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## G = C3×C27⋊C6order 486 = 2·35

### Direct product of C3 and C27⋊C6

Aliases: C3×C27⋊C6, D27⋊C32, C33.4D9, C27⋊(C3×C6), (C3×D27)⋊C3, C27⋊C34C6, (C3×C27)⋊4C6, C9.4(S3×C32), C32.8(C3×D9), C3.3(C32×D9), (C32×C9).18S3, (C3×C27⋊C3)⋊1C2, (C3×C9).22(C3×S3), SmallGroup(486,113)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C27⋊C6
G = < a,b,c | a3=b27=c6=1, ab=ba, ac=ca, cbc-1=b17 >

Subgroups: 312 in 66 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C3×C9, C33, D27, C3×D9, S3×C32, C3×C27, C3×C27, C27⋊C3, C27⋊C3, C32×C9, C3×D27, C27⋊C6, C32×D9, C3×C27⋊C3, C3×C27⋊C6
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C3×D9, S3×C32, C27⋊C6, C32×D9, C3×C27⋊C6

Smallest permutation representation of C3×C27⋊C6
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 30 19 39 10 48)(2 38)(3 46 12 37 21 28)(4 54 22 36 13 45)(5 35)(6 43 15 34 24 52)(7 51 25 33 16 42)(8 32)(9 40 18 31 27 49)(11 29)(14 53)(17 50)(20 47)(23 44)(26 41)

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 6A ··· 6H 9A ··· 9I 9J ··· 9O 27A ··· 27AA order 1 2 3 3 3 3 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 27 ··· 27 size 1 27 1 1 2 2 2 3 ··· 3 27 ··· 27 2 ··· 2 6 ··· 6 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 D9 C3×D9 C27⋊C6 C3×C27⋊C6 kernel C3×C27⋊C6 C3×C27⋊C3 C3×D27 C27⋊C6 C3×C27 C27⋊C3 C32×C9 C3×C9 C33 C32 C3 C1 # reps 1 1 2 6 2 6 1 8 3 24 3 6

Matrix representation of C3×C27⋊C6 in GL6(𝔽109)

 63 0 0 0 0 0 0 63 0 0 0 0 0 0 63 0 0 0 0 0 0 63 0 0 0 0 0 0 63 0 0 0 0 0 0 63
,
 0 63 0 0 0 0 0 0 63 0 0 0 38 0 0 0 0 0 0 0 0 0 0 66 0 0 0 45 0 0 0 0 0 0 45 0
,
 0 0 0 45 0 0 0 0 0 0 63 0 0 0 0 0 0 1 45 0 0 0 0 0 0 63 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(109))| [63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63],[0,0,38,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,45,0,0,0,66,0,0],[0,0,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,0,0,0] >;

C3×C27⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_{27}\rtimes C_6
% in TeX

G:=Group("C3xC27:C6");
// GroupNames label

G:=SmallGroup(486,113);
// by ID

G=gap.SmallGroup(486,113);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2163,2169,381,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^3=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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