non-abelian, soluble, monomial
Aliases: C7⋊S4, A4⋊D7, C22⋊D21, (C7×A4)⋊1C2, (C2×C14)⋊2S3, SmallGroup(168,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×A4 — C7⋊S4 |
Generators and relations for C7⋊S4
G = < a,b,c,d,e | a7=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Character table of C7⋊S4
| class | 1 | 2A | 2B | 3 | 4 | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | |
| size | 1 | 3 | 42 | 8 | 42 | 2 | 2 | 2 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ5 | 2 | 2 | 0 | 2 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 0 | 2 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ7 | 2 | 2 | 0 | -1 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | orthogonal lifted from D21 |
| ρ8 | 2 | 2 | 0 | -1 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ76-ζ32ζ7-ζ7 | orthogonal lifted from D21 |
| ρ9 | 2 | 2 | 0 | -1 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ3ζ74+ζ3ζ73-ζ74 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | orthogonal lifted from D21 |
| ρ10 | 2 | 2 | 0 | -1 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | ζ3ζ75-ζ3ζ72-ζ72 | orthogonal lifted from D21 |
| ρ11 | 2 | 2 | 0 | -1 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ76+ζ32ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | -ζ3ζ74+ζ3ζ73-ζ74 | orthogonal lifted from D21 |
| ρ12 | 2 | 2 | 0 | -1 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ75-ζ3ζ72-ζ72 | ζ32ζ76-ζ32ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ74+ζ3ζ73-ζ74 | -ζ32ζ76+ζ32ζ7-ζ76 | orthogonal lifted from D21 |
| ρ13 | 3 | -1 | 1 | 0 | -1 | 3 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | -1 | -1 | 0 | 1 | 3 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 6 | -2 | 0 | 0 | 0 | 3ζ74+3ζ73 | 3ζ75+3ζ72 | 3ζ76+3ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | 0 | 0 | 3ζ76+3ζ7 | 3ζ74+3ζ73 | 3ζ75+3ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ17 | 6 | -2 | 0 | 0 | 0 | 3ζ75+3ζ72 | 3ζ76+3ζ7 | 3ζ74+3ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 28 points - transitive group 28T30
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 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)
(2 7)(3 6)(4 5)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(14 21)(22 26)(23 25)(27 28)
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), (1,8)(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), (2,7)(3,6)(4,5)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(14,21)(22,26)(23,25)(27,28)>;
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), (1,8)(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), (2,7)(3,6)(4,5)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(14,21)(22,26)(23,25)(27,28) );
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)], [(1,8),(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)], [(2,7),(3,6),(4,5),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(14,21),(22,26),(23,25),(27,28)]])
G:=TransitiveGroup(28,30);
C7⋊S4 is a maximal subgroup of
D7×S4
C7⋊S4 is a maximal quotient of Q8.D21 Q8⋊D21 A4⋊Dic7
Matrix representation of C7⋊S4 ►in GL5(𝔽337)
| 103 | 274 | 0 | 0 | 0 |
| 63 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 336 |
| 0 | 0 | 336 | 0 | 1 |
| 0 | 0 | 0 | 0 | 336 |
| 336 | 1 | 0 | 0 | 0 |
| 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 336 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 0 | 1 |
| 336 | 0 | 0 | 0 | 0 |
| 336 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 |
G:=sub<GL(5,GF(337))| [103,63,0,0,0,274,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,336,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,336,0,0,0,0,336,1,336],[336,336,0,0,0,1,0,0,0,0,0,0,336,1,336,0,0,336,0,0,0,0,0,0,1],[336,336,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,336] >;
C7⋊S4 in GAP, Magma, Sage, TeX
C_7\rtimes S_4
% in TeX
G:=Group("C7:S4"); // GroupNames label
G:=SmallGroup(168,46);
// by ID
G=gap.SmallGroup(168,46);
# by ID
G:=PCGroup([5,-2,-3,-7,-2,2,41,542,1683,848,1054,1584]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C7⋊S4 in TeX
Character table of C7⋊S4 in TeX