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## G = C7×3- 1+2order 189 = 33·7

### Direct product of C7 and 3- 1+2

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

Aliases: C7×3- 1+2, C9⋊C21, C634C3, C32.C21, C21.8C32, (C3×C21).1C3, C3.2(C3×C21), SmallGroup(189,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C7×3- 1+2
 Chief series C1 — C3 — C21 — C63 — C7×3- 1+2
 Lower central C1 — C3 — C7×3- 1+2
 Upper central C1 — C21 — C7×3- 1+2

Generators and relations for C7×3- 1+2
G = < a,b,c | a7=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C7×3- 1+2
On 63 points
Generators in S63
(1 39 56 52 20 10 30)(2 40 57 53 21 11 31)(3 41 58 54 22 12 32)(4 42 59 46 23 13 33)(5 43 60 47 24 14 34)(6 44 61 48 25 15 35)(7 45 62 49 26 16 36)(8 37 63 50 27 17 28)(9 38 55 51 19 18 29)
(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)(55 56 57 58 59 60 61 62 63)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 58 61)(57 63 60)

G:=sub<Sym(63)| (1,39,56,52,20,10,30)(2,40,57,53,21,11,31)(3,41,58,54,22,12,32)(4,42,59,46,23,13,33)(5,43,60,47,24,14,34)(6,44,61,48,25,15,35)(7,45,62,49,26,16,36)(8,37,63,50,27,17,28)(9,38,55,51,19,18,29), (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)(55,56,57,58,59,60,61,62,63), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,58,61)(57,63,60)>;

G:=Group( (1,39,56,52,20,10,30)(2,40,57,53,21,11,31)(3,41,58,54,22,12,32)(4,42,59,46,23,13,33)(5,43,60,47,24,14,34)(6,44,61,48,25,15,35)(7,45,62,49,26,16,36)(8,37,63,50,27,17,28)(9,38,55,51,19,18,29), (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)(55,56,57,58,59,60,61,62,63), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,58,61)(57,63,60) );

G=PermutationGroup([[(1,39,56,52,20,10,30),(2,40,57,53,21,11,31),(3,41,58,54,22,12,32),(4,42,59,46,23,13,33),(5,43,60,47,24,14,34),(6,44,61,48,25,15,35),(7,45,62,49,26,16,36),(8,37,63,50,27,17,28),(9,38,55,51,19,18,29)], [(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),(55,56,57,58,59,60,61,62,63)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,22,25),(21,27,24),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(47,53,50),(48,51,54),(55,58,61),(57,63,60)]])

C7×3- 1+2 is a maximal subgroup of   D63⋊C3

77 conjugacy classes

 class 1 3A 3B 3C 3D 7A ··· 7F 9A ··· 9F 21A ··· 21L 21M ··· 21X 63A ··· 63AJ order 1 3 3 3 3 7 ··· 7 9 ··· 9 21 ··· 21 21 ··· 21 63 ··· 63 size 1 1 1 3 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3

77 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C7 C21 C21 3- 1+2 C7×3- 1+2 kernel C7×3- 1+2 C63 C3×C21 3- 1+2 C9 C32 C7 C1 # reps 1 6 2 6 36 12 2 12

Matrix representation of C7×3- 1+2 in GL3(𝔽127) generated by

 4 0 0 0 4 0 0 0 4
,
 0 1 0 0 0 107 1 0 0
,
 1 0 0 0 107 0 0 0 19
G:=sub<GL(3,GF(127))| [4,0,0,0,4,0,0,0,4],[0,0,1,1,0,0,0,107,0],[1,0,0,0,107,0,0,0,19] >;

C7×3- 1+2 in GAP, Magma, Sage, TeX

C_7\times 3_-^{1+2}
% in TeX

G:=Group("C7xES-(3,1)");
// GroupNames label

G:=SmallGroup(189,11);
// by ID

G=gap.SmallGroup(189,11);
# by ID

G:=PCGroup([4,-3,-3,-7,-3,252,529]);
// Polycyclic

G:=Group<a,b,c|a^7=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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