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## G = C5×S3×A4order 360 = 23·32·5

### Direct product of C5, S3 and A4

Aliases: C5×S3×A4, C3⋊(C10×A4), (C2×C6)⋊C30, C153(C2×A4), (C2×C30)⋊3C6, (C3×A4)⋊3C10, (A4×C15)⋊7C2, (C22×S3)⋊C15, C222(S3×C15), (S3×C2×C10)⋊C3, (C2×C10)⋊4(C3×S3), SmallGroup(360,143)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×S3×A4
 Chief series C1 — C3 — C2×C6 — C2×C30 — A4×C15 — C5×S3×A4
 Lower central C2×C6 — C5×S3×A4
 Upper central C1 — C5

Generators and relations for C5×S3×A4
G = < a,b,c,d,e,f | a5=b3=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Smallest permutation representation of C5×S3×A4
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 36 8)(2 37 9)(3 38 10)(4 39 6)(5 40 7)(11 24 32)(12 25 33)(13 21 34)(14 22 35)(15 23 31)(16 55 46)(17 51 47)(18 52 48)(19 53 49)(20 54 50)(26 41 59)(27 42 60)(28 43 56)(29 44 57)(30 45 58)
(6 39)(7 40)(8 36)(9 37)(10 38)(11 32)(12 33)(13 34)(14 35)(15 31)(26 41)(27 42)(28 43)(29 44)(30 45)(46 55)(47 51)(48 52)(49 53)(50 54)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 46)(7 47)(8 48)(9 49)(10 50)(11 45)(12 41)(13 42)(14 43)(15 44)(21 60)(22 56)(23 57)(24 58)(25 59)(26 33)(27 34)(28 35)(29 31)(30 32)(36 52)(37 53)(38 54)(39 55)(40 51)
(1 58)(2 59)(3 60)(4 56)(5 57)(6 43)(7 44)(8 45)(9 41)(10 42)(11 48)(12 49)(13 50)(14 46)(15 47)(16 22)(17 23)(18 24)(19 25)(20 21)(26 37)(27 38)(28 39)(29 40)(30 36)(31 51)(32 52)(33 53)(34 54)(35 55)
(11 48 45)(12 49 41)(13 50 42)(14 46 43)(15 47 44)(16 56 22)(17 57 23)(18 58 24)(19 59 25)(20 60 21)(26 33 53)(27 34 54)(28 35 55)(29 31 51)(30 32 52)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 6C 10A ··· 10H 10I 10J 10K 10L 15A 15B 15C 15D 15E ··· 15L 15M ··· 15T 30A 30B 30C 30D 30E ··· 30L order 1 2 2 2 3 3 3 3 3 5 5 5 5 6 6 6 10 ··· 10 10 10 10 10 15 15 15 15 15 ··· 15 15 ··· 15 30 30 30 30 30 ··· 30 size 1 3 3 9 2 4 4 8 8 1 1 1 1 6 12 12 3 ··· 3 9 9 9 9 2 2 2 2 4 ··· 4 8 ··· 8 6 6 6 6 12 ··· 12

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 C3×S3 C5×S3 S3×C15 A4 C2×A4 C5×A4 C10×A4 S3×A4 C5×S3×A4 kernel C5×S3×A4 A4×C15 S3×C2×C10 S3×A4 C2×C30 C3×A4 C22×S3 C2×C6 C5×A4 C2×C10 A4 C22 C5×S3 C15 S3 C3 C5 C1 # reps 1 1 2 4 2 4 8 8 1 2 4 8 1 1 4 4 1 4

Matrix representation of C5×S3×A4 in GL5(𝔽31)

 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 30 30 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 30 30 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 30 30 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 30 30 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 30 30 30 0 0 0 1 0

G:=sub<GL(5,GF(31))| [8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[30,1,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,30,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,30,0,1,0,0,30,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,1,0,30,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,1,30,0,0,0,0,30,1,0,0,0,30,0] >;

C5×S3×A4 in GAP, Magma, Sage, TeX

C_5\times S_3\times A_4
% in TeX

G:=Group("C5xS3xA4");
// GroupNames label

G:=SmallGroup(360,143);
// by ID

G=gap.SmallGroup(360,143);
# by ID

G:=PCGroup([6,-2,-3,-5,-2,2,-3,1089,460,8645]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^3=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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