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## G = C2×C7⋊A4order 168 = 23·3·7

### Direct product of C2 and C7⋊A4

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

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

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 >

Character table of C2×C7⋊A4

 class 1 2A 2B 2C 3A 3B 6A 6B 7A 7B 14A 14B 14C 14D 14E 14F 14G 14H 14I 14J 14K 14L 14M 14N size 1 1 3 3 28 28 28 28 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 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 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 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 6 ρ6 1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 6 ρ7 3 3 -1 -1 0 0 0 0 3 3 -1 -1 -1 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ8 3 -3 1 -1 0 0 0 0 3 3 -1 -1 -1 -3 -3 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from C2×A4 ρ9 3 3 -1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 -1+√-7/2 -1-√-7/2 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 -ζ74-ζ72+ζ7 -ζ76-ζ75+ζ73 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 -ζ76+ζ75-ζ73 complex lifted from C7⋊A4 ρ10 3 -3 1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 ζ76-ζ75-ζ73 1-√-7/2 1+√-7/2 -ζ74-ζ72+ζ7 ζ74-ζ72+ζ7 -ζ74+ζ72+ζ7 -ζ76+ζ75+ζ73 ζ74+ζ72-ζ7 ζ76+ζ75-ζ73 ζ76-ζ75+ζ73 -ζ76-ζ75+ζ73 -ζ76+ζ75-ζ73 complex faithful ρ11 3 -3 1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 1-√-7/2 1+√-7/2 ζ74-ζ72-ζ7 ζ74+ζ72-ζ7 ζ74-ζ72+ζ7 ζ76+ζ75-ζ73 -ζ74+ζ72+ζ7 ζ76-ζ75+ζ73 -ζ76+ζ75+ζ73 -ζ76+ζ75-ζ73 ζ76-ζ75-ζ73 complex faithful ρ12 3 3 -1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 -1+√-7/2 -1-√-7/2 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 -ζ74+ζ72-ζ7 ζ76-ζ75-ζ73 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ76-ζ75+ζ73 complex lifted from C7⋊A4 ρ13 3 -3 1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 1+√-7/2 1-√-7/2 -ζ76+ζ75-ζ73 ζ76+ζ75-ζ73 -ζ76+ζ75+ζ73 ζ74-ζ72+ζ7 ζ76-ζ75+ζ73 ζ74+ζ72-ζ7 -ζ74+ζ72+ζ7 -ζ74-ζ72+ζ7 ζ74-ζ72-ζ7 complex faithful ρ14 3 -3 1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 -ζ76+ζ75-ζ73 1-√-7/2 1+√-7/2 -ζ74+ζ72-ζ7 -ζ74+ζ72+ζ7 ζ74+ζ72-ζ7 ζ76-ζ75+ζ73 ζ74-ζ72+ζ7 -ζ76+ζ75+ζ73 ζ76+ζ75-ζ73 ζ76-ζ75-ζ73 -ζ76-ζ75+ζ73 complex faithful ρ15 3 3 -1 -1 0 0 0 0 -1-√-7/2 -1+√-7/2 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 -1+√-7/2 -1-√-7/2 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 -ζ76-ζ75+ζ73 ζ74-ζ72-ζ7 -ζ76+ζ75-ζ73 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 ζ76-ζ75-ζ73 complex lifted from C7⋊A4 ρ16 3 -3 1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 1+√-7/2 1-√-7/2 -ζ76-ζ75+ζ73 -ζ76+ζ75+ζ73 ζ76-ζ75+ζ73 -ζ74+ζ72+ζ7 ζ76+ζ75-ζ73 ζ74-ζ72+ζ7 ζ74+ζ72-ζ7 -ζ74+ζ72-ζ7 -ζ74-ζ72+ζ7 complex faithful ρ17 3 3 3 3 0 0 0 0 -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 ρ18 3 -3 -3 3 0 0 0 0 -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 C2×C7⋊C3 ρ19 3 -3 1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 1+√-7/2 1-√-7/2 ζ76-ζ75-ζ73 ζ76-ζ75+ζ73 ζ76+ζ75-ζ73 ζ74+ζ72-ζ7 -ζ76+ζ75+ζ73 -ζ74+ζ72+ζ7 ζ74-ζ72+ζ7 ζ74-ζ72-ζ7 -ζ74+ζ72-ζ7 complex faithful ρ20 3 3 -1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 -1-√-7/2 -1+√-7/2 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 ζ74-ζ72-ζ7 -ζ76-ζ75+ζ73 -ζ74+ζ72-ζ7 -ζ74-ζ72+ζ7 -ζ74+ζ72-ζ7 -ζ74-ζ72+ζ7 complex lifted from C7⋊A4 ρ21 3 -3 -3 3 0 0 0 0 -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 C2×C7⋊C3 ρ22 3 3 3 3 0 0 0 0 -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 ρ23 3 3 -1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 -1-√-7/2 -1+√-7/2 ζ76-ζ75-ζ73 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 -ζ74-ζ72+ζ7 ζ76-ζ75-ζ73 ζ74-ζ72-ζ7 -ζ74+ζ72-ζ7 ζ74-ζ72-ζ7 -ζ74+ζ72-ζ7 complex lifted from C7⋊A4 ρ24 3 3 -1 -1 0 0 0 0 -1+√-7/2 -1-√-7/2 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 -1-√-7/2 -1+√-7/2 -ζ76+ζ75-ζ73 -ζ76-ζ75+ζ73 ζ76-ζ75-ζ73 -ζ74+ζ72-ζ7 -ζ76+ζ75-ζ73 -ζ74-ζ72+ζ7 ζ74-ζ72-ζ7 -ζ74-ζ72+ζ7 ζ74-ζ72-ζ7 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

 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 1 0 0 0 1 1 0 0
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

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