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## G = C7×F7order 294 = 2·3·72

### Direct product of C7 and F7

Aliases: C7×F7, C7⋊C42, D7⋊C21, C721C6, C7⋊C3⋊C14, (C7×D7)⋊1C3, (C7×C7⋊C3)⋊1C2, SmallGroup(294,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7×F7
 Chief series C1 — C7 — C72 — C7×C7⋊C3 — C7×F7
 Lower central C7 — C7×F7
 Upper central C1 — C7

Generators and relations for C7×F7
G = < a,b,c | a7=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C7×F7
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 2 3 4 5 6 7)(8 12 9 13 10 14 11)(15 17 19 21 16 18 20)(22 28 27 26 25 24 23)(29 32 35 31 34 30 33)(36 41 39 37 42 40 38)
(1 36 8 22 15 29)(2 37 9 23 16 30)(3 38 10 24 17 31)(4 39 11 25 18 32)(5 40 12 26 19 33)(6 41 13 27 20 34)(7 42 14 28 21 35)

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

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

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

49 conjugacy classes

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

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 6 6 type + + + image C1 C2 C3 C6 C7 C14 C21 C42 F7 C7×F7 kernel C7×F7 C7×C7⋊C3 C7×D7 C72 F7 C7⋊C3 D7 C7 C7 C1 # reps 1 1 2 2 6 6 12 12 1 6

Matrix representation of C7×F7 in GL6(𝔽43)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 19 34 38 25 0 41 42 39 5 11 0 0 16 0 0 0 0 0 0 11 0 0 0 0 0 0 21 0 0 0 0 0 0 35
,
 26 33 0 0 0 0 16 9 0 0 0 0 0 0 0 1 0 0 0 3 36 34 33 27 0 0 1 0 0 0 3 0 6 10 11 17

G:=sub<GL(6,GF(43))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,41,0,0,0,0,19,42,16,0,0,0,34,39,0,11,0,0,38,5,0,0,21,0,25,11,0,0,0,35],[26,16,0,0,0,3,33,9,0,3,0,0,0,0,0,36,1,6,0,0,1,34,0,10,0,0,0,33,0,11,0,0,0,27,0,17] >;

C7×F7 in GAP, Magma, Sage, TeX

C_7\times F_7
% in TeX

G:=Group("C7xF7");
// GroupNames label

G:=SmallGroup(294,8);
// by ID

G=gap.SmallGroup(294,8);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,4035,1351]);
// Polycyclic

G:=Group<a,b,c|a^7=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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