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-23,24,4227]);
// Polycyclic

G:=Group<a,b,c|a^3=b^46=1,c^2=b^23,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic23 in TeX

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