direct product, non-abelian, soluble, monomial
Aliases: C11×S4, A4⋊C22, (C2×C22)⋊1S3, C22⋊(S3×C11), (C11×A4)⋊3C2, SmallGroup(264,31)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C11×S4 |
Generators and relations for C11×S4
G = < a,b,c,d,e | a11=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 >
Smallest permutation representation of C11×S4
►On 44 points
Generators in S44
(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 29 30 31 32 33)(34 35 36 37 38 39 40 41 42 43 44)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 12)(9 13)(10 14)(11 15)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 34)(33 35)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(12 36 23)(13 37 24)(14 38 25)(15 39 26)(16 40 27)(17 41 28)(18 42 29)(19 43 30)(20 44 31)(21 34 32)(22 35 33)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,12)(9,13)(10,14)(11,15)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,34)(33,35), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (12,36,23)(13,37,24)(14,38,25)(15,39,26)(16,40,27)(17,41,28)(18,42,29)(19,43,30)(20,44,31)(21,34,32)(22,35,33), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,12)(9,13)(10,14)(11,15)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,34)(33,35), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (12,36,23)(13,37,24)(14,38,25)(15,39,26)(16,40,27)(17,41,28)(18,42,29)(19,43,30)(20,44,31)(21,34,32)(22,35,33), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33),(34,35,36,37,38,39,40,41,42,43,44)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,12),(9,13),(10,14),(11,15),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,34),(33,35)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)], [(12,36,23),(13,37,24),(14,38,25),(15,39,26),(16,40,27),(17,41,28),(18,42,29),(19,43,30),(20,44,31),(21,34,32),(22,35,33)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4 | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 33A | ··· | 33J | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 3 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 |
| size | 1 | 3 | 6 | 8 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | ··· | 8 | 6 | ··· | 6 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C11 | C22 | S3 | S3×C11 | S4 | C11×S4 |
| kernel | C11×S4 | C11×A4 | S4 | A4 | C2×C22 | C22 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 1 | 10 | 2 | 20 |
Matrix representation of C11×S4 ►in GL3(𝔽397) generated by
| 290 | 0 | 0 |
| 0 | 290 | 0 |
| 0 | 0 | 290 |
| 0 | 396 | 1 |
| 0 | 396 | 0 |
| 1 | 396 | 0 |
| 396 | 0 | 0 |
| 396 | 0 | 1 |
| 396 | 1 | 0 |
| 1 | 0 | 396 |
| 0 | 0 | 396 |
| 0 | 1 | 396 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(397))| [290,0,0,0,290,0,0,0,290],[0,0,1,396,396,396,1,0,0],[396,396,396,0,0,1,0,1,0],[1,0,0,0,0,1,396,396,396],[1,0,0,0,0,1,0,1,0] >;
C11×S4 in GAP, Magma, Sage, TeX
C_{11}\times S_4 % in TeX
G:=Group("C11xS4"); // GroupNames label
G:=SmallGroup(264,31);
// by ID
G=gap.SmallGroup(264,31);
# by ID
G:=PCGroup([5,-2,-11,-3,-2,2,662,2643,133,1654,239]);
// Polycyclic
G:=Group<a,b,c,d,e|a^11=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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