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## G = C2×C27⋊C3order 162 = 2·34

### Direct product of C2 and C27⋊C3

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×C27⋊C3, C54⋊C3, C18.C9, C9.C18, C272C6, C32.C18, C18.2C32, (C3×C6).C9, C9.1(C3×C6), (C3×C9).4C6, C6.3(C3×C9), (C3×C18).3C3, C3.3(C3×C18), SmallGroup(162,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C27⋊C3
 Chief series C1 — C3 — C9 — C3×C9 — C27⋊C3 — C2×C27⋊C3
 Lower central C1 — C3 — C2×C27⋊C3
 Upper central C1 — C18 — C2×C27⋊C3

Generators and relations for C2×C27⋊C3
G = < a,b,c | a2=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >

Smallest permutation representation of C2×C27⋊C3
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 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)

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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)>;

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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54) );

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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54)]])

C2×C27⋊C3 is a maximal subgroup of   C27⋊C12

66 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 9A ··· 9F 9G 9H 9I 9J 18A ··· 18F 18G 18H 18I 18J 27A ··· 27R 54A ··· 54R order 1 2 3 3 3 3 6 6 6 6 9 ··· 9 9 9 9 9 18 ··· 18 18 18 18 18 27 ··· 27 54 ··· 54 size 1 1 1 1 3 3 1 1 3 3 1 ··· 1 3 3 3 3 1 ··· 1 3 3 3 3 3 ··· 3 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 C27⋊C3 C2×C27⋊C3 kernel C2×C27⋊C3 C27⋊C3 C54 C3×C18 C27 C3×C9 C18 C3×C6 C9 C32 C2 C1 # reps 1 1 6 2 6 2 12 6 12 6 6 6

Matrix representation of C2×C27⋊C3 in GL3(𝔽109) generated by

 108 0 0 0 108 0 0 0 108
,
 0 1 0 0 0 63 27 0 0
,
 1 0 0 0 63 0 0 0 45
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[0,0,27,1,0,0,0,63,0],[1,0,0,0,63,0,0,0,45] >;

C2×C27⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{27}\rtimes C_3
% in TeX

G:=Group("C2xC27:C3");
// GroupNames label

G:=SmallGroup(162,27);
// by ID

G=gap.SmallGroup(162,27);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,96,457,78]);
// Polycyclic

G:=Group<a,b,c|a^2=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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