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## G = C9×F7order 378 = 2·33·7

### Direct product of C9 and F7

Aliases: C9×F7, C639C6, C7⋊C3⋊C18, C7⋊C97C6, C7⋊C184C3, C71(C3×C18), D71(C3×C9), (C9×D7)⋊1C3, C3.1(C3×F7), C21.1(C3×C6), (C3×F7).3C3, (C3×D7).1C32, (C9×C7⋊C3)⋊3C2, (C3×C7⋊C3).5C6, SmallGroup(378,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C9×F7
 Chief series C1 — C7 — C21 — C63 — C9×C7⋊C3 — C9×F7
 Lower central C7 — C9×F7
 Upper central C1 — C9

Generators and relations for C9×F7
G = < a,b,c | a9=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C9×F7
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 35 24 47 39 16 56)(2 36 25 48 40 17 57)(3 28 26 49 41 18 58)(4 29 27 50 42 10 59)(5 30 19 51 43 11 60)(6 31 20 52 44 12 61)(7 32 21 53 45 13 62)(8 33 22 54 37 14 63)(9 34 23 46 38 15 55)
(1 4 7)(2 5 8)(3 6 9)(10 32 47 27 62 39)(11 33 48 19 63 40)(12 34 49 20 55 41)(13 35 50 21 56 42)(14 36 51 22 57 43)(15 28 52 23 58 44)(16 29 53 24 59 45)(17 30 54 25 60 37)(18 31 46 26 61 38)

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 6A ··· 6H 7 9A ··· 9F 9G ··· 9R 18A ··· 18R 21A 21B 63A ··· 63F order 1 2 3 3 3 ··· 3 6 ··· 6 7 9 ··· 9 9 ··· 9 18 ··· 18 21 21 63 ··· 63 size 1 7 1 1 7 ··· 7 7 ··· 7 6 1 ··· 1 7 ··· 7 7 ··· 7 6 6 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 6 6 6 type + + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 F7 C3×F7 C9×F7 kernel C9×F7 C9×C7⋊C3 C7⋊C18 C9×D7 C3×F7 C7⋊C9 C63 C3×C7⋊C3 F7 C7⋊C3 C9 C3 C1 # reps 1 1 4 2 2 4 2 2 18 18 1 2 6

Matrix representation of C9×F7 in GL6(𝔽127)

 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52 0 0 0 0 0 0 52
,
 126 126 126 126 126 126 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 0 0 0 19 0 0 0 19 0 0 0 19 0 0 0 0 0 0 0 0 0 0 19 0 0 0 19 0 0 0 19 0 0 0 0

G:=sub<GL(6,GF(127))| [52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[126,1,0,0,0,0,126,0,1,0,0,0,126,0,0,1,0,0,126,0,0,0,1,0,126,0,0,0,0,1,126,0,0,0,0,0],[0,0,19,0,0,0,0,0,0,0,0,19,0,19,0,0,0,0,0,0,0,0,19,0,19,0,0,0,0,0,0,0,0,19,0,0] >;

C9×F7 in GAP, Magma, Sage, TeX

C_9\times F_7
% in TeX

G:=Group("C9xF7");
// GroupNames label

G:=SmallGroup(378,7);
// by ID

G=gap.SmallGroup(378,7);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,57,8104,2709]);
// Polycyclic

G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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