direct product, non-abelian, soluble, monomial
Aliases: D7xS4, A4:1D14, C7:S4:C2, (C7xS4):C2, C7:1(C2xS4), (A4xD7):C2, (C2xC14):D6, C22:(S3xD7), (C7xA4):C22, (C22xD7):S3, SmallGroup(336,212)
Series: Derived ►Chief ►Lower central ►Upper central
| C7xA4 — D7xS4 |
Generators and relations for D7xS4
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, C2xC4, D4, C23, A4, D6, D7, D7, C14, C2xD4, C21, S4, S4, C2xA4, Dic7, C28, D14, C2xC14, C2xC14, S3xC7, C3xD7, D21, C2xS4, C4xD7, D28, C7:D4, C7xD4, C22xD7, C22xD7, S3xD7, C7xA4, D4xD7, C7xS4, C7:S4, A4xD7, D7xS4
Quotients: C1, C2, C22, S3, D6, D7, S4, D14, C2xS4, S3xD7, D7xS4
Character table of D7xS4
| 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 C2xS4 |
| 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 C2xS4 |
| 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 S3xD7 |
| 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 S3xD7 |
| 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 S3xD7 |
| 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 D7xS4 ►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] >;
D7xS4 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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