direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×Dic3, C3⋊C44, C33⋊3C4, C6.C22, C66.3C2, C22.2S3, C2.(S3×C11), SmallGroup(132,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C11×Dic3 |
Generators and relations for C11×Dic3
G = < a,b,c | a11=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×Dic3
►Regular action on 132 points
Generators in S132
(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 83 109 99 77 119)(2 84 110 89 67 120)(3 85 100 90 68 121)(4 86 101 91 69 111)(5 87 102 92 70 112)(6 88 103 93 71 113)(7 78 104 94 72 114)(8 79 105 95 73 115)(9 80 106 96 74 116)(10 81 107 97 75 117)(11 82 108 98 76 118)(12 128 37 29 46 66)(13 129 38 30 47 56)(14 130 39 31 48 57)(15 131 40 32 49 58)(16 132 41 33 50 59)(17 122 42 23 51 60)(18 123 43 24 52 61)(19 124 44 25 53 62)(20 125 34 26 54 63)(21 126 35 27 55 64)(22 127 36 28 45 65)
(1 23 99 17)(2 24 89 18)(3 25 90 19)(4 26 91 20)(5 27 92 21)(6 28 93 22)(7 29 94 12)(8 30 95 13)(9 31 96 14)(10 32 97 15)(11 33 98 16)(34 69 63 86)(35 70 64 87)(36 71 65 88)(37 72 66 78)(38 73 56 79)(39 74 57 80)(40 75 58 81)(41 76 59 82)(42 77 60 83)(43 67 61 84)(44 68 62 85)(45 103 127 113)(46 104 128 114)(47 105 129 115)(48 106 130 116)(49 107 131 117)(50 108 132 118)(51 109 122 119)(52 110 123 120)(53 100 124 121)(54 101 125 111)(55 102 126 112)
G:=sub<Sym(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), (1,83,109,99,77,119)(2,84,110,89,67,120)(3,85,100,90,68,121)(4,86,101,91,69,111)(5,87,102,92,70,112)(6,88,103,93,71,113)(7,78,104,94,72,114)(8,79,105,95,73,115)(9,80,106,96,74,116)(10,81,107,97,75,117)(11,82,108,98,76,118)(12,128,37,29,46,66)(13,129,38,30,47,56)(14,130,39,31,48,57)(15,131,40,32,49,58)(16,132,41,33,50,59)(17,122,42,23,51,60)(18,123,43,24,52,61)(19,124,44,25,53,62)(20,125,34,26,54,63)(21,126,35,27,55,64)(22,127,36,28,45,65), (1,23,99,17)(2,24,89,18)(3,25,90,19)(4,26,91,20)(5,27,92,21)(6,28,93,22)(7,29,94,12)(8,30,95,13)(9,31,96,14)(10,32,97,15)(11,33,98,16)(34,69,63,86)(35,70,64,87)(36,71,65,88)(37,72,66,78)(38,73,56,79)(39,74,57,80)(40,75,58,81)(41,76,59,82)(42,77,60,83)(43,67,61,84)(44,68,62,85)(45,103,127,113)(46,104,128,114)(47,105,129,115)(48,106,130,116)(49,107,131,117)(50,108,132,118)(51,109,122,119)(52,110,123,120)(53,100,124,121)(54,101,125,111)(55,102,126,112)>;
G:=Group( (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), (1,83,109,99,77,119)(2,84,110,89,67,120)(3,85,100,90,68,121)(4,86,101,91,69,111)(5,87,102,92,70,112)(6,88,103,93,71,113)(7,78,104,94,72,114)(8,79,105,95,73,115)(9,80,106,96,74,116)(10,81,107,97,75,117)(11,82,108,98,76,118)(12,128,37,29,46,66)(13,129,38,30,47,56)(14,130,39,31,48,57)(15,131,40,32,49,58)(16,132,41,33,50,59)(17,122,42,23,51,60)(18,123,43,24,52,61)(19,124,44,25,53,62)(20,125,34,26,54,63)(21,126,35,27,55,64)(22,127,36,28,45,65), (1,23,99,17)(2,24,89,18)(3,25,90,19)(4,26,91,20)(5,27,92,21)(6,28,93,22)(7,29,94,12)(8,30,95,13)(9,31,96,14)(10,32,97,15)(11,33,98,16)(34,69,63,86)(35,70,64,87)(36,71,65,88)(37,72,66,78)(38,73,56,79)(39,74,57,80)(40,75,58,81)(41,76,59,82)(42,77,60,83)(43,67,61,84)(44,68,62,85)(45,103,127,113)(46,104,128,114)(47,105,129,115)(48,106,130,116)(49,107,131,117)(50,108,132,118)(51,109,122,119)(52,110,123,120)(53,100,124,121)(54,101,125,111)(55,102,126,112) );
G=PermutationGroup([[(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)], [(1,83,109,99,77,119),(2,84,110,89,67,120),(3,85,100,90,68,121),(4,86,101,91,69,111),(5,87,102,92,70,112),(6,88,103,93,71,113),(7,78,104,94,72,114),(8,79,105,95,73,115),(9,80,106,96,74,116),(10,81,107,97,75,117),(11,82,108,98,76,118),(12,128,37,29,46,66),(13,129,38,30,47,56),(14,130,39,31,48,57),(15,131,40,32,49,58),(16,132,41,33,50,59),(17,122,42,23,51,60),(18,123,43,24,52,61),(19,124,44,25,53,62),(20,125,34,26,54,63),(21,126,35,27,55,64),(22,127,36,28,45,65)], [(1,23,99,17),(2,24,89,18),(3,25,90,19),(4,26,91,20),(5,27,92,21),(6,28,93,22),(7,29,94,12),(8,30,95,13),(9,31,96,14),(10,32,97,15),(11,33,98,16),(34,69,63,86),(35,70,64,87),(36,71,65,88),(37,72,66,78),(38,73,56,79),(39,74,57,80),(40,75,58,81),(41,76,59,82),(42,77,60,83),(43,67,61,84),(44,68,62,85),(45,103,127,113),(46,104,128,114),(47,105,129,115),(48,106,130,116),(49,107,131,117),(50,108,132,118),(51,109,122,119),(52,110,123,120),(53,100,124,121),(54,101,125,111),(55,102,126,112)]])
C11×Dic3 is a maximal subgroup of
D33⋊C4 C3⋊D44 C33⋊Q8 S3×C44
66 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33J | 44A | ··· | 44T | 66A | ··· | 66J |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | S3 | Dic3 | S3×C11 | C11×Dic3 |
| kernel | C11×Dic3 | C66 | C33 | Dic3 | C6 | C3 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 1 | 10 | 10 |
Matrix representation of C11×Dic3 ►in GL2(𝔽23) generated by
| 13 | 0 |
| 0 | 13 |
| 1 | 21 |
| 12 | 0 |
| 12 | 12 |
| 9 | 11 |
G:=sub<GL(2,GF(23))| [13,0,0,13],[1,12,21,0],[12,9,12,11] >;
C11×Dic3 in GAP, Magma, Sage, TeX
C_{11}\times {\rm Dic}_3 % in TeX
G:=Group("C11xDic3"); // GroupNames label
G:=SmallGroup(132,1);
// by ID
G=gap.SmallGroup(132,1);
# by ID
G:=PCGroup([4,-2,-11,-2,-3,88,1411]);
// Polycyclic
G:=Group<a,b,c|a^11=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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