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## G = C5×S3×D5order 300 = 22·3·52

### Direct product of C5, S3 and D5

Aliases: C5×S3×D5, D15⋊C10, C525D6, C155D10, (C5×S3)⋊C10, C15⋊(C2×C10), (C3×D5)⋊C10, C31(D5×C10), C51(S3×C10), (C5×D15)⋊1C2, (D5×C15)⋊3C2, (C5×C15)⋊1C22, (S3×C52)⋊1C2, SmallGroup(300,37)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×S3×D5
G = < a,b,c,d,e | a5=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 C5×S3×D5
On 30 points - transitive group 30T75
Generators in S30
(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 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)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 20)(2 16)(3 17)(4 18)(5 19)(6 25)(7 21)(8 22)(9 23)(10 24)(11 30)(12 26)(13 27)(14 28)(15 29)

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

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

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

G:=TransitiveGroup(30,75);

60 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 5E ··· 5N 6 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10R 10S 10T 10U 10V 15A 15B 15C 15D 15E ··· 15N 30A 30B 30C 30D order 1 2 2 2 3 5 5 5 5 5 ··· 5 6 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 3 5 15 2 1 1 1 1 2 ··· 2 10 3 3 3 3 5 5 5 5 6 ··· 6 15 15 15 15 2 2 2 2 4 ··· 4 10 10 10 10

60 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 C5 C10 C10 C10 S3 D5 D6 D10 C5×S3 C5×D5 S3×C10 D5×C10 S3×D5 C5×S3×D5 kernel C5×S3×D5 D5×C15 S3×C52 C5×D15 S3×D5 C5×S3 C3×D5 D15 C5×D5 C5×S3 C52 C15 D5 S3 C5 C3 C5 C1 # reps 1 1 1 1 4 4 4 4 1 2 1 2 4 8 4 8 2 8

Matrix representation of C5×S3×D5 in GL4(𝔽31) generated by

 2 0 0 0 0 2 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 0 30 0 0 1 30
,
 1 0 0 0 0 1 0 0 0 0 1 30 0 0 0 30
,
 4 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 8 0 0 4 0 0 0 0 0 30 0 0 0 0 30
G:=sub<GL(4,GF(31))| [2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30],[4,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,8,0,0,0,0,0,30,0,0,0,0,30] >;

C5×S3×D5 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_5
% in TeX

G:=Group("C5xS3xD5");
// GroupNames label

G:=SmallGroup(300,37);
// by ID

G=gap.SmallGroup(300,37);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-5,408,6004]);
// Polycyclic

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