direct product, non-abelian, soluble, monomial
Aliases: D7×S4, A4⋊1D14, C7⋊S4⋊C2, (C7×S4)⋊C2, C7⋊1(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
| C7×A4 — D7×S4 |
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 |
►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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