direct product, non-abelian, soluble, monomial
Aliases: C5xC42:S3, C42:(C5xS3), (C4xC20):2S3, C22.(C5xS4), C42:C3:2C10, (C2xC10).1S4, (C5xC42:C3):6C2, SmallGroup(480,254)
Series: Derived ►Chief ►Lower central ►Upper central
| C42:C3 — C5xC42:S3 |
Generators and relations for C5xC42:S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >
Quotients: C1, C2, C5, S3, C10, S4, C5xS3, C42:S3, C5xS4, C5xC42:S3
3C2
12C2
16C3
3C4
3C4
6C4
6C22
16S3
3C10
12C10
16C15
3D4
3Q8
6C8
6D4
4A4
3C20
3C20
6C20
16C5xS3
4S4
6C40
Smallest permutation representation of C5xC42:S3
►On 60 points
Generators in S60
(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 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)
(1 58 36 20)(2 59 37 16)(3 60 38 17)(4 56 39 18)(5 57 40 19)(6 12)(7 13)(8 14)(9 15)(10 11)(21 46 29 55)(22 47 30 51)(23 48 26 52)(24 49 27 53)(25 50 28 54)(31 44)(32 45)(33 41)(34 42)(35 43)
(1 58 36 20)(2 59 37 16)(3 60 38 17)(4 56 39 18)(5 57 40 19)(6 45 12 32)(7 41 13 33)(8 42 14 34)(9 43 15 35)(10 44 11 31)(21 29)(22 30)(23 26)(24 27)(25 28)(46 55)(47 51)(48 52)(49 53)(50 54)
(1 23 32)(2 24 33)(3 25 34)(4 21 35)(5 22 31)(6 58 48)(7 59 49)(8 60 50)(9 56 46)(10 57 47)(11 19 51)(12 20 52)(13 16 53)(14 17 54)(15 18 55)(26 45 36)(27 41 37)(28 42 38)(29 43 39)(30 44 40)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 52)(7 53)(8 54)(9 55)(10 51)(11 47)(12 48)(13 49)(14 50)(15 46)(16 59)(17 60)(18 56)(19 57)(20 58)(21 43)(22 44)(23 45)(24 41)(25 42)(26 32)(27 33)(28 34)(29 35)(30 31)
G:=sub<Sym(60)| (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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60), (1,58,36,20)(2,59,37,16)(3,60,38,17)(4,56,39,18)(5,57,40,19)(6,12)(7,13)(8,14)(9,15)(10,11)(21,46,29,55)(22,47,30,51)(23,48,26,52)(24,49,27,53)(25,50,28,54)(31,44)(32,45)(33,41)(34,42)(35,43), (1,58,36,20)(2,59,37,16)(3,60,38,17)(4,56,39,18)(5,57,40,19)(6,45,12,32)(7,41,13,33)(8,42,14,34)(9,43,15,35)(10,44,11,31)(21,29)(22,30)(23,26)(24,27)(25,28)(46,55)(47,51)(48,52)(49,53)(50,54), (1,23,32)(2,24,33)(3,25,34)(4,21,35)(5,22,31)(6,58,48)(7,59,49)(8,60,50)(9,56,46)(10,57,47)(11,19,51)(12,20,52)(13,16,53)(14,17,54)(15,18,55)(26,45,36)(27,41,37)(28,42,38)(29,43,39)(30,44,40), (1,36)(2,37)(3,38)(4,39)(5,40)(6,52)(7,53)(8,54)(9,55)(10,51)(11,47)(12,48)(13,49)(14,50)(15,46)(16,59)(17,60)(18,56)(19,57)(20,58)(21,43)(22,44)(23,45)(24,41)(25,42)(26,32)(27,33)(28,34)(29,35)(30,31)>;
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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60), (1,58,36,20)(2,59,37,16)(3,60,38,17)(4,56,39,18)(5,57,40,19)(6,12)(7,13)(8,14)(9,15)(10,11)(21,46,29,55)(22,47,30,51)(23,48,26,52)(24,49,27,53)(25,50,28,54)(31,44)(32,45)(33,41)(34,42)(35,43), (1,58,36,20)(2,59,37,16)(3,60,38,17)(4,56,39,18)(5,57,40,19)(6,45,12,32)(7,41,13,33)(8,42,14,34)(9,43,15,35)(10,44,11,31)(21,29)(22,30)(23,26)(24,27)(25,28)(46,55)(47,51)(48,52)(49,53)(50,54), (1,23,32)(2,24,33)(3,25,34)(4,21,35)(5,22,31)(6,58,48)(7,59,49)(8,60,50)(9,56,46)(10,57,47)(11,19,51)(12,20,52)(13,16,53)(14,17,54)(15,18,55)(26,45,36)(27,41,37)(28,42,38)(29,43,39)(30,44,40), (1,36)(2,37)(3,38)(4,39)(5,40)(6,52)(7,53)(8,54)(9,55)(10,51)(11,47)(12,48)(13,49)(14,50)(15,46)(16,59)(17,60)(18,56)(19,57)(20,58)(21,43)(22,44)(23,45)(24,41)(25,42)(26,32)(27,33)(28,34)(29,35)(30,31) );
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),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60)], [(1,58,36,20),(2,59,37,16),(3,60,38,17),(4,56,39,18),(5,57,40,19),(6,12),(7,13),(8,14),(9,15),(10,11),(21,46,29,55),(22,47,30,51),(23,48,26,52),(24,49,27,53),(25,50,28,54),(31,44),(32,45),(33,41),(34,42),(35,43)], [(1,58,36,20),(2,59,37,16),(3,60,38,17),(4,56,39,18),(5,57,40,19),(6,45,12,32),(7,41,13,33),(8,42,14,34),(9,43,15,35),(10,44,11,31),(21,29),(22,30),(23,26),(24,27),(25,28),(46,55),(47,51),(48,52),(49,53),(50,54)], [(1,23,32),(2,24,33),(3,25,34),(4,21,35),(5,22,31),(6,58,48),(7,59,49),(8,60,50),(9,56,46),(10,57,47),(11,19,51),(12,20,52),(13,16,53),(14,17,54),(15,18,55),(26,45,36),(27,41,37),(28,42,38),(29,43,39),(30,44,40)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,52),(7,53),(8,54),(9,55),(10,51),(11,47),(12,48),(13,49),(14,50),(15,46),(16,59),(17,60),(18,56),(19,57),(20,58),(21,43),(22,44),(23,45),(24,41),(25,42),(26,32),(27,33),(28,34),(29,35),(30,31)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 3 | 12 | 32 | 3 | 3 | 6 | 12 | 1 | 1 | 1 | 1 | 12 | 12 | 3 | 3 | 3 | 3 | 12 | 12 | 12 | 12 | 32 | 32 | 32 | 32 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C5 | C10 | S3 | C5xS3 | S4 | C42:S3 | C5xS4 | C5xC42:S3 | C42:S3 | C5xC42:S3 |
| kernel | C5xC42:S3 | C5xC42:C3 | C42:S3 | C42:C3 | C4xC20 | C42 | C2xC10 | C5 | C22 | C1 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 4 | 2 | 4 | 8 | 16 | 1 | 4 |
Matrix representation of C5xC42:S3 ►in GL3(F241) generated by
| 98 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 177 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 177 |
| 177 | 0 | 0 |
| 0 | 177 | 0 |
| 0 | 0 | 240 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 240 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 240 | 0 |
G:=sub<GL(3,GF(241))| [98,0,0,0,98,0,0,0,98],[177,0,0,0,240,0,0,0,177],[177,0,0,0,177,0,0,0,240],[0,0,1,1,0,0,0,1,0],[240,0,0,0,0,240,0,240,0] >;
C5xC42:S3 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes S_3
% in TeX
G:=Group("C5xC4^2:S3"); // GroupNames label
G:=SmallGroup(480,254);
// by ID
G=gap.SmallGroup(480,254);
# by ID
G:=PCGroup([7,-2,-5,-3,-2,2,-2,2,422,1683,185,360,1054,1173,102,15125,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=c,d*c*d^-1=b^-1*c^-1,e*c*e=b,e*d*e=d^-1>;
// generators/relations
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