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G = C3×Dic22order 264 = 23·3·11

Direct product of C3 and Dic22

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

Aliases: C3×Dic22, C333Q8, C44.1C6, C132.3C2, C6.13D22, C12.3D11, Dic11.C6, C66.13C22, C11⋊(C3×Q8), C4.(C3×D11), C22.1(C2×C6), C2.3(C6×D11), (C3×Dic11).2C2, SmallGroup(264,13)

Series: Derived Chief Lower central Upper central

C1C22 — C3×Dic22
C1C11C22C66C3×Dic11 — C3×Dic22
C11C22 — C3×Dic22
C1C6C12

Generators and relations for C3×Dic22
 G = < a,b,c | a3=b44=1, c2=b22, ab=ba, ac=ca, cbc-1=b-1 >

11C4
11C4
11Q8
11C12
11C12
11C3×Q8

Smallest permutation representation of C3×Dic22
Regular action on 264 points
Generators in S264
(1 206 243)(2 207 244)(3 208 245)(4 209 246)(5 210 247)(6 211 248)(7 212 249)(8 213 250)(9 214 251)(10 215 252)(11 216 253)(12 217 254)(13 218 255)(14 219 256)(15 220 257)(16 177 258)(17 178 259)(18 179 260)(19 180 261)(20 181 262)(21 182 263)(22 183 264)(23 184 221)(24 185 222)(25 186 223)(26 187 224)(27 188 225)(28 189 226)(29 190 227)(30 191 228)(31 192 229)(32 193 230)(33 194 231)(34 195 232)(35 196 233)(36 197 234)(37 198 235)(38 199 236)(39 200 237)(40 201 238)(41 202 239)(42 203 240)(43 204 241)(44 205 242)(45 101 155)(46 102 156)(47 103 157)(48 104 158)(49 105 159)(50 106 160)(51 107 161)(52 108 162)(53 109 163)(54 110 164)(55 111 165)(56 112 166)(57 113 167)(58 114 168)(59 115 169)(60 116 170)(61 117 171)(62 118 172)(63 119 173)(64 120 174)(65 121 175)(66 122 176)(67 123 133)(68 124 134)(69 125 135)(70 126 136)(71 127 137)(72 128 138)(73 129 139)(74 130 140)(75 131 141)(76 132 142)(77 89 143)(78 90 144)(79 91 145)(80 92 146)(81 93 147)(82 94 148)(83 95 149)(84 96 150)(85 97 151)(86 98 152)(87 99 153)(88 100 154)
(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)
(1 96 23 118)(2 95 24 117)(3 94 25 116)(4 93 26 115)(5 92 27 114)(6 91 28 113)(7 90 29 112)(8 89 30 111)(9 132 31 110)(10 131 32 109)(11 130 33 108)(12 129 34 107)(13 128 35 106)(14 127 36 105)(15 126 37 104)(16 125 38 103)(17 124 39 102)(18 123 40 101)(19 122 41 100)(20 121 42 99)(21 120 43 98)(22 119 44 97)(45 260 67 238)(46 259 68 237)(47 258 69 236)(48 257 70 235)(49 256 71 234)(50 255 72 233)(51 254 73 232)(52 253 74 231)(53 252 75 230)(54 251 76 229)(55 250 77 228)(56 249 78 227)(57 248 79 226)(58 247 80 225)(59 246 81 224)(60 245 82 223)(61 244 83 222)(62 243 84 221)(63 242 85 264)(64 241 86 263)(65 240 87 262)(66 239 88 261)(133 201 155 179)(134 200 156 178)(135 199 157 177)(136 198 158 220)(137 197 159 219)(138 196 160 218)(139 195 161 217)(140 194 162 216)(141 193 163 215)(142 192 164 214)(143 191 165 213)(144 190 166 212)(145 189 167 211)(146 188 168 210)(147 187 169 209)(148 186 170 208)(149 185 171 207)(150 184 172 206)(151 183 173 205)(152 182 174 204)(153 181 175 203)(154 180 176 202)

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

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

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

75 conjugacy classes

class 1  2 3A3B4A4B4C6A6B11A···11E12A12B12C12D12E12F22A···22E33A···33J44A···44J66A···66J132A···132T
order12334446611···1112121212121222···2233···3344···4466···66132···132
size111122222112···222222222222···22···22···22···22···2

75 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8D11C3×Q8D22C3×D11Dic22C6×D11C3×Dic22
kernelC3×Dic22C3×Dic11C132Dic22Dic11C44C33C12C11C6C4C3C2C1
# reps121242152510101020

Matrix representation of C3×Dic22 in GL2(𝔽43) generated by

360
036
,
3020
3240
,
2233
2721
G:=sub<GL(2,GF(43))| [36,0,0,36],[30,32,20,40],[22,27,33,21] >;

C3×Dic22 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{22}
% in TeX

G:=Group("C3xDic22");
// GroupNames label

G:=SmallGroup(264,13);
// by ID

G=gap.SmallGroup(264,13);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-11,60,141,66,6004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^44=1,c^2=b^22,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic22 in TeX

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