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## G = C9×C7⋊C3order 189 = 33·7

### Direct product of C9 and C7⋊C3

Aliases: C9×C7⋊C3, C631C3, C21.1C32, C7⋊C94C3, C71(C3×C9), C3.1(C3×C7⋊C3), (C3×C7⋊C3).3C3, SmallGroup(189,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C9×C7⋊C3
 Chief series C1 — C7 — C21 — C3×C7⋊C3 — C9×C7⋊C3
 Lower central C7 — C9×C7⋊C3
 Upper central C1 — C9

Generators and relations for C9×C7⋊C3
G = < a,b,c | a9=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C9×C7⋊C3
On 63 points
Generators in S63
(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)
(1 48 24 12 34 44 56)(2 49 25 13 35 45 57)(3 50 26 14 36 37 58)(4 51 27 15 28 38 59)(5 52 19 16 29 39 60)(6 53 20 17 30 40 61)(7 54 21 18 31 41 62)(8 46 22 10 32 42 63)(9 47 23 11 33 43 55)
(1 7 4)(2 8 5)(3 9 6)(10 60 45)(11 61 37)(12 62 38)(13 63 39)(14 55 40)(15 56 41)(16 57 42)(17 58 43)(18 59 44)(19 35 46)(20 36 47)(21 28 48)(22 29 49)(23 30 50)(24 31 51)(25 32 52)(26 33 53)(27 34 54)

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

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

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

C9×C7⋊C3 is a maximal subgroup of   C95F7

45 conjugacy classes

 class 1 3A 3B 3C ··· 3H 7A 7B 9A ··· 9F 9G ··· 9R 21A 21B 21C 21D 63A ··· 63L order 1 3 3 3 ··· 3 7 7 9 ··· 9 9 ··· 9 21 21 21 21 63 ··· 63 size 1 1 1 7 ··· 7 3 3 1 ··· 1 7 ··· 7 3 3 3 3 3 ··· 3

45 irreducible representations

 dim 1 1 1 1 1 3 3 3 type + image C1 C3 C3 C3 C9 C7⋊C3 C3×C7⋊C3 C9×C7⋊C3 kernel C9×C7⋊C3 C7⋊C9 C63 C3×C7⋊C3 C7⋊C3 C9 C3 C1 # reps 1 4 2 2 18 2 4 12

Matrix representation of C9×C7⋊C3 in GL3(𝔽127) generated by

 22 0 0 0 22 0 0 0 22
,
 104 105 1 1 0 0 0 1 0
,
 19 0 0 37 108 108 0 19 0
G:=sub<GL(3,GF(127))| [22,0,0,0,22,0,0,0,22],[104,1,0,105,0,1,1,0,0],[19,37,0,0,108,19,0,108,0] >;

C9×C7⋊C3 in GAP, Magma, Sage, TeX

C_9\times C_7\rtimes C_3
% in TeX

G:=Group("C9xC7:C3");
// GroupNames label

G:=SmallGroup(189,3);
// by ID

G=gap.SmallGroup(189,3);
# by ID

G:=PCGroup([4,-3,-3,-3,-7,29,867]);
// Polycyclic

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

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