direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C6, C15⋊3C23, C30⋊3C22, C10⋊(C2×C6), C5⋊(C22×C6), (C2×C10)⋊5C6, (C2×C30)⋊5C2, SmallGroup(120,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C6 |
Generators and relations for D5×C2×C6
G = < a,b,c,d | a2=b6=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 152 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C6, C23, D5, C10, C2×C6, C2×C6, C15, D10, C2×C10, C22×C6, C3×D5, C30, C22×D5, C6×D5, C2×C30, D5×C2×C6
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, D10, C22×C6, C3×D5, C22×D5, C6×D5, D5×C2×C6
►On 60 points
Generators in S60
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(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 36 39 53 43)(2 31 40 54 44)(3 32 41 49 45)(4 33 42 50 46)(5 34 37 51 47)(6 35 38 52 48)(7 13 55 21 30)(8 14 56 22 25)(9 15 57 23 26)(10 16 58 24 27)(11 17 59 19 28)(12 18 60 20 29)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)(13 58)(14 59)(15 60)(16 55)(17 56)(18 57)(25 28)(26 29)(27 30)(31 51)(32 52)(33 53)(34 54)(35 49)(36 50)(37 40)(38 41)(39 42)
G:=sub<Sym(60)| (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,36)(20,31)(21,32)(22,33)(23,34)(24,35)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41), (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), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,13,55,21,30)(8,14,56,22,25)(9,15,57,23,26)(10,16,58,24,27)(11,17,59,19,28)(12,18,60,20,29), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23)(13,58)(14,59)(15,60)(16,55)(17,56)(18,57)(25,28)(26,29)(27,30)(31,51)(32,52)(33,53)(34,54)(35,49)(36,50)(37,40)(38,41)(39,42)>;
G:=Group( (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,36)(20,31)(21,32)(22,33)(23,34)(24,35)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41), (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), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,13,55,21,30)(8,14,56,22,25)(9,15,57,23,26)(10,16,58,24,27)(11,17,59,19,28)(12,18,60,20,29), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23)(13,58)(14,59)(15,60)(16,55)(17,56)(18,57)(25,28)(26,29)(27,30)(31,51)(32,52)(33,53)(34,54)(35,49)(36,50)(37,40)(38,41)(39,42) );
G=PermutationGroup([[(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,45),(14,46),(15,47),(16,48),(17,43),(18,44),(19,36),(20,31),(21,32),(22,33),(23,34),(24,35),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41)], [(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)], [(1,36,39,53,43),(2,31,40,54,44),(3,32,41,49,45),(4,33,42,50,46),(5,34,37,51,47),(6,35,38,52,48),(7,13,55,21,30),(8,14,56,22,25),(9,15,57,23,26),(10,16,58,24,27),(11,17,59,19,28),(12,18,60,20,29)], [(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,24),(8,19),(9,20),(10,21),(11,22),(12,23),(13,58),(14,59),(15,60),(16,55),(17,56),(18,57),(25,28),(26,29),(27,30),(31,51),(32,52),(33,53),(34,54),(35,49),(36,50),(37,40),(38,41),(39,42)]])
D5×C2×C6 is a maximal subgroup of
D10⋊Dic3 D10.D6
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 |
| kernel | D5×C2×C6 | C6×D5 | C2×C30 | C22×D5 | D10 | C2×C10 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 2 | 6 | 4 | 12 |
Matrix representation of D5×C2×C6 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 14 |
| 0 | 0 | 30 | 13 |
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 30 | 1 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,14,13],[30,0,0,0,0,1,0,0,0,0,30,30,0,0,0,1] >;
D5×C2×C6 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_6
% in TeX
G:=Group("D5xC2xC6"); // GroupNames label
G:=SmallGroup(120,44);
// by ID
G=gap.SmallGroup(120,44);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations