direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic11, C11⋊C12, C33⋊2C4, C22.C6, C66.2C2, C6.2D11, C2.(C3×D11), SmallGroup(132,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C3×Dic11 |
Generators and relations for C3×Dic11
G = < a,b,c | a3=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic11
►Regular action on 132 points
Generators in S132
(1 61 40)(2 62 41)(3 63 42)(4 64 43)(5 65 44)(6 66 23)(7 45 24)(8 46 25)(9 47 26)(10 48 27)(11 49 28)(12 50 29)(13 51 30)(14 52 31)(15 53 32)(16 54 33)(17 55 34)(18 56 35)(19 57 36)(20 58 37)(21 59 38)(22 60 39)(67 111 89)(68 112 90)(69 113 91)(70 114 92)(71 115 93)(72 116 94)(73 117 95)(74 118 96)(75 119 97)(76 120 98)(77 121 99)(78 122 100)(79 123 101)(80 124 102)(81 125 103)(82 126 104)(83 127 105)(84 128 106)(85 129 107)(86 130 108)(87 131 109)(88 132 110)
(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 67 12 78)(2 88 13 77)(3 87 14 76)(4 86 15 75)(5 85 16 74)(6 84 17 73)(7 83 18 72)(8 82 19 71)(9 81 20 70)(10 80 21 69)(11 79 22 68)(23 106 34 95)(24 105 35 94)(25 104 36 93)(26 103 37 92)(27 102 38 91)(28 101 39 90)(29 100 40 89)(30 99 41 110)(31 98 42 109)(32 97 43 108)(33 96 44 107)(45 127 56 116)(46 126 57 115)(47 125 58 114)(48 124 59 113)(49 123 60 112)(50 122 61 111)(51 121 62 132)(52 120 63 131)(53 119 64 130)(54 118 65 129)(55 117 66 128)
G:=sub<Sym(132)| (1,61,40)(2,62,41)(3,63,42)(4,64,43)(5,65,44)(6,66,23)(7,45,24)(8,46,25)(9,47,26)(10,48,27)(11,49,28)(12,50,29)(13,51,30)(14,52,31)(15,53,32)(16,54,33)(17,55,34)(18,56,35)(19,57,36)(20,58,37)(21,59,38)(22,60,39)(67,111,89)(68,112,90)(69,113,91)(70,114,92)(71,115,93)(72,116,94)(73,117,95)(74,118,96)(75,119,97)(76,120,98)(77,121,99)(78,122,100)(79,123,101)(80,124,102)(81,125,103)(82,126,104)(83,127,105)(84,128,106)(85,129,107)(86,130,108)(87,131,109)(88,132,110), (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), (1,67,12,78)(2,88,13,77)(3,87,14,76)(4,86,15,75)(5,85,16,74)(6,84,17,73)(7,83,18,72)(8,82,19,71)(9,81,20,70)(10,80,21,69)(11,79,22,68)(23,106,34,95)(24,105,35,94)(25,104,36,93)(26,103,37,92)(27,102,38,91)(28,101,39,90)(29,100,40,89)(30,99,41,110)(31,98,42,109)(32,97,43,108)(33,96,44,107)(45,127,56,116)(46,126,57,115)(47,125,58,114)(48,124,59,113)(49,123,60,112)(50,122,61,111)(51,121,62,132)(52,120,63,131)(53,119,64,130)(54,118,65,129)(55,117,66,128)>;
G:=Group( (1,61,40)(2,62,41)(3,63,42)(4,64,43)(5,65,44)(6,66,23)(7,45,24)(8,46,25)(9,47,26)(10,48,27)(11,49,28)(12,50,29)(13,51,30)(14,52,31)(15,53,32)(16,54,33)(17,55,34)(18,56,35)(19,57,36)(20,58,37)(21,59,38)(22,60,39)(67,111,89)(68,112,90)(69,113,91)(70,114,92)(71,115,93)(72,116,94)(73,117,95)(74,118,96)(75,119,97)(76,120,98)(77,121,99)(78,122,100)(79,123,101)(80,124,102)(81,125,103)(82,126,104)(83,127,105)(84,128,106)(85,129,107)(86,130,108)(87,131,109)(88,132,110), (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), (1,67,12,78)(2,88,13,77)(3,87,14,76)(4,86,15,75)(5,85,16,74)(6,84,17,73)(7,83,18,72)(8,82,19,71)(9,81,20,70)(10,80,21,69)(11,79,22,68)(23,106,34,95)(24,105,35,94)(25,104,36,93)(26,103,37,92)(27,102,38,91)(28,101,39,90)(29,100,40,89)(30,99,41,110)(31,98,42,109)(32,97,43,108)(33,96,44,107)(45,127,56,116)(46,126,57,115)(47,125,58,114)(48,124,59,113)(49,123,60,112)(50,122,61,111)(51,121,62,132)(52,120,63,131)(53,119,64,130)(54,118,65,129)(55,117,66,128) );
G=PermutationGroup([[(1,61,40),(2,62,41),(3,63,42),(4,64,43),(5,65,44),(6,66,23),(7,45,24),(8,46,25),(9,47,26),(10,48,27),(11,49,28),(12,50,29),(13,51,30),(14,52,31),(15,53,32),(16,54,33),(17,55,34),(18,56,35),(19,57,36),(20,58,37),(21,59,38),(22,60,39),(67,111,89),(68,112,90),(69,113,91),(70,114,92),(71,115,93),(72,116,94),(73,117,95),(74,118,96),(75,119,97),(76,120,98),(77,121,99),(78,122,100),(79,123,101),(80,124,102),(81,125,103),(82,126,104),(83,127,105),(84,128,106),(85,129,107),(86,130,108),(87,131,109),(88,132,110)], [(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)], [(1,67,12,78),(2,88,13,77),(3,87,14,76),(4,86,15,75),(5,85,16,74),(6,84,17,73),(7,83,18,72),(8,82,19,71),(9,81,20,70),(10,80,21,69),(11,79,22,68),(23,106,34,95),(24,105,35,94),(25,104,36,93),(26,103,37,92),(27,102,38,91),(28,101,39,90),(29,100,40,89),(30,99,41,110),(31,98,42,109),(32,97,43,108),(33,96,44,107),(45,127,56,116),(46,126,57,115),(47,125,58,114),(48,124,59,113),(49,123,60,112),(50,122,61,111),(51,121,62,132),(52,120,63,131),(53,119,64,130),(54,118,65,129),(55,117,66,128)]])
C3×Dic11 is a maximal subgroup of
D33⋊C4 C11⋊D12 C33⋊Q8 C12×D11
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 11A | ··· | 11E | 12A | 12B | 12C | 12D | 22A | ··· | 22E | 33A | ··· | 33J | 66A | ··· | 66J |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 1 | 1 | 2 | ··· | 2 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D11 | Dic11 | C3×D11 | C3×Dic11 |
| kernel | C3×Dic11 | C66 | Dic11 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 |
Matrix representation of C3×Dic11 ►in GL2(𝔽43) generated by
| 36 | 0 |
| 0 | 36 |
| 1 | 22 |
| 21 | 33 |
| 23 | 16 |
| 26 | 20 |
G:=sub<GL(2,GF(43))| [36,0,0,36],[1,21,22,33],[23,26,16,20] >;
C3×Dic11 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{11} % in TeX
G:=Group("C3xDic11"); // GroupNames label
G:=SmallGroup(132,2);
// by ID
G=gap.SmallGroup(132,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-11,24,1923]);
// Polycyclic
G:=Group<a,b,c|a^3=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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