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G = C2×C27⋊C6  order 324 = 22·34

Direct product of C2 and C27⋊C6

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C27⋊C6, C54⋊C6, D27⋊C6, D54⋊C3, C32.D18, C27⋊(C2×C6), C27⋊C3⋊C22, C9.3(S3×C6), (C3×C9).3D6, C3.3(C6×D9), C6.6(C3×D9), (C3×C6).4D9, C18.6(C3×S3), (C3×C18).12S3, (C2×C27⋊C3)⋊C2, SmallGroup(324,67)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C27 — C2×C27⋊C6
Chief series
C1 — C3 — C9 — C27 — C27⋊C3 — C27⋊C6 — C2×C27⋊C6
Lower central
C27 — C2×C27⋊C6
Upper central
C1 — C2

Generators and relations for C2×C27⋊C6
 G = < a,b,c | a2=b27=c6=1, ab=ba, ac=ca, cbc-1=b17 >

C1
C2
27C2
27C2
C3
3C3
27C22
C6
3C6
9S3
9S3
27C6
27C6
C9
C32
2C9
9D6
27C2×C6
C3×C6
C18
2C18
3D9
3D9
9C3×S3
9C3×S3
C27
C3×C9
2C27
3D18
9S3×C6
C54
D27
D27
C3×C18
2C54
3C3×D9
3C3×D9
C27⋊C3
D54
3C6×D9
C27⋊C6
C27⋊C6
C2×C27⋊C3
C2×C27⋊C6

Smallest permutation representation of C2×C27⋊C6
►On 54 points
Generators in S54
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)

(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)

(2 9 11 27 20 18)(3 17 21 26 12 8)(4 25)(5 6 14 24 23 15)(7 22)(10 19)(13 16)(28 43)(29 51 38 42 47 33)(30 32 48 41 39 50)(31 40)(34 37)(35 45 44 36 53 54)(46 52)
 
G:=sub<Sym(54)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48), (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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16)(28,43)(29,51,38,42,47,33)(30,32,48,41,39,50)(31,40)(34,37)(35,45,44,36,53,54)(46,52)>;
 
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48), (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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16)(28,43)(29,51,38,42,47,33)(30,32,48,41,39,50)(31,40)(34,37)(35,45,44,36,53,54)(46,52) );
 
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48)], [(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)], [(2,9,11,27,20,18),(3,17,21,26,12,8),(4,25),(5,6,14,24,23,15),(7,22),(10,19),(13,16),(28,43),(29,51,38,42,47,33),(30,32,48,41,39,50),(31,40),(34,37),(35,45,44,36,53,54),(46,52)]])
 

42 conjugacy classes

class 1 2A2B2C3A3B3C6A6B6C6D6E6F6G9A9B9C9D9E18A18B18C18D18E27A···27I54A···54I
order1222333666666699999181818181827···2754···54
size1127272332332727272722266222666···66···6

42 irreducible representations

dim1111112222222266
type+++++++++
imageC1C2C2C3C6C6S3D6C3×S3D9S3×C6D18C3×D9C6×D9C27⋊C6C2×C27⋊C6
kernelC2×C27⋊C6C27⋊C6C2×C27⋊C3D54D27C54C3×C18C3×C9C18C3×C6C9C32C6C3C2C1
# reps1212421123236633

Matrix representation of C2×C27⋊C6 ►in GL8(𝔽109)

1080000000
0108000000
00100000
00010000
00001000
00000100
00000010
00000001
,
223000000
4986000000
000010810800
00001000
000000108108
00000010
0059270000
0082320000
,
3280000000
3077000000
00100000
001081080000
0000002759
0000003282
0000328200
0000507700

G:=sub<GL(8,GF(109))| [108,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[22,49,0,0,0,0,0,0,3,86,0,0,0,0,0,0,0,0,0,0,0,0,59,82,0,0,0,0,0,0,27,32,0,0,108,1,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,1,0,0,0,0,0,0,108,0,0,0],[32,30,0,0,0,0,0,0,80,77,0,0,0,0,0,0,0,0,1,108,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,0,32,50,0,0,0,0,0,0,82,77,0,0,0,0,27,32,0,0,0,0,0,0,59,82,0,0] >;
 

C2×C27⋊C6 in GAP, Magma, Sage, TeX 

C_2\times C_{27}\rtimes C_6
 
% in TeX
 
G:=Group("C2xC27:C6");
 
// GroupNames label
 
G:=SmallGroup(324,67);
 
// by ID
 
G=gap.SmallGroup(324,67);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1443,735,381,5404,208,7781]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^27=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C27⋊C6 in TeX

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