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## G = S32×D5order 360 = 23·32·5

### Direct product of S3, S3 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — S32×D5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C3×S3×D5 — S32×D5
 Lower central C3×C15 — S32×D5
 Upper central C1

Generators and relations for S32×D5
G = < a,b,c,d,e,f | a3=b2=c3=d2=e5=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 996 in 138 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C3, C3, C22, C5, S3, S3, C6, C23, C32, D5, D5, C10, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, D10, C2×C10, C22×S3, C5×S3, C5×S3, C3×D5, C3×D5, D15, D15, C30, S32, S32, S3×C6, C2×C3⋊S3, C22×D5, C3×C15, S3×D5, S3×D5, C6×D5, S3×C10, D30, C2×S32, C32×D5, S3×C15, C3×D15, C5×C3⋊S3, C3⋊D15, C2×S3×D5, C3×S3×D5, D5×C3⋊S3, C5×S32, S3×D15, D15⋊S3, S32×D5
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, S32, C22×D5, S3×D5, C2×S32, C2×S3×D5, S32×D5

Permutation representations of S32×D5
On 30 points - transitive group 30T84
Generators in S30
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)

G:=sub<Sym(30)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)>;

G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29) );

G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,26),(7,27),(8,28),(9,29),(10,30),(11,21),(12,22),(13,23),(14,24),(15,25)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29)]])

G:=TransitiveGroup(30,84);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 15E 15F 30A 30B 30C 30D order 1 2 2 2 2 2 2 2 3 3 3 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 15 15 15 15 15 15 30 30 30 30 size 1 3 3 5 9 15 15 45 2 2 4 2 2 6 6 10 10 20 30 30 6 6 6 6 18 18 4 4 4 4 8 8 12 12 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D10 D10 S32 S3×D5 C2×S32 C2×S3×D5 S32×D5 kernel S32×D5 C3×S3×D5 D5×C3⋊S3 C5×S32 S3×D15 D15⋊S3 S3×D5 S32 C5×S3 C3×D5 D15 C3×S3 C3⋊S3 D5 S3 C5 C3 C1 # reps 1 2 1 1 2 1 2 2 2 2 2 4 2 1 4 1 4 2

Matrix representation of S32×D5 in GL6(𝔽31)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 30 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 13 1 0 0 0 0 17 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 1 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,17,0,0,0,0,1,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S32×D5 in GAP, Magma, Sage, TeX

S_3^2\times D_5
% in TeX

G:=Group("S3^2xD5");
// GroupNames label

G:=SmallGroup(360,137);
// by ID

G=gap.SmallGroup(360,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,111,730,10373]);
// Polycyclic

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

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