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G = C27⋊C12order 324 = 22·34

The semidirect product of C27 and C12 acting via C12/C2=C6

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

C1C27 — C27⋊C12
C1C3C9C27C54C2×C27⋊C3 — C27⋊C12
C27 — C27⋊C12
C1C2

Generators and relations for C27⋊C12
 G = < a,b | a27=b12=1, bab-1=a8 >

3C3
27C4
3C6
2C9
9Dic3
27C12
2C18
2C27
3Dic9
9C3×Dic3
2C54
3C3×Dic9

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 3A3B3C4A4B6A6B6C9A9B9C9D9E12A12B12C12D18A18B18C18D18E27A···27I54A···54I
order12333446669999912121212181818181827···2754···54
size1123327272332226627272727222666···66···6

42 irreducible representations

dim1111112222222266
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3C3×S3D9C3×Dic3Dic9C3×D9C3×Dic9C27⋊C6C27⋊C12
kernelC27⋊C12C2×C27⋊C3Dic27C27⋊C3C54C27C3×C18C3×C9C18C3×C6C9C32C6C3C2C1
# reps1122241123236633

Matrix representation of C27⋊C12 in GL8(𝔽109)

1081000000
1080000000
0000827700
0000325000
0000008277
0000003250
00100000
00010000
,
3017000000
4779000000
0017990000
007920000
0000007993
0000006330
0000467900
0000166300

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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Subgroup lattice of C27⋊C12 in TeX

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