metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C33⋊3D4, C11⋊2D12, D66⋊4C2, D6⋊2D11, Dic11⋊S3, C22.6D6, C6.6D22, C66.6C22, (S3×C22)⋊2C2, C3⋊1(C11⋊D4), C2.6(S3×D11), (C3×Dic11)⋊3C2, SmallGroup(264,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11⋊D12
G = < a,b,c | a11=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C11⋊D12
►On 132 points
Generators in S132
(1 35 76 124 102 114 62 55 42 24 94)(2 95 13 43 56 63 115 103 125 77 36)(3 25 78 126 104 116 64 57 44 14 96)(4 85 15 45 58 65 117 105 127 79 26)(5 27 80 128 106 118 66 59 46 16 86)(6 87 17 47 60 67 119 107 129 81 28)(7 29 82 130 108 120 68 49 48 18 88)(8 89 19 37 50 69 109 97 131 83 30)(9 31 84 132 98 110 70 51 38 20 90)(10 91 21 39 52 71 111 99 121 73 32)(11 33 74 122 100 112 72 53 40 22 92)(12 93 23 41 54 61 113 101 123 75 34)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 84)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 125)(38 124)(39 123)(40 122)(41 121)(42 132)(43 131)(44 130)(45 129)(46 128)(47 127)(48 126)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 108)(58 107)(59 106)(60 105)(61 111)(62 110)(63 109)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)
G:=sub<Sym(132)| (1,35,76,124,102,114,62,55,42,24,94)(2,95,13,43,56,63,115,103,125,77,36)(3,25,78,126,104,116,64,57,44,14,96)(4,85,15,45,58,65,117,105,127,79,26)(5,27,80,128,106,118,66,59,46,16,86)(6,87,17,47,60,67,119,107,129,81,28)(7,29,82,130,108,120,68,49,48,18,88)(8,89,19,37,50,69,109,97,131,83,30)(9,31,84,132,98,110,70,51,38,20,90)(10,91,21,39,52,71,111,99,121,73,32)(11,33,74,122,100,112,72,53,40,22,92)(12,93,23,41,54,61,113,101,123,75,34), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,84)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,125)(38,124)(39,123)(40,122)(41,121)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,108)(58,107)(59,106)(60,105)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)>;
G:=Group( (1,35,76,124,102,114,62,55,42,24,94)(2,95,13,43,56,63,115,103,125,77,36)(3,25,78,126,104,116,64,57,44,14,96)(4,85,15,45,58,65,117,105,127,79,26)(5,27,80,128,106,118,66,59,46,16,86)(6,87,17,47,60,67,119,107,129,81,28)(7,29,82,130,108,120,68,49,48,18,88)(8,89,19,37,50,69,109,97,131,83,30)(9,31,84,132,98,110,70,51,38,20,90)(10,91,21,39,52,71,111,99,121,73,32)(11,33,74,122,100,112,72,53,40,22,92)(12,93,23,41,54,61,113,101,123,75,34), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,84)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,125)(38,124)(39,123)(40,122)(41,121)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,108)(58,107)(59,106)(60,105)(61,111)(62,110)(63,109)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112) );
G=PermutationGroup([[(1,35,76,124,102,114,62,55,42,24,94),(2,95,13,43,56,63,115,103,125,77,36),(3,25,78,126,104,116,64,57,44,14,96),(4,85,15,45,58,65,117,105,127,79,26),(5,27,80,128,106,118,66,59,46,16,86),(6,87,17,47,60,67,119,107,129,81,28),(7,29,82,130,108,120,68,49,48,18,88),(8,89,19,37,50,69,109,97,131,83,30),(9,31,84,132,98,110,70,51,38,20,90),(10,91,21,39,52,71,111,99,121,73,32),(11,33,74,122,100,112,72,53,40,22,92),(12,93,23,41,54,61,113,101,123,75,34)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,84),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,125),(38,124),(39,123),(40,122),(41,121),(42,132),(43,131),(44,130),(45,129),(46,128),(47,127),(48,126),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,108),(58,107),(59,106),(60,105),(61,111),(62,110),(63,109),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 22F | ··· | 22O | 33A | ··· | 33E | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 6 | 66 | 2 | 22 | 2 | 2 | ··· | 2 | 22 | 22 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D11 | D12 | D22 | C11⋊D4 | S3×D11 | C11⋊D12 |
| kernel | C11⋊D12 | C3×Dic11 | S3×C22 | D66 | Dic11 | C33 | C22 | D6 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 5 | 5 |
Matrix representation of C11⋊D12 ►in GL6(𝔽397)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 396 | 95 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 353 | 0 | 0 | 0 | 0 |
| 48 | 333 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 95 | 396 | 0 | 0 |
| 0 | 0 | 0 | 0 | 396 | 250 |
| 0 | 0 | 0 | 0 | 316 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 39 | 396 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 95 | 396 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 147 |
| 0 | 0 | 0 | 0 | 81 | 395 |
G:=sub<GL(6,GF(397))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,396,0,0,0,0,1,95,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,48,0,0,0,0,353,333,0,0,0,0,0,0,1,95,0,0,0,0,0,396,0,0,0,0,0,0,396,316,0,0,0,0,250,2],[1,39,0,0,0,0,0,396,0,0,0,0,0,0,1,95,0,0,0,0,0,396,0,0,0,0,0,0,2,81,0,0,0,0,147,395] >;
C11⋊D12 in GAP, Magma, Sage, TeX
C_{11}\rtimes D_{12} % in TeX
G:=Group("C11:D12"); // GroupNames label
G:=SmallGroup(264,10);
// by ID
G=gap.SmallGroup(264,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,61,168,6004]);
// Polycyclic
G:=Group<a,b,c|a^11=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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