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## G = C2×C10×S4order 480 = 25·3·5

### Direct product of C2×C10 and S4

Aliases: C2×C10×S4, C23⋊(S3×C10), A4⋊(C22×C10), C242(C5×S3), (C5×A4)⋊4C23, (C23×C10)⋊2S3, (C22×C10)⋊2D6, (C22×A4)⋊3C10, (C10×A4)⋊4C22, C22⋊(S3×C2×C10), (C2×A4)⋊(C2×C10), (A4×C2×C10)⋊7C2, (C2×C10)⋊2(C22×S3), SmallGroup(480,1198)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C2×C10×S4
 Chief series C1 — C22 — A4 — C5×A4 — C5×S4 — C10×S4 — C2×C10×S4
 Lower central A4 — C2×C10×S4
 Upper central C1 — C2×C10

Generators and relations for C2×C10×S4
G = < a,b,c,d,e,f | a2=b10=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 840 in 262 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, A4, D6, C2×C6, C15, C22×C4, C2×D4, C24, C24, C20, C2×C10, C2×C10, S4, C2×A4, C22×S3, C5×S3, C30, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C2×S4, C22×A4, C5×A4, S3×C10, C2×C30, C22×C20, D4×C10, C23×C10, C23×C10, C22×S4, C5×S4, C10×A4, S3×C2×C10, D4×C2×C10, C10×S4, A4×C2×C10, C2×C10×S4
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C2×C10, S4, C22×S3, C5×S3, C22×C10, C2×S4, S3×C10, C22×S4, C5×S4, S3×C2×C10, C10×S4, C2×C10×S4

Smallest permutation representation of C2×C10×S4
On 60 points
Generators in S60
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 48)(12 49)(13 50)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 51)(40 52)
(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 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 60)
(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 35 44)(2 36 45)(3 37 46)(4 38 47)(5 39 48)(6 40 49)(7 31 50)(8 32 41)(9 33 42)(10 34 43)(11 30 51)(12 21 52)(13 22 53)(14 23 54)(15 24 55)(16 25 56)(17 26 57)(18 27 58)(19 28 59)(20 29 60)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 31)(19 32)(20 33)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 10AC ··· 10AR 15A 15B 15C 15D 20A ··· 20P 30A ··· 30L order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 3 3 3 3 6 6 6 6 8 6 6 6 6 1 1 1 1 8 8 8 1 ··· 1 3 ··· 3 6 ··· 6 8 8 8 8 6 ··· 6 8 ··· 8

100 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 S4 C2×S4 C5×S4 C10×S4 kernel C2×C10×S4 C10×S4 A4×C2×C10 C22×S4 C2×S4 C22×A4 C23×C10 C22×C10 C24 C23 C2×C10 C10 C22 C2 # reps 1 6 1 4 24 4 1 3 4 12 2 6 8 24

Matrix representation of C2×C10×S4 in GL5(𝔽61)

 60 0 0 0 0 0 60 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 52 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 1 1 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 60 0 60
,
 60 60 0 0 0 1 0 0 0 0 0 0 60 60 59 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 60 60 59 0 0 0 0 1

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,1,0,0,0,60,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,0,0,0,60,0,0,0,0,0,60],[60,1,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,60,0,0,0,0,59,0,1],[60,1,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,60,0,0,0,0,59,1] >;

C2×C10×S4 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times S_4
% in TeX

G:=Group("C2xC10xS4");
// GroupNames label

G:=SmallGroup(480,1198);
// by ID

G=gap.SmallGroup(480,1198);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-3,-2,2,2804,10085,285,5886,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^10=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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