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G = C3×S3×D5order 180 = 22·32·5

Direct product of C3, S3 and D5

Aliases: C3×S3×D5, D15⋊C6, C154D6, C323D10, (C5×S3)⋊C6, C15⋊(C2×C6), C51(S3×C6), (C3×D5)⋊C6, C31(C6×D5), (S3×C15)⋊2C2, (C3×D15)⋊1C2, (C3×C15)⋊1C22, (C32×D5)⋊1C2, SmallGroup(180,26)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S3×D5
G = < a,b,c,d,e | a3=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Permutation representations of C3×S3×D5
On 30 points - transitive group 30T44
Generators in S30
(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 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(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 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)| (1,9,14)(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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,2,3,4,5)(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( (1,9,14)(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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,2,3,4,5)(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([[(1,9,14),(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)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(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)], [(1,2,3,4,5),(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(30,44);

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D order 1 2 2 2 3 3 3 3 3 5 5 6 6 6 6 6 6 6 6 6 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 3 5 15 1 1 2 2 2 2 2 3 3 5 5 10 10 10 15 15 6 6 2 2 2 2 4 ··· 4 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D5 D6 C3×S3 D10 C3×D5 S3×C6 C6×D5 S3×D5 C3×S3×D5 kernel C3×S3×D5 C32×D5 S3×C15 C3×D15 S3×D5 C5×S3 C3×D5 D15 C3×D5 C3×S3 C15 D5 C32 S3 C5 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 2 1 2 2 4 2 4 2 4

Matrix representation of C3×S3×D5 in GL4(𝔽31) generated by

 5 0 0 0 0 5 0 0 0 0 1 0 0 0 0 1
,
 5 0 0 0 0 25 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
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 30 0
,
 1 0 0 0 0 1 0 0 0 0 1 12 0 0 0 30
G:=sub<GL(4,GF(31))| [5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,25,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,30,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,12,30] >;

C3×S3×D5 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_5
% in TeX

G:=Group("C3xS3xD5");
// GroupNames label

G:=SmallGroup(180,26);
// by ID

G=gap.SmallGroup(180,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,248,3604]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=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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