direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×Dic33, C33⋊1C12, C66.1C6, C66.4S3, C6.4D33, C33⋊2Dic3, C32⋊2Dic11, (C3×C33)⋊4C4, C22.(C3×S3), C2.(C3×D33), C6.(C3×D11), C11⋊(C3×Dic3), C3⋊(C3×Dic11), (C3×C66).2C2, (C3×C6).1D11, SmallGroup(396,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C3×Dic33 |
Generators and relations for C3×Dic33
G = < a,b,c | a3=b66=1, c2=b33, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic33
►On 132 points
Generators in S132
(1 45 23)(2 46 24)(3 47 25)(4 48 26)(5 49 27)(6 50 28)(7 51 29)(8 52 30)(9 53 31)(10 54 32)(11 55 33)(12 56 34)(13 57 35)(14 58 36)(15 59 37)(16 60 38)(17 61 39)(18 62 40)(19 63 41)(20 64 42)(21 65 43)(22 66 44)(67 89 111)(68 90 112)(69 91 113)(70 92 114)(71 93 115)(72 94 116)(73 95 117)(74 96 118)(75 97 119)(76 98 120)(77 99 121)(78 100 122)(79 101 123)(80 102 124)(81 103 125)(82 104 126)(83 105 127)(84 106 128)(85 107 129)(86 108 130)(87 109 131)(88 110 132)
(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)
(1 111 34 78)(2 110 35 77)(3 109 36 76)(4 108 37 75)(5 107 38 74)(6 106 39 73)(7 105 40 72)(8 104 41 71)(9 103 42 70)(10 102 43 69)(11 101 44 68)(12 100 45 67)(13 99 46 132)(14 98 47 131)(15 97 48 130)(16 96 49 129)(17 95 50 128)(18 94 51 127)(19 93 52 126)(20 92 53 125)(21 91 54 124)(22 90 55 123)(23 89 56 122)(24 88 57 121)(25 87 58 120)(26 86 59 119)(27 85 60 118)(28 84 61 117)(29 83 62 116)(30 82 63 115)(31 81 64 114)(32 80 65 113)(33 79 66 112)
G:=sub<Sym(132)| (1,45,23)(2,46,24)(3,47,25)(4,48,26)(5,49,27)(6,50,28)(7,51,29)(8,52,30)(9,53,31)(10,54,32)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,61,39)(18,62,40)(19,63,41)(20,64,42)(21,65,43)(22,66,44)(67,89,111)(68,90,112)(69,91,113)(70,92,114)(71,93,115)(72,94,116)(73,95,117)(74,96,118)(75,97,119)(76,98,120)(77,99,121)(78,100,122)(79,101,123)(80,102,124)(81,103,125)(82,104,126)(83,105,127)(84,106,128)(85,107,129)(86,108,130)(87,109,131)(88,110,132), (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), (1,111,34,78)(2,110,35,77)(3,109,36,76)(4,108,37,75)(5,107,38,74)(6,106,39,73)(7,105,40,72)(8,104,41,71)(9,103,42,70)(10,102,43,69)(11,101,44,68)(12,100,45,67)(13,99,46,132)(14,98,47,131)(15,97,48,130)(16,96,49,129)(17,95,50,128)(18,94,51,127)(19,93,52,126)(20,92,53,125)(21,91,54,124)(22,90,55,123)(23,89,56,122)(24,88,57,121)(25,87,58,120)(26,86,59,119)(27,85,60,118)(28,84,61,117)(29,83,62,116)(30,82,63,115)(31,81,64,114)(32,80,65,113)(33,79,66,112)>;
G:=Group( (1,45,23)(2,46,24)(3,47,25)(4,48,26)(5,49,27)(6,50,28)(7,51,29)(8,52,30)(9,53,31)(10,54,32)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,61,39)(18,62,40)(19,63,41)(20,64,42)(21,65,43)(22,66,44)(67,89,111)(68,90,112)(69,91,113)(70,92,114)(71,93,115)(72,94,116)(73,95,117)(74,96,118)(75,97,119)(76,98,120)(77,99,121)(78,100,122)(79,101,123)(80,102,124)(81,103,125)(82,104,126)(83,105,127)(84,106,128)(85,107,129)(86,108,130)(87,109,131)(88,110,132), (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), (1,111,34,78)(2,110,35,77)(3,109,36,76)(4,108,37,75)(5,107,38,74)(6,106,39,73)(7,105,40,72)(8,104,41,71)(9,103,42,70)(10,102,43,69)(11,101,44,68)(12,100,45,67)(13,99,46,132)(14,98,47,131)(15,97,48,130)(16,96,49,129)(17,95,50,128)(18,94,51,127)(19,93,52,126)(20,92,53,125)(21,91,54,124)(22,90,55,123)(23,89,56,122)(24,88,57,121)(25,87,58,120)(26,86,59,119)(27,85,60,118)(28,84,61,117)(29,83,62,116)(30,82,63,115)(31,81,64,114)(32,80,65,113)(33,79,66,112) );
G=PermutationGroup([[(1,45,23),(2,46,24),(3,47,25),(4,48,26),(5,49,27),(6,50,28),(7,51,29),(8,52,30),(9,53,31),(10,54,32),(11,55,33),(12,56,34),(13,57,35),(14,58,36),(15,59,37),(16,60,38),(17,61,39),(18,62,40),(19,63,41),(20,64,42),(21,65,43),(22,66,44),(67,89,111),(68,90,112),(69,91,113),(70,92,114),(71,93,115),(72,94,116),(73,95,117),(74,96,118),(75,97,119),(76,98,120),(77,99,121),(78,100,122),(79,101,123),(80,102,124),(81,103,125),(82,104,126),(83,105,127),(84,106,128),(85,107,129),(86,108,130),(87,109,131),(88,110,132)], [(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)], [(1,111,34,78),(2,110,35,77),(3,109,36,76),(4,108,37,75),(5,107,38,74),(6,106,39,73),(7,105,40,72),(8,104,41,71),(9,103,42,70),(10,102,43,69),(11,101,44,68),(12,100,45,67),(13,99,46,132),(14,98,47,131),(15,97,48,130),(16,96,49,129),(17,95,50,128),(18,94,51,127),(19,93,52,126),(20,92,53,125),(21,91,54,124),(22,90,55,123),(23,89,56,122),(24,88,57,121),(25,87,58,120),(26,86,59,119),(27,85,60,118),(28,84,61,117),(29,83,62,116),(30,82,63,115),(31,81,64,114),(32,80,65,113),(33,79,66,112)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 11A | ··· | 11E | 12A | 12B | 12C | 12D | 22A | ··· | 22E | 33A | ··· | 33AN | 66A | ··· | 66AN |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 33 | 33 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 33 | 33 | 33 | 33 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | D11 | C3×Dic3 | Dic11 | C3×D11 | D33 | C3×Dic11 | Dic33 | C3×D33 | C3×Dic33 |
| kernel | C3×Dic33 | C3×C66 | Dic33 | C3×C33 | C66 | C33 | C66 | C33 | C22 | C3×C6 | C11 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 5 | 2 | 5 | 10 | 10 | 10 | 10 | 20 | 20 |
Matrix representation of C3×Dic33 ►in GL3(𝔽397) generated by
| 362 | 0 | 0 |
| 0 | 34 | 0 |
| 0 | 0 | 34 |
| 396 | 0 | 0 |
| 0 | 110 | 0 |
| 0 | 0 | 314 |
| 63 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(397))| [362,0,0,0,34,0,0,0,34],[396,0,0,0,110,0,0,0,314],[63,0,0,0,0,1,0,1,0] >;
C3×Dic33 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{33} % in TeX
G:=Group("C3xDic33"); // GroupNames label
G:=SmallGroup(396,13);
// by ID
G=gap.SmallGroup(396,13);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-11,30,483,9004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^66=1,c^2=b^33,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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