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## G = C3×D27order 162 = 2·34

### Direct product of C3 and D27

Aliases: C3×D27, C273C6, C32.2D9, (C3×C27)⋊2C2, C9.2(C3×S3), (C3×C9).5S3, C3.2(C3×D9), SmallGroup(162,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C3×D27
 Chief series C1 — C3 — C9 — C27 — C3×C27 — C3×D27
 Lower central C27 — C3×D27
 Upper central C1 — C3

Generators and relations for C3×D27
G = < a,b,c | a3=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C3×D27
On 54 points
Generators in S54
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 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)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)

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

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

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

C3×D27 is a maximal subgroup of   C273C18  C81⋊C6  C322D27  He3.3D9  He3.4D9  He3.5D9
C3×D27 is a maximal quotient of   C32⋊D27  C81⋊C6

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9I 27A ··· 27AA order 1 2 3 3 3 3 3 6 6 9 ··· 9 27 ··· 27 size 1 27 1 1 2 2 2 27 27 2 ··· 2 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 C3×S3 D9 D27 C3×D9 C3×D27 kernel C3×D27 C3×C27 D27 C27 C3×C9 C9 C32 C3 C3 C1 # reps 1 1 2 2 1 2 3 9 6 18

Matrix representation of C3×D27 in GL2(𝔽109) generated by

 63 0 0 63
,
 22 0 0 5
,
 0 1 1 0
G:=sub<GL(2,GF(109))| [63,0,0,63],[22,0,0,5],[0,1,1,0] >;

C3×D27 in GAP, Magma, Sage, TeX

C_3\times D_{27}
% in TeX

G:=Group("C3xD27");
// GroupNames label

G:=SmallGroup(162,7);
// by ID

G=gap.SmallGroup(162,7);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,452,237,1803,138,2704]);
// Polycyclic

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

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