direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C15, C5⋊C30, C52⋊3C6, C15⋊2C10, (C5×C15)⋊3C2, SmallGroup(150,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C15 |
Generators and relations for D5×C15
G = < a,b,c | a15=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of D5×C15
►On 30 points - transitive group 30T39
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 13 10 7 4)(2 14 11 8 5)(3 15 12 9 6)(16 19 22 25 28)(17 20 23 26 29)(18 21 24 27 30)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 16)(11 17)(12 18)(13 19)(14 20)(15 21)
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), (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21)>;
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), (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21) );
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)], [(1,13,10,7,4),(2,14,11,8,5),(3,15,12,9,6),(16,19,22,25,28),(17,20,23,26,29),(18,21,24,27,30)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,16),(11,17),(12,18),(13,19),(14,20),(15,21)]])
G:=TransitiveGroup(30,39);
D5×C15 is a maximal subgroup of
D5.D15
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15AB | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | D5 | C3×D5 | C5×D5 | D5×C15 |
| kernel | D5×C15 | C5×C15 | C5×D5 | C3×D5 | C52 | C15 | D5 | C5 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 2 | 4 | 8 | 16 |
Matrix representation of D5×C15 ►in GL2(𝔽31) generated by
| 18 | 0 |
| 0 | 18 |
| 16 | 0 |
| 3 | 2 |
| 29 | 30 |
| 3 | 2 |
G:=sub<GL(2,GF(31))| [18,0,0,18],[16,3,0,2],[29,3,30,2] >;
D5×C15 in GAP, Magma, Sage, TeX
D_5\times C_{15} % in TeX
G:=Group("D5xC15"); // GroupNames label
G:=SmallGroup(150,8);
// by ID
G=gap.SmallGroup(150,8);
# by ID
G:=PCGroup([4,-2,-3,-5,-5,1923]);
// Polycyclic
G:=Group<a,b,c|a^15=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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