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G = C5xS3xA4order 360 = 23·32·5

Direct product of C5, S3 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C5xS3xA4, C3:(C10xA4), (C2xC6):C30, C15:3(C2xA4), (C2xC30):3C6, (C3xA4):3C10, (A4xC15):7C2, (C22xS3):C15, C22:2(S3xC15), (S3xC2xC10):C3, (C2xC10):4(C3xS3), SmallGroup(360,143)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C5xS3xA4
C1C3C2xC6C2xC30A4xC15 — C5xS3xA4
C2xC6 — C5xS3xA4
C1C5

Generators and relations for C5xS3xA4
 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 >

Subgroups: 188 in 52 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C3, C5, S3, C6, C10, A4, C15, C3xS3, C2xA4, C5xS3, C30, C5xA4, S3xA4, S3xC15, C10xA4, C5xS3xA4
3C2
3C2
9C2
4C3
8C3
9C22
9C22
3C6
3S3
12C6
4C32
3C10
3C10
9C10
4C15
8C15
3C23
2A4
3D6
3D6
4C3xS3
9C2xC10
9C2xC10
3C30
3C5xS3
12C30
4C3xC15
3C2xA4
3C22xC10
2C5xA4
3S3xC10
3S3xC10
4S3xC15
3C10xA4

Smallest permutation representation of C5xS3xA4
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 2A2B2C3A3B3C3D3E5A5B5C5D6A6B6C10A···10H10I10J10K10L15A15B15C15D15E···15L15M···15T30A30B30C30D30E···30L
order122233333555566610···10101010101515151515···1515···153030303030···30
size1339244881111612123···3999922224···48···8666612···12

60 irreducible representations

dim111111112222333366
type++++++
imageC1C2C3C5C6C10C15C30S3C3xS3C5xS3S3xC15A4C2xA4C5xA4C10xA4S3xA4C5xS3xA4
kernelC5xS3xA4A4xC15S3xC2xC10S3xA4C2xC30C3xA4C22xS3C2xC6C5xA4C2xC10A4C22C5xS3C15S3C3C5C1
# reps112424881248114414

Matrix representation of C5xS3xA4 in GL5(F31)

80000
08000
00800
00080
00008
,
3030000
10000
00100
00010
00001
,
10000
3030000
00100
00010
00001
,
10000
01000
00303030
00001
00010
,
10000
01000
00010
00100
00303030
,
10000
01000
00100
00303030
00010

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] >;

C5xS3xA4 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

Export

Subgroup lattice of C5xS3xA4 in TeX

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