D7:A4 - GroupNames

G = D₇⋊A₄ order 168 = 2³·3·7

The semidirect product of D₇ and A₄ acting via A₄/C₂²=C₃

metabelian, soluble, monomial, A-group

Aliases: D₇⋊A₄, C₂²⋊F₇, C₇⋊A₄⋊C₂, C₇⋊(C₂×A₄), (C₂×C₁₄)⋊₂C₆, (C₂²×D₇)⋊₂C₃, SmallGroup(168,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇⋊A₄

Chief series

C₁ — C₇ — C₂×C₁₄ — C₇⋊A₄ — D₇⋊A₄

Lower central

C₂×C₁₄ — D₇⋊A₄

Upper central

C₁

Generators and relations for D₇⋊A₄
G = < a,b,c,d,e | a⁷=b²=c²=d²=e³=1, bab=a^-1, ac=ca, ad=da, eae^-1=a², bc=cb, bd=db, ebe^-1=ab, ece^-1=cd=dc, ede^-1=c >

C₁

₃C₂

₇C₂

₂₁C₂

₂₈C₃

C₇

C₂²

₂₁C₂²

₂₈C₆

D₇

₃C₁₄

₃D₇

₄C₇⋊C₃

₇C₂³

₇A₄

C₂×C₁₄

₃D₁₄

₄F₇

₇C₂×A₄

C₂²×D₇

C₇⋊A₄

D₇⋊A₄

Character table of D₇⋊A₄

class	1	2A	2B	2C	3A	3B	6A	6B	7	14A	14B	14C
size	1	3	7	21	28	28	28	28	6	6	6	6
ρ₁	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	linear of order 2
ρ₃	1	1	-1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	1	1	1	linear of order 6
ρ₄	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	1	1	linear of order 3
ρ₅	1	1	-1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	1	1	1	linear of order 6
ρ₆	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	1	1	linear of order 3
ρ₇	3	-1	-3	1	0	0	0	0	3	-1	-1	-1	orthogonal lifted from C₂×A₄
ρ₈	3	-1	3	-1	0	0	0	0	3	-1	-1	-1	orthogonal lifted from A₄
ρ₉	6	6	0	0	0	0	0	0	-1	-1	-1	-1	orthogonal lifted from F₇
ρ₁₀	6	-2	0	0	0	0	0	0	-1	2ζ₇⁶+2ζ₇+1	2ζ₇⁵+2ζ₇²+1	2ζ₇⁴+2ζ₇³+1	orthogonal faithful
ρ₁₁	6	-2	0	0	0	0	0	0	-1	2ζ₇⁴+2ζ₇³+1	2ζ₇⁶+2ζ₇+1	2ζ₇⁵+2ζ₇²+1	orthogonal faithful
ρ₁₂	6	-2	0	0	0	0	0	0	-1	2ζ₇⁵+2ζ₇²+1	2ζ₇⁴+2ζ₇³+1	2ζ₇⁶+2ζ₇+1	orthogonal faithful

Permutation representations of D₇⋊A₄
►On 28 points - transitive group 28T28

Generators in S₂₈

(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)

(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)

(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)

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

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

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

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

G:=TransitiveGroup(28,28);

D₇⋊A₄ is a maximal quotient of Q₈.F₇ Q₈⋊F₇ Dic₇⋊A₄

Matrix representation of D₇⋊A₄ ►in GL₆(𝔽₄₃)

42	1	0	0	0	0
42	0	1	0	0	0
42	0	0	1	0	0
42	0	0	0	1	0
42	0	0	0	0	1
42	0	0	0	0	0

42	0	0	0	0	0
42	0	0	0	0	1
42	0	0	0	1	0
42	0	0	1	0	0
42	0	1	0	0	0
42	1	0	0	0	0

28	3	38	0	5	40
0	31	41	38	5	2
40	3	26	41	0	2
2	0	41	26	3	40
2	5	38	41	31	0
40	5	0	38	3	28

31	38	2	0	41	5
0	26	40	2	41	3
5	38	28	40	0	3
3	0	40	28	38	5
3	41	2	40	26	0
5	41	0	2	38	31

0	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	0	1	0

G:=sub<GL(6,GF(43))| [42,42,42,42,42,42,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[42,42,42,42,42,42,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0],[28,0,40,2,2,40,3,31,3,0,5,5,38,41,26,41,38,0,0,38,41,26,41,38,5,5,0,3,31,3,40,2,2,40,0,28],[31,0,5,3,3,5,38,26,38,0,41,41,2,40,28,40,2,0,0,2,40,28,40,2,41,41,0,38,26,38,5,3,3,5,0,31],[0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0] >;

D₇⋊A₄ in GAP, Magma, Sage, TeX

D_7\rtimes A_4

% in TeX

G:=Group("D7:A4");

// GroupNames label

G:=SmallGroup(168,49);

// by ID

G=gap.SmallGroup(168,49);

# by ID

G:=PCGroup([5,-2,-3,-2,2,-7,97,188,3604,1209]);

// Polycyclic

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

// generators/relations

Export

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

G = D7⋊A4 order 168 = 23·3·7

The semidirect product of D7 and A4 acting via A4/C22=C3

G = D₇⋊A₄ order 168 = 2³·3·7

The semidirect product of D₇ and A₄ acting via A₄/C₂²=C₃