non-abelian, soluble, monomial
Aliases: C20⋊1S4, C22⋊D60, A4⋊1D20, C23.3D30, C4⋊(C5⋊S4), C5⋊1(C4⋊S4), (C5×A4)⋊4D4, (C4×A4)⋊1D5, (A4×C20)⋊1C2, (C2×C10)⋊3D12, C10.18(C2×S4), (C22×C20)⋊2S3, (C22×C4)⋊2D15, (C2×A4).10D10, (C22×C10).15D6, (C10×A4).10C22, (C2×C5⋊S4)⋊1C2, C2.4(C2×C5⋊S4), SmallGroup(480,1026)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1140 in 112 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22, C22 [×8], C5, S3 [×2], C6, C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×2], C10, C10 [×2], C12, A4, D6 [×2], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20, C20, D10 [×6], C2×C10, C2×C10 [×2], D12, S4 [×2], C2×A4, D15 [×2], C30, C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5 [×2], C22×C10, C4×A4, C2×S4 [×2], C60, C5×A4, D30 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C4⋊S4, D60, C5⋊S4 [×2], C10×A4, C20⋊7D4, A4×C20, C2×C5⋊S4 [×2], C20⋊S4
Quotients:
C1, C2 [×3], C22, S3, D4, D5, D6, D10, D12, S4, D15, D20, C2×S4, D30, C4⋊S4, D60, C5⋊S4, C2×C5⋊S4, C20⋊S4
Generators and relations
G = < a,b,c,d,e | a20=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 >
►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)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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 32 59)(2 33 60)(3 34 41)(4 35 42)(5 36 43)(6 37 44)(7 38 45)(8 39 46)(9 40 47)(10 21 48)(11 22 49)(12 23 50)(13 24 51)(14 25 52)(15 26 53)(16 27 54)(17 28 55)(18 29 56)(19 30 57)(20 31 58)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)
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)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,32,59)(2,33,60)(3,34,41)(4,35,42)(5,36,43)(6,37,44)(7,38,45)(8,39,46)(9,40,47)(10,21,48)(11,22,49)(12,23,50)(13,24,51)(14,25,52)(15,26,53)(16,27,54)(17,28,55)(18,29,56)(19,30,57)(20,31,58), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)>;
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)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,32,59)(2,33,60)(3,34,41)(4,35,42)(5,36,43)(6,37,44)(7,38,45)(8,39,46)(9,40,47)(10,21,48)(11,22,49)(12,23,50)(13,24,51)(14,25,52)(15,26,53)(16,27,54)(17,28,55)(18,29,56)(19,30,57)(20,31,58), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51) );
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),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,32,59),(2,33,60),(3,34,41),(4,35,42),(5,36,43),(6,37,44),(7,38,45),(8,39,46),(9,40,47),(10,21,48),(11,22,49),(12,23,50),(13,24,51),(14,25,52),(15,26,53),(16,27,54),(17,28,55),(18,29,56),(19,30,57),(20,31,58)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 15 | 52 | 0 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 30 | 4 | 0 | 0 | 0 |
| 57 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 |
| 15 | 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [15,9,0,0,0,52,15,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[30,57,0,0,0,4,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[9,15,0,0,0,15,52,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60] >;
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 60 | 60 | 8 | 2 | 6 | 60 | 60 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
46 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | D12 | D15 | D20 | D30 | D60 | S4 | C2×S4 | C4⋊S4 | C5⋊S4 | C2×C5⋊S4 | C20⋊S4 |
| kernel | C20⋊S4 | A4×C20 | C2×C5⋊S4 | C22×C20 | C5×A4 | C4×A4 | C22×C10 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes S_4 % in TeX
G:=Group("C20:S4"); // GroupNames label
G:=SmallGroup(480,1026);
// by ID
G=gap.SmallGroup(480,1026);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,36,451,3364,10085,1286,5886,2232]);
// 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,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