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## G = C6×C7⋊C3order 126 = 2·32·7

### Direct product of C6 and C7⋊C3

Aliases: C6×C7⋊C3, C42⋊C3, C14⋊C32, C214C6, C72(C3×C6), SmallGroup(126,10)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C7⋊C3
G = < a,b,c | a6=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Character table of C6×C7⋊C3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 7A 7B 14A 14B 21A 21B 21C 21D 42A 42B 42C 42D size 1 1 1 1 7 7 7 7 7 7 1 1 7 7 7 7 7 7 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ65 ζ6 -1 ζ65 -1 ζ6 ζ6 ζ65 1 1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 -1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 -1 -1 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ8 1 -1 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ6 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ9 1 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ10 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ11 1 -1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ65 ζ6 ζ6 ζ6 ζ65 -1 ζ65 -1 1 1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ12 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ13 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ14 1 -1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 -1 -1 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ15 1 -1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ6 ζ65 ζ65 ζ65 ζ6 -1 ζ6 -1 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ16 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ17 1 -1 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ65 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 1 1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ18 1 -1 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ6 ζ65 -1 ζ6 -1 ζ65 ζ65 ζ6 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ19 3 3 3 3 0 0 0 0 0 0 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ20 3 -3 3 3 0 0 0 0 0 0 -3 -3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ21 3 3 3 3 0 0 0 0 0 0 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ22 3 -3 3 3 0 0 0 0 0 0 -3 -3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ23 3 -3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ3ζ76-ζ3ζ75-ζ3ζ73 complex faithful ρ24 3 -3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ32ζ74-ζ32ζ72-ζ32ζ7 complex faithful ρ25 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex lifted from C3×C7⋊C3 ρ26 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex lifted from C3×C7⋊C3 ρ27 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 complex lifted from C3×C7⋊C3 ρ28 3 -3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ3ζ74-ζ3ζ72-ζ3ζ7 complex faithful ρ29 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex lifted from C3×C7⋊C3 ρ30 3 -3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ32ζ76-ζ32ζ75-ζ32ζ73 complex faithful

Smallest permutation representation of C6×C7⋊C3
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 15 8)(2 17 12)(3 19 9)(4 21 13)(5 16 10)(6 18 14)(7 20 11)(22 36 29)(23 38 33)(24 40 30)(25 42 34)(26 37 31)(27 39 35)(28 41 32)

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

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

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

C6×C7⋊C3 is a maximal subgroup of   C6.F7

Matrix representation of C6×C7⋊C3 in GL3(𝔽43) generated by

 37 0 0 0 37 0 0 0 37
,
 24 25 1 1 0 0 0 1 0
,
 36 0 0 3 7 7 0 36 0
G:=sub<GL(3,GF(43))| [37,0,0,0,37,0,0,0,37],[24,1,0,25,0,1,1,0,0],[36,3,0,0,7,36,0,7,0] >;

C6×C7⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_7\rtimes C_3
% in TeX

G:=Group("C6xC7:C3");
// GroupNames label

G:=SmallGroup(126,10);
// by ID

G=gap.SmallGroup(126,10);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,295]);
// Polycyclic

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