metacyclic, supersoluble, monomial
Aliases: C27⋊C12, C54.C6, Dic27⋊C3, C32.Dic9, C27⋊C3⋊C4, C2.(C27⋊C6), C6.3(C3×D9), (C3×C6).2D9, C18.4(C3×S3), (C3×C18).11S3, C3.3(C3×Dic9), (C3×C9).3Dic3, C9.2(C3×Dic3), (C2×C27⋊C3).C2, SmallGroup(324,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C27⋊C12 |
Generators and relations for C27⋊C12
G = < a,b | a27=b12=1, bab-1=a8 >
Smallest permutation representation of C27⋊C12
►On 108 points
Generators in S108
(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)
(1 96 42 69)(2 86 34 68 11 104 43 59 20 95 52 77)(3 103 53 67 21 85 44 76 12 94 35 58)(4 93 45 66)(5 83 37 65 14 101 46 56 23 92 28 74)(6 100 29 64 24 82 47 73 15 91 38 55)(7 90 48 63)(8 107 40 62 17 98 49 80 26 89 31 71)(9 97 32 61 27 106 50 70 18 88 41 79)(10 87 51 60)(13 84 54 57)(16 108 30 81)(19 105 33 78)(22 102 36 75)(25 99 39 72)
G:=sub<Sym(108)| (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), (1,96,42,69)(2,86,34,68,11,104,43,59,20,95,52,77)(3,103,53,67,21,85,44,76,12,94,35,58)(4,93,45,66)(5,83,37,65,14,101,46,56,23,92,28,74)(6,100,29,64,24,82,47,73,15,91,38,55)(7,90,48,63)(8,107,40,62,17,98,49,80,26,89,31,71)(9,97,32,61,27,106,50,70,18,88,41,79)(10,87,51,60)(13,84,54,57)(16,108,30,81)(19,105,33,78)(22,102,36,75)(25,99,39,72)>;
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), (1,96,42,69)(2,86,34,68,11,104,43,59,20,95,52,77)(3,103,53,67,21,85,44,76,12,94,35,58)(4,93,45,66)(5,83,37,65,14,101,46,56,23,92,28,74)(6,100,29,64,24,82,47,73,15,91,38,55)(7,90,48,63)(8,107,40,62,17,98,49,80,26,89,31,71)(9,97,32,61,27,106,50,70,18,88,41,79)(10,87,51,60)(13,84,54,57)(16,108,30,81)(19,105,33,78)(22,102,36,75)(25,99,39,72) );
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)], [(1,96,42,69),(2,86,34,68,11,104,43,59,20,95,52,77),(3,103,53,67,21,85,44,76,12,94,35,58),(4,93,45,66),(5,83,37,65,14,101,46,56,23,92,28,74),(6,100,29,64,24,82,47,73,15,91,38,55),(7,90,48,63),(8,107,40,62,17,98,49,80,26,89,31,71),(9,97,32,61,27,106,50,70,18,88,41,79),(10,87,51,60),(13,84,54,57),(16,108,30,81),(19,105,33,78),(22,102,36,75),(25,99,39,72)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 9A | 9B | 9C | 9D | 9E | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 27A | ··· | 27I | 54A | ··· | 54I |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 2 | 3 | 3 | 27 | 27 | 2 | 3 | 3 | 2 | 2 | 2 | 6 | 6 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | - | + | - | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | D9 | C3×Dic3 | Dic9 | C3×D9 | C3×Dic9 | C27⋊C6 | C27⋊C12 |
| kernel | C27⋊C12 | C2×C27⋊C3 | Dic27 | C27⋊C3 | C54 | C27 | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 3 | 2 | 3 | 6 | 6 | 3 | 3 |
Matrix representation of C27⋊C12 ►in GL8(𝔽109)
| 108 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 108 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 82 | 77 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 82 | 77 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 50 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 30 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 47 | 79 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 99 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 92 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 79 | 93 |
| 0 | 0 | 0 | 0 | 0 | 0 | 63 | 30 |
| 0 | 0 | 0 | 0 | 46 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 63 | 0 | 0 |
G:=sub<GL(8,GF(109))| [108,108,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,82,32,0,0,0,0,0,0,77,50,0,0,0,0,0,0,0,0,82,32,0,0,0,0,0,0,77,50,0,0],[30,47,0,0,0,0,0,0,17,79,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,99,92,0,0,0,0,0,0,0,0,0,0,46,16,0,0,0,0,0,0,79,63,0,0,0,0,79,63,0,0,0,0,0,0,93,30,0,0] >;
C27⋊C12 in GAP, Magma, Sage, TeX
C_{27}\rtimes C_{12} % in TeX
G:=Group("C27:C12"); // GroupNames label
G:=SmallGroup(324,12);
// by ID
G=gap.SmallGroup(324,12);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,1443,1449,381,5404,208,7781]);
// Polycyclic
G:=Group<a,b|a^27=b^12=1,b*a*b^-1=a^8>;
// generators/relations
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