A4xD7 - GroupNames

G = A₄×D₇ order 168 = 2³·3·7

Direct product of A₄ and D₇

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×D₇, C₇⋊₃(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 >

C₁

₃C₂

₇C₂

₂₁C₂

₄C₃

C₇

C₂²

₂₁C₂²

₂₈C₆

D₇

₃C₁₄

₃D₇

₄C₂₁

₇C₂³

A₄

C₂×C₁₄

₃D₁₄

₄C₃×D₇

₇C₂×A₄

C₂²×D₇

C₇×A₄

A₄×D₇

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

Export

Subgroup lattice of A₄×D₇ in TeX
Character table of A₄×D₇ in TeX

G = A4×D7 order 168 = 23·3·7

Direct product of A4 and D7

G = A₄×D₇ order 168 = 2³·3·7

Direct product of A₄ and D₇