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## G = C6×F7order 252 = 22·32·7

### Direct product of C6 and F7

Aliases: C6×F7, C7⋊C62, C422C6, D14⋊C32, C14⋊(C3×C6), D7⋊(C3×C6), (C6×D7)⋊C3, C213(C2×C6), (C3×D7)⋊2C6, (C2×C7⋊C3)⋊C6, C7⋊C3⋊(C2×C6), (C6×C7⋊C3)⋊2C2, (C3×C7⋊C3)⋊3C22, SmallGroup(252,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C6×F7
 Chief series C1 — C7 — C21 — C3×C7⋊C3 — C3×F7 — C6×F7
 Lower central C7 — C6×F7
 Upper central C1 — C6

Generators and relations for C6×F7
G = < a,b,c | a6=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 216 in 60 conjugacy classes, 34 normal (13 characteristic)
C1, C2, C2, C3, C3, C22, C6, C6, C7, C32, C2×C6, D7, C14, C3×C6, C7⋊C3, C21, D14, C62, F7, C2×C7⋊C3, C3×D7, C42, C3×C7⋊C3, C2×F7, C6×D7, C3×F7, C6×C7⋊C3, C6×F7
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C62, F7, C2×F7, C3×F7, C6×F7

Smallest permutation representation of C6×F7
On 42 points
Generators in S42
(1 29 15 22 8 36)(2 30 16 23 9 37)(3 31 17 24 10 38)(4 32 18 25 11 39)(5 33 19 26 12 40)(6 34 20 27 13 41)(7 35 21 28 14 42)
(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)
(1 22)(2 25 3 28 5 27)(4 24 7 26 6 23)(8 29)(9 32 10 35 12 34)(11 31 14 33 13 30)(15 36)(16 39 17 42 19 41)(18 38 21 40 20 37)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 6A 6B 6C ··· 6X 7 14 21A 21B 42A 42B order 1 2 2 2 3 3 3 ··· 3 6 6 6 ··· 6 7 14 21 21 42 42 size 1 1 7 7 1 1 7 ··· 7 1 1 7 ··· 7 6 6 6 6 6 6

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 F7 C2×F7 C3×F7 C6×F7 kernel C6×F7 C3×F7 C6×C7⋊C3 C2×F7 C6×D7 F7 C2×C7⋊C3 C3×D7 C42 C6 C3 C2 C1 # reps 1 2 1 6 2 12 6 4 2 1 1 2 2

Matrix representation of C6×F7 in GL6(𝔽43)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 42 42 42 42 42 42 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
,
 42 0 0 0 0 0 0 0 0 0 0 42 0 0 0 42 0 0 0 42 0 0 0 0 1 1 1 1 1 1 0 0 0 0 42 0

G:=sub<GL(6,GF(43))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[42,0,0,0,1,0,0,0,0,42,1,0,0,0,0,0,1,0,0,0,42,0,1,0,0,0,0,0,1,42,0,42,0,0,1,0] >;

C6×F7 in GAP, Magma, Sage, TeX

C_6\times F_7
% in TeX

G:=Group("C6xF7");
// GroupNames label

G:=SmallGroup(252,28);
// by ID

G=gap.SmallGroup(252,28);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,5404,914]);
// Polycyclic

G:=Group<a,b,c|a^6=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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