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## G = D7×C7⋊C3order 294 = 2·3·72

### Direct product of D7 and C7⋊C3

Aliases: D7×C7⋊C3, C722C6, C72(C3×D7), (C7×D7)⋊2C3, (C7×C7⋊C3)⋊2C2, C75(C2×C7⋊C3), SmallGroup(294,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — D7×C7⋊C3
 Chief series C1 — C7 — C72 — C7×C7⋊C3 — D7×C7⋊C3
 Lower central C72 — D7×C7⋊C3
 Upper central C1

Generators and relations for D7×C7⋊C3
G = < a,b,c,d | a7=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of D7×C7⋊C3

 class 1 2 3A 3B 6A 6B 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 14A 14B 21A 21B 21C 21D 21E 21F size 1 7 7 7 49 49 2 2 2 3 3 6 6 6 6 6 6 21 21 14 14 14 14 14 14 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 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 linear of order 2 ρ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 1 -1 -1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 1 -1 -1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ7 2 0 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 2 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 0 0 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ8 2 0 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 2 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 0 0 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ9 2 0 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 2 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 0 0 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ10 2 0 -1-√-3 -1+√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 2 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 0 0 ζ3ζ76+ζ3ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 complex lifted from C3×D7 ρ11 2 0 -1-√-3 -1+√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 2 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 0 0 ζ3ζ75+ζ3ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 complex lifted from C3×D7 ρ12 2 0 -1+√-3 -1-√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 2 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 0 0 ζ32ζ75+ζ32ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 complex lifted from C3×D7 ρ13 2 0 -1+√-3 -1-√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 2 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 0 0 ζ32ζ76+ζ32ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 complex lifted from C3×D7 ρ14 2 0 -1+√-3 -1-√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 2 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 0 0 ζ32ζ74+ζ32ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 complex lifted from C3×D7 ρ15 2 0 -1-√-3 -1+√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 2 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 0 0 ζ3ζ74+ζ3ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 complex lifted from C3×D7 ρ16 3 -3 0 0 0 0 3 3 3 -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 0 0 0 0 0 0 complex lifted from C2×C7⋊C3 ρ17 3 3 0 0 0 0 3 3 3 -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 0 0 0 0 0 0 complex lifted from C7⋊C3 ρ18 3 -3 0 0 0 0 3 3 3 -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 0 0 0 0 0 0 complex lifted from C2×C7⋊C3 ρ19 3 3 0 0 0 0 3 3 3 -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 0 0 0 0 0 0 complex lifted from C7⋊C3 ρ20 6 0 0 0 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 -1-√-7 -1+√-7 -ζ76-ζ72+ζ7 -ζ75-ζ74+ζ72 ζ75-ζ73-ζ72 ζ76-ζ75-ζ7 ζ74-ζ73-ζ7 -ζ76-ζ74+ζ73 0 0 0 0 0 0 0 0 complex faithful ρ21 6 0 0 0 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 -1-√-7 -1+√-7 -ζ75-ζ74+ζ72 ζ74-ζ73-ζ7 -ζ76-ζ74+ζ73 ζ75-ζ73-ζ72 -ζ76-ζ72+ζ7 ζ76-ζ75-ζ7 0 0 0 0 0 0 0 0 complex faithful ρ22 6 0 0 0 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 -1+√-7 -1-√-7 -ζ76-ζ74+ζ73 ζ76-ζ75-ζ7 -ζ76-ζ72+ζ7 ζ74-ζ73-ζ7 ζ75-ζ73-ζ72 -ζ75-ζ74+ζ72 0 0 0 0 0 0 0 0 complex faithful ρ23 6 0 0 0 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 -1-√-7 -1+√-7 ζ74-ζ73-ζ7 -ζ76-ζ72+ζ7 ζ76-ζ75-ζ7 -ζ76-ζ74+ζ73 -ζ75-ζ74+ζ72 ζ75-ζ73-ζ72 0 0 0 0 0 0 0 0 complex faithful ρ24 6 0 0 0 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 -1+√-7 -1-√-7 ζ75-ζ73-ζ72 -ζ76-ζ74+ζ73 ζ74-ζ73-ζ7 -ζ75-ζ74+ζ72 ζ76-ζ75-ζ7 -ζ76-ζ72+ζ7 0 0 0 0 0 0 0 0 complex faithful ρ25 6 0 0 0 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 -1+√-7 -1-√-7 ζ76-ζ75-ζ7 ζ75-ζ73-ζ72 -ζ75-ζ74+ζ72 -ζ76-ζ72+ζ7 -ζ76-ζ74+ζ73 ζ74-ζ73-ζ7 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of D7×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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 33)(9 32)(10 31)(11 30)(12 29)(13 35)(14 34)(15 40)(16 39)(17 38)(18 37)(19 36)(20 42)(21 41)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)(22 28 27 26 25 24 23)(29 34 32 30 35 33 31)(36 39 42 38 41 37 40)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,33)(9,32)(10,31)(11,30)(12,29)(13,35)(14,34)(15,40)(16,39)(17,38)(18,37)(19,36)(20,42)(21,41), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18)(22,28,27,26,25,24,23)(29,34,32,30,35,33,31)(36,39,42,38,41,37,40), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)>;

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

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

Matrix representation of D7×C7⋊C3 in GL5(𝔽43)

 8 42 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 42 0 0 0 20 35 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 10 19 0 0 33 33 42 0 0 34 34 25
,
 6 0 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(5,GF(43))| [8,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,20,0,0,0,42,35,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,33,34,0,0,10,33,34,0,0,19,42,25],[6,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

D7×C7⋊C3 in GAP, Magma, Sage, TeX

D_7\times C_7\rtimes C_3
% in TeX

G:=Group("D7xC7:C3");
// GroupNames label

G:=SmallGroup(294,9);
// by ID

G=gap.SmallGroup(294,9);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,434,679]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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