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## G = C2×C27⋊C6order 324 = 22·34

### Direct product of C2 and C27⋊C6

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 >

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 2A 2B 2C 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D 9E 18A 18B 18C 18D 18E 27A ··· 27I 54A ··· 54I order 1 2 2 2 3 3 3 6 6 6 6 6 6 6 9 9 9 9 9 18 18 18 18 18 27 ··· 27 54 ··· 54 size 1 1 27 27 2 3 3 2 3 3 27 27 27 27 2 2 2 6 6 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 C2 C3 C6 C6 S3 D6 C3×S3 D9 S3×C6 D18 C3×D9 C6×D9 C27⋊C6 C2×C27⋊C6 kernel C2×C27⋊C6 C27⋊C6 C2×C27⋊C3 D54 D27 C54 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 2 3 2 3 6 6 3 3

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

 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 3 0 0 0 0 0 0 49 86 0 0 0 0 0 0 0 0 0 0 108 108 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 108 108 0 0 0 0 0 0 1 0 0 0 59 27 0 0 0 0 0 0 82 32 0 0 0 0
,
 32 80 0 0 0 0 0 0 30 77 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 108 108 0 0 0 0 0 0 0 0 0 0 27 59 0 0 0 0 0 0 32 82 0 0 0 0 32 82 0 0 0 0 0 0 50 77 0 0

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

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