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## G = C3×S3×F5order 360 = 23·32·5

### Direct product of C3, S3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×S3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C32×D5 — C32×F5 — C3×S3×F5
 Lower central C15 — C3×S3×F5
 Upper central C1 — C3

Generators and relations for C3×S3×F5
G = < a,b,c,d,e | a3=b3=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 292 in 70 conjugacy classes, 28 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C5, S3, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3×C6, F5, F5, D10, C4×S3, C2×C12, C5×S3, C3×D5, C3×D5, D15, C30, C3×Dic3, C3×C12, S3×C6, C2×F5, C3×C15, C3×F5, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C12, C32×D5, S3×C15, C3×D15, S3×F5, C6×F5, C32×F5, C3×C3⋊F5, C3×S3×D5, C3×S3×F5
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, F5, C4×S3, C2×C12, S3×C6, C2×F5, C3×F5, S3×C12, S3×F5, C6×F5, C3×S3×F5

Permutation representations of C3×S3×F5
On 30 points - transitive group 30T91
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

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

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

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

G:=TransitiveGroup(30,91);

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 F5 C2×F5 C3×F5 C6×F5 S3×F5 C3×S3×F5 kernel C3×S3×F5 C32×F5 C3×C3⋊F5 C3×S3×D5 S3×F5 S3×C15 C3×D15 C3×F5 C3⋊F5 S3×D5 C5×S3 D15 C3×F5 C3×D5 F5 C15 D5 C5 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 4 1 1 2 2 1 2

Matrix representation of C3×S3×F5 in GL6(𝔽61)

 47 0 0 0 0 0 0 47 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
,
 47 0 0 0 0 0 19 13 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
,
 60 5 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 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 1 0 0 60 0 0 0 1 0 60 0 0 0 0 1 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,19,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C3×S3×F5 in GAP, Magma, Sage, TeX

C_3\times S_3\times F_5
% in TeX

G:=Group("C3xS3xF5");
// GroupNames label

G:=SmallGroup(360,126);
// by ID

G=gap.SmallGroup(360,126);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,730,5189,887]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=e^4=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^-1=d^3>;
// generators/relations

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