direct product, non-abelian, soluble, monomial
Aliases: C7×S4, A4⋊C14, C22⋊(S3×C7), (C7×A4)⋊3C2, (C2×C14)⋊1S3, SmallGroup(168,45)
Series: Derived ►Chief ►Lower central ►Upper central
| 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, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Permutation representations of C7×S4
►On 28 points - transitive group 28T31
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)
(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), (1,8)(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), (1,8)(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)], [(1,8),(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(28,31);
35 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4 | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21F | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 3 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 |
| size | 1 | 3 | 6 | 8 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | ··· | 8 | 6 | ··· | 6 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C7 | C14 | S3 | S3×C7 | S4 | C7×S4 |
| kernel | C7×S4 | C7×A4 | S4 | A4 | C2×C14 | C22 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 6 | 2 | 12 |
Matrix representation of C7×S4 ►in GL3(𝔽337) generated by
| 79 | 0 | 0 |
| 0 | 79 | 0 |
| 0 | 0 | 79 |
| 0 | 0 | 1 |
| 336 | 336 | 336 |
| 1 | 0 | 0 |
| 336 | 336 | 336 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 336 | 336 | 336 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(337))| [79,0,0,0,79,0,0,0,79],[0,336,1,0,336,0,1,336,0],[336,0,0,336,0,1,336,1,0],[1,336,0,0,336,1,0,336,0],[1,0,0,0,0,1,0,1,0] >;
C7×S4 in GAP, Magma, Sage, TeX
C_7\times S_4
% in TeX
G:=Group("C7xS4"); // GroupNames label
G:=SmallGroup(168,45);
// by ID
G=gap.SmallGroup(168,45);
# by ID
G:=PCGroup([5,-2,-7,-3,-2,2,422,1683,133,1054,239]);
// 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,a*e=e*a,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
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