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G = S32×C10order 360 = 23·32·5

Direct product of C10, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C10
 Chief series C1 — C3 — C32 — C3×C15 — S3×C15 — C5×S32 — S32×C10
 Lower central C32 — S32×C10
 Upper central C1 — C10

Generators and relations for S32×C10
G = < a,b,c,d,e | a10=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 412 in 138 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C5, S3, S3, C6, C6, C23, C32, C10, C10, D6, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, C2×C10, C22×S3, C5×S3, C5×S3, C30, C30, S32, S3×C6, C2×C3⋊S3, C22×C10, C3×C15, S3×C10, S3×C10, C2×C30, C2×S32, S3×C15, C5×C3⋊S3, C3×C30, S3×C2×C10, C5×S32, S3×C30, C10×C3⋊S3, S32×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C2×C10, C22×S3, C5×S3, S32, C22×C10, S3×C10, C2×S32, S3×C2×C10, C5×S32, S32×C10

Smallest permutation representation of S32×C10
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 39 45)(2 40 46)(3 31 47)(4 32 48)(5 33 49)(6 34 50)(7 35 41)(8 36 42)(9 37 43)(10 38 44)(11 26 57)(12 27 58)(13 28 59)(14 29 60)(15 30 51)(16 21 52)(17 22 53)(18 23 54)(19 24 55)(20 25 56)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)
(1 45 39)(2 46 40)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 41 35)(8 42 36)(9 43 37)(10 44 38)(11 26 57)(12 27 58)(13 28 59)(14 29 60)(15 30 51)(16 21 52)(17 22 53)(18 23 54)(19 24 55)(20 25 56)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D6 D6 C5×S3 S3×C10 S3×C10 S32 C2×S32 C5×S32 S32×C10 kernel S32×C10 C5×S32 S3×C30 C10×C3⋊S3 C2×S32 S32 S3×C6 C2×C3⋊S3 S3×C10 C5×S3 C30 D6 S3 C6 C10 C5 C2 C1 # reps 1 4 2 1 4 16 8 4 2 4 2 8 16 8 1 1 4 4

Matrix representation of S32×C10 in GL4(𝔽31) generated by

 30 0 0 0 0 30 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 30 1 0 0 30 0
,
 1 0 0 0 0 1 0 0 0 0 1 30 0 0 0 30
,
 30 1 0 0 30 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30],[30,30,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S32×C10 in GAP, Magma, Sage, TeX

S_3^2\times C_{10}
% in TeX

G:=Group("S3^2xC10");
// GroupNames label

G:=SmallGroup(360,153);
// by ID

G=gap.SmallGroup(360,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-3,-3,1210,8645]);
// Polycyclic

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

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