direct product, non-abelian, soluble, monomial
Aliases: C20×S4, (C2×S4).C10, C22⋊(S3×C20), A4⋊C4⋊2C10, (C4×A4)⋊2C10, (A4×C20)⋊6C2, A4⋊1(C2×C20), C2.1(C10×S4), (C10×S4).2C2, C10.28(C2×S4), (C22×C20)⋊1S3, C23.2(S3×C10), (C22×C10).9D6, (C10×A4).19C22, (C5×A4⋊C4)⋊5C2, (C5×A4)⋊9(C2×C4), (C2×C10)⋊6(C4×S3), (C22×C4)⋊1(C5×S3), (C2×A4).2(C2×C10), SmallGroup(480,1014)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C20×S4 |
Subgroups: 344 in 112 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×5], C22, C22 [×6], C5, S3 [×2], C6, C2×C4 [×7], D4 [×4], C23, C23, C10, C10 [×4], Dic3, C12, A4, D6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C20, C20 [×5], C2×C10, C2×C10 [×6], C4×S3, S4 [×2], C2×A4, C5×S3 [×2], C30, C4×D4, C2×C20 [×7], C5×D4 [×4], C22×C10, C22×C10, A4⋊C4, C4×A4, C2×S4, C5×Dic3, C60, C5×A4, S3×C10, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C4×S4, S3×C20, C5×S4 [×2], C10×A4, D4×C20, C5×A4⋊C4, A4×C20, C10×S4, C20×S4
Quotients:
C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, C10 [×3], D6, C20 [×2], C2×C10, C4×S3, S4, C5×S3, C2×C20, C2×S4, S3×C10, C4×S4, S3×C20, C5×S4, C10×S4, C20×S4
Generators and relations
G = < a,b,c,d,e | a20=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 >
►On 60 points
(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 49 27)(2 50 28)(3 51 29)(4 52 30)(5 53 31)(6 54 32)(7 55 33)(8 56 34)(9 57 35)(10 58 36)(11 59 37)(12 60 38)(13 41 39)(14 42 40)(15 43 21)(16 44 22)(17 45 23)(18 46 24)(19 47 25)(20 48 26)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,49,27)(2,50,28)(3,51,29)(4,52,30)(5,53,31)(6,54,32)(7,55,33)(8,56,34)(9,57,35)(10,58,36)(11,59,37)(12,60,38)(13,41,39)(14,42,40)(15,43,21)(16,44,22)(17,45,23)(18,46,24)(19,47,25)(20,48,26), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)>;
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,49,27)(2,50,28)(3,51,29)(4,52,30)(5,53,31)(6,54,32)(7,55,33)(8,56,34)(9,57,35)(10,58,36)(11,59,37)(12,60,38)(13,41,39)(14,42,40)(15,43,21)(16,44,22)(17,45,23)(18,46,24)(19,47,25)(20,48,26), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52) );
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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,49,27),(2,50,28),(3,51,29),(4,52,30),(5,53,31),(6,54,32),(7,55,33),(8,56,34),(9,57,35),(10,58,36),(11,59,37),(12,60,38),(13,41,39),(14,42,40),(15,43,21),(16,44,22),(17,45,23),(18,46,24),(19,47,25),(20,48,26)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52)])
Matrix representation ►G ⊆ GL3(𝔽61) generated by
| 38 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 38 |
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 60 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 60 | 0 |
G:=sub<GL(3,GF(61))| [38,0,0,0,38,0,0,0,38],[60,0,0,0,60,0,0,0,1],[1,0,0,0,60,0,0,0,60],[0,1,0,0,0,1,1,0,0],[60,0,0,0,0,60,0,60,0] >;
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AN | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 8 | 1 | 1 | 3 | 3 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C20 | S4 | C2×S4 | C4×S4 | C5×S4 | C10×S4 | C20×S4 |
| kernel | C20×S4 | C5×A4⋊C4 | A4×C20 | C10×S4 | C5×S4 | C4×S4 | A4⋊C4 | C4×A4 | C2×S4 | S4 | C22×C20 | C22×C10 | C2×C10 | C22×C4 | C23 | C22 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_{20}\times S_4 % in TeX
G:=Group("C20xS4"); // GroupNames label
G:=SmallGroup(480,1014);
// by ID
G=gap.SmallGroup(480,1014);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,148,2804,10085,285,5886,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=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