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## G = C20×S4order 480 = 25·3·5

### Direct product of C20 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C20×S4
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C10×S4 — C20×S4
 Lower central A4 — C20×S4
 Upper central C1 — C20

Generators and relations for C20×S4
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 >

Subgroups: 344 in 112 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, A4, D6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C4×S3, S4, C2×A4, C5×S3, C30, C4×D4, C2×C20, C5×D4, C22×C10, C22×C10, A4⋊C4, C4×A4, C2×S4, C5×Dic3, C60, C5×A4, S3×C10, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C4×S4, S3×C20, C5×S4, C10×A4, D4×C20, C5×A4⋊C4, A4×C20, C10×S4, C20×S4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, D6, C20, C2×C10, C4×S3, S4, C5×S3, C2×C20, C2×S4, S3×C10, C4×S4, S3×C20, C5×S4, C10×S4, C20×S4

Smallest permutation representation of C20×S4
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 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 23)(2 50 24)(3 51 25)(4 52 26)(5 53 27)(6 54 28)(7 55 29)(8 56 30)(9 57 31)(10 58 32)(11 59 33)(12 60 34)(13 41 35)(14 42 36)(15 43 37)(16 44 38)(17 45 39)(18 46 40)(19 47 21)(20 48 22)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)

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), (1,49,23)(2,50,24)(3,51,25)(4,52,26)(5,53,27)(6,54,28)(7,55,29)(8,56,30)(9,57,31)(10,58,32)(11,59,33)(12,60,34)(13,41,35)(14,42,36)(15,43,37)(16,44,38)(17,45,39)(18,46,40)(19,47,21)(20,48,22), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)>;

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), (1,49,23)(2,50,24)(3,51,25)(4,52,26)(5,53,27)(6,54,28)(7,55,29)(8,56,30)(9,57,31)(10,58,32)(11,59,33)(12,60,34)(13,41,35)(14,42,36)(15,43,37)(16,44,38)(17,45,39)(18,46,40)(19,47,21)(20,48,22), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56) );

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)], [(1,49,23),(2,50,24),(3,51,25),(4,52,26),(5,53,27),(6,54,28),(7,55,29),(8,56,30),(9,57,31),(10,58,32),(11,59,33),(12,60,34),(13,41,35),(14,42,36),(15,43,37),(16,44,38),(17,45,39),(18,46,40),(19,47,21),(20,48,22)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,57),(22,58),(23,59),(24,60),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)]])

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

Matrix representation of C20×S4 in 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] >;

C20×S4 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

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