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G = C2×C27⋊C6order 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

C1C27 — C2×C27⋊C6
C1C3C9C27C27⋊C3C27⋊C6 — C2×C27⋊C6
C27 — C2×C27⋊C6
C1C2

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

27C2
27C2
3C3
27C22
3C6
9S3
9S3
27C6
27C6
2C9
9D6
27C2×C6
2C18
3D9
3D9
9C3×S3
9C3×S3
2C27
3D18
9S3×C6
2C54
3C3×D9
3C3×D9
3C6×D9

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

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

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

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

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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