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## G = C3×C11⋊C8order 264 = 23·3·11

### Direct product of C3 and C11⋊C8

Aliases: C3×C11⋊C8, C11⋊C24, C332C8, C22.C12, C66.2C4, C44.2C6, C132.4C2, C12.4D11, C6.2Dic11, C4.2(C3×D11), C2.(C3×Dic11), SmallGroup(264,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C3×C11⋊C8
 Chief series C1 — C11 — C22 — C44 — C132 — C3×C11⋊C8
 Lower central C11 — C3×C11⋊C8
 Upper central C1 — C12

Generators and relations for C3×C11⋊C8
G = < a,b,c | a3=b11=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

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

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

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

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

84 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 11A ··· 11E 12A 12B 12C 12D 22A ··· 22E 24A ··· 24H 33A ··· 33J 44A ··· 44J 66A ··· 66J 132A ··· 132T order 1 2 3 3 4 4 6 6 8 8 8 8 11 ··· 11 12 12 12 12 22 ··· 22 24 ··· 24 33 ··· 33 44 ··· 44 66 ··· 66 132 ··· 132 size 1 1 1 1 1 1 1 1 11 11 11 11 2 ··· 2 1 1 1 1 2 ··· 2 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C24 D11 Dic11 C3×D11 C11⋊C8 C3×Dic11 C3×C11⋊C8 kernel C3×C11⋊C8 C132 C11⋊C8 C66 C44 C33 C22 C11 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 5 5 10 10 10 20

Matrix representation of C3×C11⋊C8 in GL2(𝔽1321) generated by

 297 0 0 297
,
 1054 1320 1055 1320
,
 704 1182 304 617
G:=sub<GL(2,GF(1321))| [297,0,0,297],[1054,1055,1320,1320],[704,304,1182,617] >;

C3×C11⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC11:C8");
// GroupNames label

G:=SmallGroup(264,2);
// by ID

G=gap.SmallGroup(264,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-11,30,42,6004]);
// Polycyclic

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

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