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## G = C11×C3⋊C8order 264 = 23·3·11

### Direct product of C11 and C3⋊C8

Aliases: C11×C3⋊C8, C3⋊C88, C333C8, C6.C44, C66.3C4, C44.4S3, C132.6C2, C12.2C22, C22.2Dic3, C4.2(S3×C11), C2.(C11×Dic3), SmallGroup(264,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C11×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C132 — C11×C3⋊C8
 Lower central C3 — C11×C3⋊C8
 Upper central C1 — C44

Generators and relations for C11×C3⋊C8
G = < a,b,c | a11=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×C3⋊C8
Regular action on 264 points
Generators in S264
(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)
(1 238 152)(2 239 153)(3 240 154)(4 241 144)(5 242 145)(6 232 146)(7 233 147)(8 234 148)(9 235 149)(10 236 150)(11 237 151)(12 58 47)(13 59 48)(14 60 49)(15 61 50)(16 62 51)(17 63 52)(18 64 53)(19 65 54)(20 66 55)(21 56 45)(22 57 46)(23 105 34)(24 106 35)(25 107 36)(26 108 37)(27 109 38)(28 110 39)(29 100 40)(30 101 41)(31 102 42)(32 103 43)(33 104 44)(67 88 90)(68 78 91)(69 79 92)(70 80 93)(71 81 94)(72 82 95)(73 83 96)(74 84 97)(75 85 98)(76 86 99)(77 87 89)(111 243 256)(112 244 257)(113 245 258)(114 246 259)(115 247 260)(116 248 261)(117 249 262)(118 250 263)(119 251 264)(120 252 254)(121 253 255)(122 167 155)(123 168 156)(124 169 157)(125 170 158)(126 171 159)(127 172 160)(128 173 161)(129 174 162)(130 175 163)(131 176 164)(132 166 165)(133 230 217)(134 231 218)(135 221 219)(136 222 220)(137 223 210)(138 224 211)(139 225 212)(140 226 213)(141 227 214)(142 228 215)(143 229 216)(177 188 199)(178 189 200)(179 190 201)(180 191 202)(181 192 203)(182 193 204)(183 194 205)(184 195 206)(185 196 207)(186 197 208)(187 198 209)
(1 254 122 20 187 77 221 24)(2 255 123 21 177 67 222 25)(3 256 124 22 178 68 223 26)(4 257 125 12 179 69 224 27)(5 258 126 13 180 70 225 28)(6 259 127 14 181 71 226 29)(7 260 128 15 182 72 227 30)(8 261 129 16 183 73 228 31)(9 262 130 17 184 74 229 32)(10 263 131 18 185 75 230 33)(11 264 132 19 186 76 231 23)(34 237 251 166 54 197 99 218)(35 238 252 167 55 198 89 219)(36 239 253 168 45 188 90 220)(37 240 243 169 46 189 91 210)(38 241 244 170 47 190 92 211)(39 242 245 171 48 191 93 212)(40 232 246 172 49 192 94 213)(41 233 247 173 50 193 95 214)(42 234 248 174 51 194 96 215)(43 235 249 175 52 195 97 216)(44 236 250 176 53 196 98 217)(56 199 88 136 107 153 121 156)(57 200 78 137 108 154 111 157)(58 201 79 138 109 144 112 158)(59 202 80 139 110 145 113 159)(60 203 81 140 100 146 114 160)(61 204 82 141 101 147 115 161)(62 205 83 142 102 148 116 162)(63 206 84 143 103 149 117 163)(64 207 85 133 104 150 118 164)(65 208 86 134 105 151 119 165)(66 209 87 135 106 152 120 155)

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

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

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

132 conjugacy classes

 class 1 2 3 4A 4B 6 8A 8B 8C 8D 11A ··· 11J 12A 12B 22A ··· 22J 33A ··· 33J 44A ··· 44T 66A ··· 66J 88A ··· 88AN 132A ··· 132T order 1 2 3 4 4 6 8 8 8 8 11 ··· 11 12 12 22 ··· 22 33 ··· 33 44 ··· 44 66 ··· 66 88 ··· 88 132 ··· 132 size 1 1 2 1 1 2 3 3 3 3 1 ··· 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C8 C11 C22 C44 C88 S3 Dic3 C3⋊C8 S3×C11 C11×Dic3 C11×C3⋊C8 kernel C11×C3⋊C8 C132 C66 C33 C3⋊C8 C12 C6 C3 C44 C22 C11 C4 C2 C1 # reps 1 1 2 4 10 10 20 40 1 1 2 10 10 20

Matrix representation of C11×C3⋊C8 in GL3(𝔽1321) generated by

 1 0 0 0 205 0 0 0 205
,
 1 0 0 0 1320 1320 0 1 0
,
 235 0 0 0 1137 177 0 361 184
G:=sub<GL(3,GF(1321))| [1,0,0,0,205,0,0,0,205],[1,0,0,0,1320,1,0,1320,0],[235,0,0,0,1137,361,0,177,184] >;

C11×C3⋊C8 in GAP, Magma, Sage, TeX

C_{11}\times C_3\rtimes C_8
% in TeX

G:=Group("C11xC3:C8");
// GroupNames label

G:=SmallGroup(264,1);
// by ID

G=gap.SmallGroup(264,1);
# by ID

G:=PCGroup([5,-2,-11,-2,-2,-3,110,42,4404]);
// Polycyclic

G:=Group<a,b,c|a^11=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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