direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×D52, C15⋊4D10, (C5×D5)⋊C6, C5⋊D5⋊2C6, C5⋊1(C6×D5), C52⋊2(C2×C6), (D5×C15)⋊2C2, (C5×C15)⋊5C22, (C3×C5⋊D5)⋊3C2, SmallGroup(300,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×D52 |
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 >
5C2
5C2
25C2
2C5
2C5
25C22
5C6
5C6
25C6
5C10
5C10
5D5
5D5
10D5
10D5
2C15
2C15
25C2×C6
5D10
5D10
5C30
5C30
10C3×D5
10C3×D5
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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