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## G = A4×D7order 168 = 23·3·7

### Direct product of A4 and D7

Aliases: A4×D7, C73(C2×A4), C22⋊(C3×D7), (C7×A4)⋊2C2, (C2×C14)⋊1C6, (C22×D7)⋊1C3, SmallGroup(168,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — A4×D7
 Chief series C1 — C7 — C2×C14 — C7×A4 — A4×D7
 Lower central C2×C14 — A4×D7
 Upper central C1

Generators and relations for A4×D7
G = < a,b,c,d,e | a2=b2=c3=d7=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

3C2
7C2
21C2
4C3
21C22
21C22
28C6
3C14
3D7
4C21
7C23
3D14
3D14

Character table of A4×D7

 class 1 2A 2B 2C 3A 3B 6A 6B 7A 7B 7C 14A 14B 14C 21A 21B 21C 21D 21E 21F size 1 3 7 21 4 4 28 28 2 2 2 6 6 6 8 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 6 ρ5 1 1 -1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ7 2 2 0 0 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ8 2 2 0 0 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ9 2 2 0 0 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ10 2 2 0 0 -1-√-3 -1+√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ74+ζ3ζ73 complex lifted from C3×D7 ρ11 2 2 0 0 -1-√-3 -1+√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ75+ζ3ζ72 complex lifted from C3×D7 ρ12 2 2 0 0 -1+√-3 -1-√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 ζ32ζ76+ζ32ζ7 complex lifted from C3×D7 ρ13 2 2 0 0 -1+√-3 -1-√-3 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ74+ζ3ζ73 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 ζ32ζ75+ζ32ζ72 complex lifted from C3×D7 ρ14 2 2 0 0 -1-√-3 -1+√-3 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ75+ζ32ζ72 ζ32ζ74+ζ32ζ73 ζ32ζ76+ζ32ζ7 ζ3ζ76+ζ3ζ7 complex lifted from C3×D7 ρ15 2 2 0 0 -1+√-3 -1-√-3 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ32ζ76+ζ32ζ7 ζ32ζ75+ζ32ζ72 ζ3ζ76+ζ3ζ7 ζ3ζ75+ζ3ζ72 ζ3ζ74+ζ3ζ73 ζ32ζ74+ζ32ζ73 complex lifted from C3×D7 ρ16 3 -1 -3 1 0 0 0 0 3 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 -1 3 -1 0 0 0 0 3 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ18 6 -2 0 0 0 0 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 0 0 0 orthogonal faithful ρ19 6 -2 0 0 0 0 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 0 0 0 orthogonal faithful ρ20 6 -2 0 0 0 0 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 0 0 0 orthogonal faithful

Permutation representations of A4×D7
On 28 points - transitive group 28T29
Generators in S28
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(8 15 22)(9 16 23)(10 17 24)(11 18 25)(12 19 26)(13 20 27)(14 21 28)
(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)

G:=sub<Sym(28)| (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (8,15,22)(9,16,23)(10,17,24)(11,18,25)(12,19,26)(13,20,27)(14,21,28), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)>;

G:=Group( (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (8,15,22)(9,16,23)(10,17,24)(11,18,25)(12,19,26)(13,20,27)(14,21,28), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27) );

G=PermutationGroup([[(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)], [(8,15,22),(9,16,23),(10,17,24),(11,18,25),(12,19,26),(13,20,27),(14,21,28)], [(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27)]])

G:=TransitiveGroup(28,29);

A4×D7 is a maximal quotient of   Dic7.2A4

Matrix representation of A4×D7 in GL5(𝔽43)

 1 0 0 0 0 0 1 0 0 0 0 0 1 26 0 0 0 0 42 0 0 0 23 41 42
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 26 0 0 23 42 41 0 0 0 0 42
,
 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 23 42 42 0 0 0 1 0
,
 27 2 0 0 0 42 35 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 27 40 0 0 0 42 16 0 0 0 0 0 42 0 0 0 0 0 42 0 0 0 0 0 42

G:=sub<GL(5,GF(43))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,23,0,0,26,42,41,0,0,0,0,42],[1,0,0,0,0,0,1,0,0,0,0,0,1,23,0,0,0,0,42,0,0,0,26,41,42],[36,0,0,0,0,0,36,0,0,0,0,0,1,23,0,0,0,0,42,1,0,0,0,42,0],[27,42,0,0,0,2,35,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[27,42,0,0,0,40,16,0,0,0,0,0,42,0,0,0,0,0,42,0,0,0,0,0,42] >;

A4×D7 in GAP, Magma, Sage, TeX

A_4\times D_7
% in TeX

G:=Group("A4xD7");
// GroupNames label

G:=SmallGroup(168,48);
// by ID

G=gap.SmallGroup(168,48);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-7,142,68,3604]);
// Polycyclic

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

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