direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C11×D12, C33⋊6D4, C44⋊3S3, C12⋊1C22, C132⋊5C2, D6⋊1C22, C22.15D6, C66.20C22, C4⋊(S3×C11), C3⋊1(D4×C11), (S3×C22)⋊4C2, C2.4(S3×C22), C6.3(C2×C22), SmallGroup(264,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×D12
G = < a,b,c | a11=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D12
►On 132 points
Generators in S132
(1 37 116 49 73 22 102 70 32 132 91)(2 38 117 50 74 23 103 71 33 121 92)(3 39 118 51 75 24 104 72 34 122 93)(4 40 119 52 76 13 105 61 35 123 94)(5 41 120 53 77 14 106 62 36 124 95)(6 42 109 54 78 15 107 63 25 125 96)(7 43 110 55 79 16 108 64 26 126 85)(8 44 111 56 80 17 97 65 27 127 86)(9 45 112 57 81 18 98 66 28 128 87)(10 46 113 58 82 19 99 67 29 129 88)(11 47 114 59 83 20 100 68 30 130 89)(12 48 115 60 84 21 101 69 31 131 90)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 106)(98 105)(99 104)(100 103)(101 102)(107 108)(109 110)(111 120)(112 119)(113 118)(114 117)(115 116)(121 130)(122 129)(123 128)(124 127)(125 126)(131 132)
G:=sub<Sym(132)| (1,37,116,49,73,22,102,70,32,132,91)(2,38,117,50,74,23,103,71,33,121,92)(3,39,118,51,75,24,104,72,34,122,93)(4,40,119,52,76,13,105,61,35,123,94)(5,41,120,53,77,14,106,62,36,124,95)(6,42,109,54,78,15,107,63,25,125,96)(7,43,110,55,79,16,108,64,26,126,85)(8,44,111,56,80,17,97,65,27,127,86)(9,45,112,57,81,18,98,66,28,128,87)(10,46,113,58,82,19,99,67,29,129,88)(11,47,114,59,83,20,100,68,30,130,89)(12,48,115,60,84,21,101,69,31,131,90), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,106)(98,105)(99,104)(100,103)(101,102)(107,108)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,132)>;
G:=Group( (1,37,116,49,73,22,102,70,32,132,91)(2,38,117,50,74,23,103,71,33,121,92)(3,39,118,51,75,24,104,72,34,122,93)(4,40,119,52,76,13,105,61,35,123,94)(5,41,120,53,77,14,106,62,36,124,95)(6,42,109,54,78,15,107,63,25,125,96)(7,43,110,55,79,16,108,64,26,126,85)(8,44,111,56,80,17,97,65,27,127,86)(9,45,112,57,81,18,98,66,28,128,87)(10,46,113,58,82,19,99,67,29,129,88)(11,47,114,59,83,20,100,68,30,130,89)(12,48,115,60,84,21,101,69,31,131,90), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,106)(98,105)(99,104)(100,103)(101,102)(107,108)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,132) );
G=PermutationGroup([[(1,37,116,49,73,22,102,70,32,132,91),(2,38,117,50,74,23,103,71,33,121,92),(3,39,118,51,75,24,104,72,34,122,93),(4,40,119,52,76,13,105,61,35,123,94),(5,41,120,53,77,14,106,62,36,124,95),(6,42,109,54,78,15,107,63,25,125,96),(7,43,110,55,79,16,108,64,26,126,85),(8,44,111,56,80,17,97,65,27,127,86),(9,45,112,57,81,18,98,66,28,128,87),(10,46,113,58,82,19,99,67,29,129,88),(11,47,114,59,83,20,100,68,30,130,89),(12,48,115,60,84,21,101,69,31,131,90)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,106),(98,105),(99,104),(100,103),(101,102),(107,108),(109,110),(111,120),(112,119),(113,118),(114,117),(115,116),(121,130),(122,129),(123,128),(124,127),(125,126),(131,132)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 11A | ··· | 11J | 12A | 12B | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | D4 | D6 | D12 | S3×C11 | D4×C11 | S3×C22 | C11×D12 |
| kernel | C11×D12 | C132 | S3×C22 | D12 | C12 | D6 | C44 | C33 | C22 | C11 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 1 | 1 | 2 | 10 | 10 | 10 | 20 |
Matrix representation of C11×D12 ►in GL2(𝔽23) generated by
| 13 | 0 |
| 0 | 13 |
| 7 | 6 |
| 19 | 0 |
| 0 | 17 |
| 19 | 0 |
G:=sub<GL(2,GF(23))| [13,0,0,13],[7,19,6,0],[0,19,17,0] >;
C11×D12 in GAP, Magma, Sage, TeX
C_{11}\times D_{12} % in TeX
G:=Group("C11xD12"); // GroupNames label
G:=SmallGroup(264,20);
// by ID
G=gap.SmallGroup(264,20);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-3,461,226,4404]);
// Polycyclic
G:=Group<a,b,c|a^11=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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