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## G = C3×D52order 300 = 22·3·52

### Direct product of C3, D5 and D5

Aliases: C3×D52, C154D10, (C5×D5)⋊C6, C5⋊D52C6, C51(C6×D5), C522(C2×C6), (D5×C15)⋊2C2, (C5×C15)⋊5C22, (C3×C5⋊D5)⋊3C2, SmallGroup(300,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C3×D52
 Chief series C1 — C5 — C52 — C5×C15 — D5×C15 — C3×D52
 Lower central C52 — C3×D52
 Upper central C1 — C3

Generators and relations for C3×D52
G = < a,b,c,d,e | a3=b5=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×D52
On 30 points - transitive group 30T74
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 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 18)(2 17)(3 16)(4 20)(5 19)(6 23)(7 22)(8 21)(9 25)(10 24)(11 28)(12 27)(13 26)(14 30)(15 29)
(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 18)(2 19)(3 20)(4 16)(5 17)(6 23)(7 24)(8 25)(9 21)(10 22)(11 28)(12 29)(13 30)(14 26)(15 27)

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

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

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

G:=TransitiveGroup(30,74);

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 5E 5F 5G 5H 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 15A ··· 15H 15I ··· 15P 30A ··· 30H order 1 2 2 2 3 3 5 5 5 5 5 5 5 5 6 6 6 6 6 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 5 5 25 1 1 2 2 2 2 4 4 4 4 5 5 5 5 25 25 10 10 10 10 2 ··· 2 4 ··· 4 10 ··· 10

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 D52 C3×D52 kernel C3×D52 D5×C15 C3×C5⋊D5 D52 C5×D5 C5⋊D5 C3×D5 C15 D5 C5 C3 C1 # reps 1 2 1 2 4 2 4 4 8 8 4 8

Matrix representation of C3×D52 in GL4(𝔽31) generated by

 5 0 0 0 0 5 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 30 0 0 1 12
,
 30 0 0 0 0 30 0 0 0 0 30 19 0 0 0 1
,
 0 30 0 0 1 18 0 0 0 0 1 0 0 0 0 1
,
 30 13 0 0 0 1 0 0 0 0 30 0 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],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,12],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,19,1],[0,1,0,0,30,18,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,13,1,0,0,0,0,30,0,0,0,0,30] >;

C3×D52 in GAP, Magma, Sage, TeX

C_3\times D_5^2
% in TeX

G:=Group("C3xD5^2");
// GroupNames label

G:=SmallGroup(300,36);
// by ID

G=gap.SmallGroup(300,36);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,488,6004]);
// Polycyclic

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