direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C12, C20⋊2C6, C60⋊5C2, C12○Dic5, Dic5⋊2C6, D10.2C6, C6.14D10, C30.14C22, C5⋊2(C2×C12), C15⋊8(C2×C4), C4○(C3×Dic5), C2.1(C6×D5), C10.2(C2×C6), C12○(C3×Dic5), (C6×D5).4C2, (C3×Dic5)⋊5C2, SmallGroup(120,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C12 |
Generators and relations for D5×C12
G = < a,b,c | a12=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C12
►On 60 points
Generators in S60
(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 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)
(1 55 27 16 39)(2 56 28 17 40)(3 57 29 18 41)(4 58 30 19 42)(5 59 31 20 43)(6 60 32 21 44)(7 49 33 22 45)(8 50 34 23 46)(9 51 35 24 47)(10 52 36 13 48)(11 53 25 14 37)(12 54 26 15 38)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)
G:=sub<Sym(60)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,55,27,16,39)(2,56,28,17,40)(3,57,29,18,41)(4,58,30,19,42)(5,59,31,20,43)(6,60,32,21,44)(7,49,33,22,45)(8,50,34,23,46)(9,51,35,24,47)(10,52,36,13,48)(11,53,25,14,37)(12,54,26,15,38), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,55,27,16,39)(2,56,28,17,40)(3,57,29,18,41)(4,58,30,19,42)(5,59,31,20,43)(6,60,32,21,44)(7,49,33,22,45)(8,50,34,23,46)(9,51,35,24,47)(10,52,36,13,48)(11,53,25,14,37)(12,54,26,15,38), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60)], [(1,55,27,16,39),(2,56,28,17,40),(3,57,29,18,41),(4,58,30,19,42),(5,59,31,20,43),(6,60,32,21,44),(7,49,33,22,45),(8,50,34,23,46),(9,51,35,24,47),(10,52,36,13,48),(11,53,25,14,37),(12,54,26,15,38)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,37),(12,38),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,49),(23,50),(24,51)]])
D5×C12 is a maximal subgroup of
C20.32D6 C60.C4 C12.F5 C60⋊C4 D6.D10 D12⋊5D5 C12.28D10
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D5 | D10 | C3×D5 | C4×D5 | C6×D5 | D5×C12 |
| kernel | D5×C12 | C3×Dic5 | C60 | C6×D5 | C4×D5 | C3×D5 | Dic5 | C20 | D10 | D5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of D5×C12 ►in GL2(𝔽61) generated by
| 21 | 0 |
| 0 | 21 |
| 0 | 1 |
| 60 | 17 |
| 1 | 0 |
| 17 | 60 |
G:=sub<GL(2,GF(61))| [21,0,0,21],[0,60,1,17],[1,17,0,60] >;
D5×C12 in GAP, Magma, Sage, TeX
D_5\times C_{12} % in TeX
G:=Group("D5xC12"); // GroupNames label
G:=SmallGroup(120,17);
// by ID
G=gap.SmallGroup(120,17);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-5,66,2404]);
// Polycyclic
G:=Group<a,b,c|a^12=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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