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## G = C14×C7⋊C3order 294 = 2·3·72

### Direct product of C14 and C7⋊C3

Aliases: C14×C7⋊C3, C14⋊C21, C72C42, C728C6, (C7×C14)⋊1C3, SmallGroup(294,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C14×C7⋊C3
 Chief series C1 — C7 — C72 — C7×C7⋊C3 — C14×C7⋊C3
 Lower central C7 — C14×C7⋊C3
 Upper central C1 — C14

Generators and relations for C14×C7⋊C3
G = < a,b,c | a14=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C14×C7⋊C3
On 42 points
Generators in S42
(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 3 5 7 9 11 13)(2 4 6 8 10 12 14)(15 19 23 27 17 21 25)(16 20 24 28 18 22 26)(29 37 31 39 33 41 35)(30 38 32 40 34 42 36)
(1 30 16)(2 31 17)(3 32 18)(4 33 19)(5 34 20)(6 35 21)(7 36 22)(8 37 23)(9 38 24)(10 39 25)(11 40 26)(12 41 27)(13 42 28)(14 29 15)

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 6A 6B 7A ··· 7F 7G ··· 7T 14A ··· 14F 14G ··· 14T 21A ··· 21L 42A ··· 42L order 1 2 3 3 6 6 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 7 7 7 7 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 7 ··· 7 7 ··· 7

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C6 C7 C14 C21 C42 C7⋊C3 C2×C7⋊C3 C7×C7⋊C3 C14×C7⋊C3 kernel C14×C7⋊C3 C7×C7⋊C3 C7×C14 C72 C2×C7⋊C3 C7⋊C3 C14 C7 C14 C7 C2 C1 # reps 1 1 2 2 6 6 12 12 2 2 12 12

Matrix representation of C14×C7⋊C3 in GL3(𝔽43) generated by

 22 0 0 0 22 0 0 0 22
,
 11 9 6 0 35 0 0 0 21
,
 6 0 0 0 0 1 10 7 37
G:=sub<GL(3,GF(43))| [22,0,0,0,22,0,0,0,22],[11,0,0,9,35,0,6,0,21],[6,0,10,0,0,7,0,1,37] >;

C14×C7⋊C3 in GAP, Magma, Sage, TeX

C_{14}\times C_7\rtimes C_3
% in TeX

G:=Group("C14xC7:C3");
// GroupNames label

G:=SmallGroup(294,15);
// by ID

G=gap.SmallGroup(294,15);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,679]);
// Polycyclic

G:=Group<a,b,c|a^14=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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