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## G = D7×S4order 336 = 24·3·7

### Direct product of D7 and S4

Aliases: D7×S4, A41D14, C7⋊S4⋊C2, (C7×S4)⋊C2, C71(C2×S4), (A4×D7)⋊C2, (C2×C14)⋊D6, C22⋊(S3×D7), (C7×A4)⋊C22, (C22×D7)⋊S3, SmallGroup(336,212)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×A4 — D7×S4
 Chief series C1 — C22 — C2×C14 — C7×A4 — A4×D7 — D7×S4
 Lower central C7×A4 — D7×S4
 Upper central C1

Generators and relations for D7×S4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 604 in 66 conjugacy classes, 13 normal (all characteristic)
C1, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, D4, C23, A4, D6, D7, D7, C14, C2×D4, C21, S4, S4, C2×A4, Dic7, C28, D14, C2×C14, C2×C14, S3×C7, C3×D7, D21, C2×S4, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C22×D7, S3×D7, C7×A4, D4×D7, C7×S4, C7⋊S4, A4×D7, D7×S4
Quotients: C1, C2, C22, S3, D6, D7, S4, D14, C2×S4, S3×D7, D7×S4

Character table of D7×S4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 6 7A 7B 7C 14A 14B 14C 14D 14E 14F 21A 21B 21C 28A 28B 28C size 1 3 6 7 21 42 8 6 42 56 2 2 2 6 6 6 12 12 12 16 16 16 12 12 12 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 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 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 -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 -1 linear of order 2 ρ5 2 2 0 -2 -2 0 -1 0 0 1 2 2 2 2 2 2 0 0 0 -1 -1 -1 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 2 2 0 -1 0 0 -1 2 2 2 2 2 2 0 0 0 -1 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 2 0 0 0 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ8 2 2 -2 0 0 0 2 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ9 2 2 -2 0 0 0 2 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ10 2 2 -2 0 0 0 2 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ11 2 2 2 0 0 0 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ12 2 2 2 0 0 0 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ13 3 -1 -1 -3 1 1 0 1 -1 0 3 3 3 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 orthogonal lifted from C2×S4 ρ14 3 -1 1 -3 1 -1 0 -1 1 0 3 3 3 -1 -1 -1 1 1 1 0 0 0 -1 -1 -1 orthogonal lifted from C2×S4 ρ15 3 -1 -1 3 -1 -1 0 1 1 0 3 3 3 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 orthogonal lifted from S4 ρ16 3 -1 1 3 -1 1 0 -1 -1 0 3 3 3 -1 -1 -1 1 1 1 0 0 0 -1 -1 -1 orthogonal lifted from S4 ρ17 4 4 0 0 0 0 -2 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 orthogonal lifted from S3×D7 ρ18 4 4 0 0 0 0 -2 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 orthogonal lifted from S3×D7 ρ19 4 4 0 0 0 0 -2 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 orthogonal lifted from S3×D7 ρ20 6 -2 2 0 0 0 0 -2 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal faithful ρ21 6 -2 2 0 0 0 0 -2 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal faithful ρ22 6 -2 -2 0 0 0 0 2 0 0 3ζ75+3ζ72 3ζ74+3ζ73 3ζ76+3ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal faithful ρ23 6 -2 2 0 0 0 0 -2 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal faithful ρ24 6 -2 -2 0 0 0 0 2 0 0 3ζ74+3ζ73 3ζ76+3ζ7 3ζ75+3ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal faithful ρ25 6 -2 -2 0 0 0 0 2 0 0 3ζ76+3ζ7 3ζ75+3ζ72 3ζ74+3ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal faithful

Permutation representations of D7×S4
On 28 points - transitive group 28T45
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 14)(9 13)(10 12)(15 17)(18 21)(19 20)(22 25)(23 24)(26 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 26)(16 27)(17 28)(18 22)(19 23)(20 24)(21 25)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)
(8 24 20)(9 25 21)(10 26 15)(11 27 16)(12 28 17)(13 22 18)(14 23 19)
(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)

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

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

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

G:=TransitiveGroup(28,45);

Matrix representation of D7×S4 in GL5(𝔽337)

 336 1 0 0 0 108 228 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 229 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 336 1 0 0 0 336 0 0 0 1 336 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 336 0 0 1 0 336 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 336 0 0 0 0 0 336 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(337))| [336,108,0,0,0,1,228,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,229,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,336,336,336,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,336,336,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[336,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

D7×S4 in GAP, Magma, Sage, TeX

D_7\times S_4
% in TeX

G:=Group("D7xS4");
// GroupNames label

G:=SmallGroup(336,212);
// by ID

G=gap.SmallGroup(336,212);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,2,80,1731,2530,1276,1523,2285]);
// Polycyclic

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

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