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G = C20⋊S4order 480 = 25·3·5

1st semidirect product of C20 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C201S4, C22⋊D60, A41D20, C23.3D30, C4⋊(C5⋊S4), C51(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

C1C22C10×A4 — C20⋊S4
C1C22C2×C10C5×A4C10×A4C2×C5⋊S4 — C20⋊S4
C5×A4C10×A4 — C20⋊S4
C1C2C4

Generators and relations for C20⋊S4
 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 >

Subgroups: 1140 in 112 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, A4, D6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, D12, S4, C2×A4, D15, C30, C4⋊D4, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×A4, C2×S4, C60, C5×A4, D30, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C4⋊S4, D60, C5⋊S4, C10×A4, C207D4, A4×C20, C2×C5⋊S4, C20⋊S4
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, D12, S4, D15, D20, C2×S4, D30, C4⋊S4, D60, C5⋊S4, C2×C5⋊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)(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 23 56)(2 24 57)(3 25 58)(4 26 59)(5 27 60)(6 28 41)(7 29 42)(8 30 43)(9 31 44)(10 32 45)(11 33 46)(12 34 47)(13 35 48)(14 36 49)(15 37 50)(16 38 51)(17 39 52)(18 40 53)(19 21 54)(20 22 55)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 60)(40 59)

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

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

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

46 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B 6 10A10B10C10D10E10F12A12B15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D60A···60H
order1222223444455610101010101012121515151520202020202020203030303060···60
size1133606082660602282266668888882222666688888···8

46 irreducible representations

dim1112222222222336666
type+++++++++++++++++++
imageC1C2C2S3D4D5D6D10D12D15D20D30D60S4C2×S4C4⋊S4C5⋊S4C2×C5⋊S4C20⋊S4
kernelC20⋊S4A4×C20C2×C5⋊S4C22×C20C5×A4C4×A4C22×C10C2×A4C2×C10C22×C4A4C23C22C20C10C5C4C2C1
# reps1121121224448221224

Matrix representation of C20⋊S4 in GL5(𝔽61)

1552000
915000
006000
000600
000060
,
10000
01000
006000
006001
006010
,
10000
01000
000160
001060
000060
,
304000
5730000
00010
00001
00100
,
915000
1552000
000600
006000
000060

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] >;

C20⋊S4 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

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