Copied to
clipboard

G = C2×C7⋊A4order 168 = 23·3·7

Direct product of C2 and C7⋊A4

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

Aliases: C2×C7⋊A4, C14⋊A4, C72(C2×A4), (C2×C14)⋊7C6, C232(C7⋊C3), (C22×C14)⋊3C3, C22⋊(C2×C7⋊C3), SmallGroup(168,53)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C2×C7⋊A4
C1C7C2×C14C7⋊A4 — C2×C7⋊A4
C2×C14 — C2×C7⋊A4
C1C2

Generators and relations for C2×C7⋊A4
 G = < a,b,c,d,e | a2=b7=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, ece-1=cd=dc, ede-1=c >

3C2
3C2
28C3
3C22
3C22
28C6
3C14
3C14
4C7⋊C3
7A4
3C2×C14
3C2×C14
4C2×C7⋊C3
7C2×A4

Character table of C2×C7⋊A4

 class 12A2B2C3A3B6A6B7A7B14A14B14C14D14E14F14G14H14I14J14K14L14M14N
 size 1133282828283333333333333333
ρ1111111111111111111111111    trivial
ρ21-1-1111-1-111111-1-11-1-1-1-1-1-111    linear of order 2
ρ31111ζ32ζ3ζ3ζ321111111111111111    linear of order 3
ρ41111ζ3ζ32ζ32ζ31111111111111111    linear of order 3
ρ51-1-11ζ3ζ32ζ6ζ6511111-1-11-1-1-1-1-1-111    linear of order 6
ρ61-1-11ζ32ζ3ζ65ζ611111-1-11-1-1-1-1-1-111    linear of order 6
ρ733-1-1000033-1-1-133-1-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ83-31-1000033-1-1-1-3-3-1111111-1-1    orthogonal lifted from C2×A4
ρ933-1-10000-1--7/2-1+-7/274727ζ74727ζ767573-1+-7/2-1--7/27472774727ζ74727ζ76757374727767573767573767573767573    complex lifted from C7⋊A4
ρ103-31-10000-1--7/2-1+-7/274727ζ74727ζ7675731--7/21+-7/274727ζ7472774727767573ζ74727ζ767573ζ767573767573767573    complex faithful
ρ113-31-10000-1--7/2-1+-7/274727747277675731--7/21+-7/2ζ74727ζ74727ζ74727ζ76757374727ζ767573767573767573ζ767573    complex faithful
ρ1233-1-10000-1--7/2-1+-7/2ζ7472774727767573-1+-7/2-1--7/274727ζ747277472776757374727ζ767573767573ζ767573767573    complex lifted from C7⋊A4
ρ133-31-10000-1+-7/2-1--7/2767573ζ767573747271+-7/21--7/2767573ζ767573767573ζ74727ζ767573ζ747277472774727ζ74727    complex faithful
ρ143-31-10000-1--7/2-1+-7/2ζ74727747277675731--7/21+-7/27472774727ζ74727ζ767573ζ74727767573ζ767573ζ767573767573    complex faithful
ρ1533-1-10000-1--7/2-1+-7/27472774727767573-1+-7/2-1--7/2ζ747277472774727767573ζ74727767573ζ767573767573ζ767573    complex lifted from C7⋊A4
ρ163-31-10000-1+-7/2-1--7/2ζ767573767573ζ747271+-7/21--7/2767573767573ζ76757374727ζ767573ζ74727ζ747277472774727    complex faithful
ρ1733330000-1+-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--7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2    complex lifted from C7⋊C3
ρ183-3-330000-1+-7/2-1--7/2-1--7/2-1--7/2-1+-7/21+-7/21--7/2-1--7/21+-7/21+-7/21--7/21+-7/21--7/21--7/2-1+-7/2-1+-7/2    complex lifted from C2×C7⋊C3
ρ193-31-10000-1+-7/2-1--7/2767573767573747271+-7/21--7/2ζ767573ζ767573ζ767573ζ7472776757374727ζ74727ζ7472774727    complex faithful
ρ2033-1-10000-1+-7/2-1--7/2ζ767573767573ζ74727-1--7/2-1+-7/2767573ζ767573767573ζ7472776757374727747277472774727    complex lifted from C7⋊A4
ρ213-3-330000-1--7/2-1+-7/2-1+-7/2-1+-7/2-1--7/21--7/21+-7/2-1+-7/21--7/21--7/21+-7/21--7/21+-7/21+-7/2-1--7/2-1--7/2    complex lifted from C2×C7⋊C3
ρ2233330000-1--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+-7/2-1--7/2-1--7/2-1--7/2-1--7/2    complex lifted from C7⋊C3
ρ2333-1-10000-1+-7/2-1--7/276757376757374727-1--7/2-1+-7/2ζ76757376757376757374727ζ767573ζ7472774727ζ7472774727    complex lifted from C7⋊A4
ρ2433-1-10000-1+-7/2-1--7/2767573ζ76757374727-1--7/2-1+-7/2767573767573ζ7675737472776757374727ζ7472774727ζ74727    complex lifted from C7⋊A4

Smallest permutation representation of C2×C7⋊A4
On 42 points
Generators in S42
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 30 18)(2 32 15)(3 34 19)(4 29 16)(5 31 20)(6 33 17)(7 35 21)(8 37 25)(9 39 22)(10 41 26)(11 36 23)(12 38 27)(13 40 24)(14 42 28)

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

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

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

C2×C7⋊A4 is a maximal subgroup of   Dic7⋊A4
C2×C7⋊A4 is a maximal quotient of   C28.A4

Matrix representation of C2×C7⋊A4 in GL3(𝔽43) generated by

4200
0420
0042
,
2100
0110
0035
,
4200
0420
001
,
4200
010
0042
,
010
001
100
G:=sub<GL(3,GF(43))| [42,0,0,0,42,0,0,0,42],[21,0,0,0,11,0,0,0,35],[42,0,0,0,42,0,0,0,1],[42,0,0,0,1,0,0,0,42],[0,0,1,1,0,0,0,1,0] >;

C2×C7⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes A_4
% in TeX

G:=Group("C2xC7:A4");
// GroupNames label

G:=SmallGroup(168,53);
// by ID

G=gap.SmallGroup(168,53);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-7,97,188,609]);
// Polycyclic

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

Export

Subgroup lattice of C2×C7⋊A4 in TeX
Character table of C2×C7⋊A4 in TeX

׿
×
𝔽