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## G = C2×D7⋊A4order 336 = 24·3·7

### Direct product of C2 and D7⋊A4

Aliases: C2×D7⋊A4, D14⋊A4, C23⋊F7, C14⋊(C2×A4), D7⋊(C2×A4), C7⋊A4⋊C22, C7⋊(C22×A4), C22⋊(C2×F7), (C23×D7)⋊3C3, (C22×C14)⋊4C6, (C22×D7)⋊5C6, (C2×C7⋊A4)⋊C2, (C2×C14)⋊3(C2×C6), SmallGroup(336,218)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D7⋊A4
G = < a,b,c,d,e,f | a2=b7=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >

Subgroups: 640 in 78 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, A4, C2×C6, D7, D7, C14, C14, C24, C7⋊C3, C2×A4, D14, D14, C2×C14, C2×C14, F7, C2×C7⋊C3, C22×A4, C22×D7, C22×D7, C22×C14, C2×F7, C7⋊A4, C23×D7, D7⋊A4, C2×C7⋊A4, C2×D7⋊A4
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, F7, C22×A4, C2×F7, D7⋊A4, C2×D7⋊A4

Character table of C2×D7⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A 6B 6C 6D 6E 6F 7 14A 14B 14C 14D 14E 14F 14G size 1 1 3 3 7 7 21 21 28 28 28 28 28 28 28 28 6 6 6 6 6 6 6 6 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 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ4 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 ρ5 1 -1 -1 1 -1 1 -1 1 ζ32 ζ3 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ6 1 -1 -1 1 1 -1 1 -1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ7 1 -1 -1 1 1 -1 1 -1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ8 1 -1 -1 1 -1 1 -1 1 ζ3 ζ32 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ9 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 1 1 1 1 1 1 1 1 linear of order 6 ρ10 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 1 1 1 1 1 1 1 1 linear of order 6 ρ11 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ12 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ13 3 -3 1 -1 -3 3 1 -1 0 0 0 0 0 0 0 0 3 -1 -1 1 1 1 -3 -1 orthogonal lifted from C2×A4 ρ14 3 3 -1 -1 -3 -3 1 1 0 0 0 0 0 0 0 0 3 -1 -1 -1 -1 -1 3 -1 orthogonal lifted from C2×A4 ρ15 3 3 -1 -1 3 3 -1 -1 0 0 0 0 0 0 0 0 3 -1 -1 -1 -1 -1 3 -1 orthogonal lifted from A4 ρ16 3 -3 1 -1 3 -3 -1 1 0 0 0 0 0 0 0 0 3 -1 -1 1 1 1 -3 -1 orthogonal lifted from C2×A4 ρ17 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ18 6 -6 -6 6 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 orthogonal lifted from C2×F7 ρ19 6 6 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 -1 2ζ74+2ζ73+1 orthogonal lifted from D7⋊A4 ρ20 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 -2ζ74-2ζ73-1 -2ζ76-2ζ7-1 -2ζ75-2ζ72-1 1 2ζ74+2ζ73+1 orthogonal faithful ρ21 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 -2ζ76-2ζ7-1 -2ζ75-2ζ72-1 -2ζ74-2ζ73-1 1 2ζ76+2ζ7+1 orthogonal faithful ρ22 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 -2ζ75-2ζ72-1 -2ζ74-2ζ73-1 -2ζ76-2ζ7-1 1 2ζ75+2ζ72+1 orthogonal faithful ρ23 6 6 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 -1 2ζ76+2ζ7+1 orthogonal lifted from D7⋊A4 ρ24 6 6 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 -1 2ζ75+2ζ72+1 orthogonal lifted from D7⋊A4

Smallest permutation representation of C2×D7⋊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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(35 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 32 16)(2 29 18)(3 33 20)(4 30 15)(5 34 17)(6 31 19)(7 35 21)(8 39 23)(9 36 25)(10 40 27)(11 37 22)(12 41 24)(13 38 26)(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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42), (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), (1,32,16)(2,29,18)(3,33,20)(4,30,15)(5,34,17)(6,31,19)(7,35,21)(8,39,23)(9,36,25)(10,40,27)(11,37,22)(12,41,24)(13,38,26)(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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42), (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), (1,32,16)(2,29,18)(3,33,20)(4,30,15)(5,34,17)(6,31,19)(7,35,21)(8,39,23)(9,36,25)(10,40,27)(11,37,22)(12,41,24)(13,38,26)(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(35,42)], [(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)], [(1,32,16),(2,29,18),(3,33,20),(4,30,15),(5,34,17),(6,31,19),(7,35,21),(8,39,23),(9,36,25),(10,40,27),(11,37,22),(12,41,24),(13,38,26),(14,42,28)]])

Matrix representation of C2×D7⋊A4 in GL6(𝔽43)

 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42
,
 19 42 0 0 0 0 1 0 0 0 0 0 0 0 24 16 0 0 0 0 27 27 0 0 0 0 0 0 16 24 0 0 0 0 19 42
,
 24 1 0 0 0 0 27 19 0 0 0 0 0 0 19 27 0 0 0 0 1 24 0 0 0 0 0 0 27 19 0 0 0 0 16 16
,
 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 42 0 0 0 0 0 0 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 0 0 0 0 0 0 42
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0

G:=sub<GL(6,GF(43))| [42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[19,1,0,0,0,0,42,0,0,0,0,0,0,0,24,27,0,0,0,0,16,27,0,0,0,0,0,0,16,19,0,0,0,0,24,42],[24,27,0,0,0,0,1,19,0,0,0,0,0,0,19,1,0,0,0,0,27,24,0,0,0,0,0,0,27,16,0,0,0,0,19,16],[42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×D7⋊A4 in GAP, Magma, Sage, TeX

C_2\times D_7\rtimes A_4
% in TeX

G:=Group("C2xD7:A4");
// GroupNames label

G:=SmallGroup(336,218);
// by ID

G=gap.SmallGroup(336,218);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-7,159,286,10373,1745]);
// Polycyclic

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

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