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G = C20xS4order 480 = 25·3·5

Direct product of C20 and S4

direct product, non-abelian, soluble, monomial

Aliases: C20xS4, (C2xS4).C10, C22:(S3xC20), A4:C4:2C10, (A4xC20):6C2, (C4xA4):2C10, A4:1(C2xC20), C2.1(C10xS4), (C10xS4).2C2, C10.28(C2xS4), (C22xC20):1S3, C23.2(S3xC10), (C22xC10).9D6, (C10xA4).19C22, (C5xA4:C4):5C2, (C5xA4):9(C2xC4), (C2xC10):6(C4xS3), (C22xC4):1(C5xS3), (C2xA4).2(C2xC10), SmallGroup(480,1014)

Series: Derived Chief Lower central Upper central

C1C22A4 — C20xS4
C1C22A4C2xA4C10xA4C10xS4 — C20xS4
A4 — C20xS4
C1C20

Generators and relations for C20xS4
 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, C2xC4, D4, C23, C23, C10, C10, Dic3, C12, A4, D6, C15, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C20, C20, C2xC10, C2xC10, C4xS3, S4, C2xA4, C5xS3, C30, C4xD4, C2xC20, C5xD4, C22xC10, C22xC10, A4:C4, C4xA4, C2xS4, C5xDic3, C60, C5xA4, S3xC10, C4xC20, C5xC22:C4, C5xC4:C4, C22xC20, C22xC20, D4xC10, C4xS4, S3xC20, C5xS4, C10xA4, D4xC20, C5xA4:C4, A4xC20, C10xS4, C20xS4
Quotients: C1, C2, C4, C22, C5, S3, C2xC4, C10, D6, C20, C2xC10, C4xS3, S4, C5xS3, C2xC20, C2xS4, S3xC10, C4xS4, S3xC20, C5xS4, C10xS4, C20xS4

Smallest permutation representation of C20xS4
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 2A2B2C2D2E 3 4A4B4C4D4E···4J5A5B5C5D 6 10A10B10C10D10E···10L10M···10T12A12B15A15B15C15D20A···20H20I···20P20Q···20AN30A30B30C30D60A···60H
order122222344444···4555561010101010···1010···1012121515151520···2020···2020···203030303060···60
size113366811336···61111811113···36···68888881···13···36···688888···8

100 irreducible representations

dim1111111111222222333333
type++++++++
imageC1C2C2C2C4C5C10C10C10C20S3D6C4xS3C5xS3S3xC10S3xC20S4C2xS4C4xS4C5xS4C10xS4C20xS4
kernelC20xS4C5xA4:C4A4xC20C10xS4C5xS4C4xS4A4:C4C4xA4C2xS4S4C22xC20C22xC10C2xC10C22xC4C23C22C20C10C5C4C2C1
# reps111144444161124482248816

Matrix representation of C20xS4 in GL3(F61) generated by

3800
0380
0038
,
6000
0600
001
,
100
0600
0060
,
001
100
010
,
6000
0060
0600
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] >;

C20xS4 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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