direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic11, D6.D11, C6.2D22, C22.2D6, Dic33⋊3C2, C66.2C22, (S3×C11)⋊C4, C11⋊3(C4×S3), C33⋊2(C2×C4), (S3×C22).C2, C2.2(S3×D11), C3⋊1(C2×Dic11), (C3×Dic11)⋊1C2, SmallGroup(264,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — S3×Dic11 |
Generators and relations for S3×Dic11
G = < a,b,c,d | a3=b2=c22=1, d2=c11, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×Dic11
►On 132 points
Generators in S132
(1 99 116)(2 100 117)(3 101 118)(4 102 119)(5 103 120)(6 104 121)(7 105 122)(8 106 123)(9 107 124)(10 108 125)(11 109 126)(12 110 127)(13 89 128)(14 90 129)(15 91 130)(16 92 131)(17 93 132)(18 94 111)(19 95 112)(20 96 113)(21 97 114)(22 98 115)(23 60 86)(24 61 87)(25 62 88)(26 63 67)(27 64 68)(28 65 69)(29 66 70)(30 45 71)(31 46 72)(32 47 73)(33 48 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 81)(41 56 82)(42 57 83)(43 58 84)(44 59 85)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)
(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)
(1 63 12 52)(2 62 13 51)(3 61 14 50)(4 60 15 49)(5 59 16 48)(6 58 17 47)(7 57 18 46)(8 56 19 45)(9 55 20 66)(10 54 21 65)(11 53 22 64)(23 130 34 119)(24 129 35 118)(25 128 36 117)(26 127 37 116)(27 126 38 115)(28 125 39 114)(29 124 40 113)(30 123 41 112)(31 122 42 111)(32 121 43 132)(33 120 44 131)(67 110 78 99)(68 109 79 98)(69 108 80 97)(70 107 81 96)(71 106 82 95)(72 105 83 94)(73 104 84 93)(74 103 85 92)(75 102 86 91)(76 101 87 90)(77 100 88 89)
G:=sub<Sym(132)| (1,99,116)(2,100,117)(3,101,118)(4,102,119)(5,103,120)(6,104,121)(7,105,122)(8,106,123)(9,107,124)(10,108,125)(11,109,126)(12,110,127)(13,89,128)(14,90,129)(15,91,130)(16,92,131)(17,93,132)(18,94,111)(19,95,112)(20,96,113)(21,97,114)(22,98,115)(23,60,86)(24,61,87)(25,62,88)(26,63,67)(27,64,68)(28,65,69)(29,66,70)(30,45,71)(31,46,72)(32,47,73)(33,48,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,81)(41,56,82)(42,57,83)(43,58,84)(44,59,85), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116), (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), (1,63,12,52)(2,62,13,51)(3,61,14,50)(4,60,15,49)(5,59,16,48)(6,58,17,47)(7,57,18,46)(8,56,19,45)(9,55,20,66)(10,54,21,65)(11,53,22,64)(23,130,34,119)(24,129,35,118)(25,128,36,117)(26,127,37,116)(27,126,38,115)(28,125,39,114)(29,124,40,113)(30,123,41,112)(31,122,42,111)(32,121,43,132)(33,120,44,131)(67,110,78,99)(68,109,79,98)(69,108,80,97)(70,107,81,96)(71,106,82,95)(72,105,83,94)(73,104,84,93)(74,103,85,92)(75,102,86,91)(76,101,87,90)(77,100,88,89)>;
G:=Group( (1,99,116)(2,100,117)(3,101,118)(4,102,119)(5,103,120)(6,104,121)(7,105,122)(8,106,123)(9,107,124)(10,108,125)(11,109,126)(12,110,127)(13,89,128)(14,90,129)(15,91,130)(16,92,131)(17,93,132)(18,94,111)(19,95,112)(20,96,113)(21,97,114)(22,98,115)(23,60,86)(24,61,87)(25,62,88)(26,63,67)(27,64,68)(28,65,69)(29,66,70)(30,45,71)(31,46,72)(32,47,73)(33,48,74)(34,49,75)(35,50,76)(36,51,77)(37,52,78)(38,53,79)(39,54,80)(40,55,81)(41,56,82)(42,57,83)(43,58,84)(44,59,85), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116), (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), (1,63,12,52)(2,62,13,51)(3,61,14,50)(4,60,15,49)(5,59,16,48)(6,58,17,47)(7,57,18,46)(8,56,19,45)(9,55,20,66)(10,54,21,65)(11,53,22,64)(23,130,34,119)(24,129,35,118)(25,128,36,117)(26,127,37,116)(27,126,38,115)(28,125,39,114)(29,124,40,113)(30,123,41,112)(31,122,42,111)(32,121,43,132)(33,120,44,131)(67,110,78,99)(68,109,79,98)(69,108,80,97)(70,107,81,96)(71,106,82,95)(72,105,83,94)(73,104,84,93)(74,103,85,92)(75,102,86,91)(76,101,87,90)(77,100,88,89) );
G=PermutationGroup([[(1,99,116),(2,100,117),(3,101,118),(4,102,119),(5,103,120),(6,104,121),(7,105,122),(8,106,123),(9,107,124),(10,108,125),(11,109,126),(12,110,127),(13,89,128),(14,90,129),(15,91,130),(16,92,131),(17,93,132),(18,94,111),(19,95,112),(20,96,113),(21,97,114),(22,98,115),(23,60,86),(24,61,87),(25,62,88),(26,63,67),(27,64,68),(28,65,69),(29,66,70),(30,45,71),(31,46,72),(32,47,73),(33,48,74),(34,49,75),(35,50,76),(36,51,77),(37,52,78),(38,53,79),(39,54,80),(40,55,81),(41,56,82),(42,57,83),(43,58,84),(44,59,85)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,111),(106,112),(107,113),(108,114),(109,115),(110,116)], [(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)], [(1,63,12,52),(2,62,13,51),(3,61,14,50),(4,60,15,49),(5,59,16,48),(6,58,17,47),(7,57,18,46),(8,56,19,45),(9,55,20,66),(10,54,21,65),(11,53,22,64),(23,130,34,119),(24,129,35,118),(25,128,36,117),(26,127,37,116),(27,126,38,115),(28,125,39,114),(29,124,40,113),(30,123,41,112),(31,122,42,111),(32,121,43,132),(33,120,44,131),(67,110,78,99),(68,109,79,98),(69,108,80,97),(70,107,81,96),(71,106,82,95),(72,105,83,94),(73,104,84,93),(74,103,85,92),(75,102,86,91),(76,101,87,90),(77,100,88,89)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 22F | ··· | 22O | 33A | ··· | 33E | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 3 | 3 | 2 | 11 | 11 | 33 | 33 | 2 | 2 | ··· | 2 | 22 | 22 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D11 | C4×S3 | Dic11 | D22 | S3×D11 | S3×Dic11 |
| kernel | S3×Dic11 | C3×Dic11 | Dic33 | S3×C22 | S3×C11 | Dic11 | C22 | D6 | C11 | S3 | C6 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 2 | 10 | 5 | 5 | 5 |
Matrix representation of S3×Dic11 ►in GL4(𝔽397) generated by
| 0 | 396 | 0 | 0 |
| 1 | 396 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 396 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 396 | 0 |
| 0 | 0 | 0 | 396 |
| 396 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 335 | 396 |
| 0 | 0 | 333 | 95 |
| 334 | 0 | 0 | 0 |
| 0 | 334 | 0 | 0 |
| 0 | 0 | 45 | 322 |
| 0 | 0 | 117 | 352 |
G:=sub<GL(4,GF(397))| [0,1,0,0,396,396,0,0,0,0,1,0,0,0,0,1],[396,0,0,0,1,1,0,0,0,0,396,0,0,0,0,396],[396,0,0,0,0,396,0,0,0,0,335,333,0,0,396,95],[334,0,0,0,0,334,0,0,0,0,45,117,0,0,322,352] >;
S3×Dic11 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{11} % in TeX
G:=Group("S3xDic11"); // GroupNames label
G:=SmallGroup(264,6);
// by ID
G=gap.SmallGroup(264,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,168,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^22=1,d^2=c^11,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export