direct product, metabelian, supersoluble, monomial
Aliases: D5×Dic10, C20.20D10, D10.15D10, Dic5.9D10, C4.5D52, (C5×D5)⋊Q8, C5⋊2(Q8×D5), C52⋊2(C2×Q8), (C4×D5).1D5, (D5×C20).4C2, C5⋊1(C2×Dic10), C52⋊4Q8⋊4C2, C52⋊2Q8⋊2C2, (C5×Dic10)⋊4C2, (C5×C10).1C23, (D5×Dic5).1C2, C10.1(C22×D5), (C5×C20).16C22, (D5×C10).9C22, C52⋊6C4.1C22, (C5×Dic5).1C22, C2.4(C2×D52), SmallGroup(400,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×Dic10
G = < a,b,c,d | a5=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 468 in 88 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, Q8, D5, C10, C10, C2×Q8, Dic5, Dic5, Dic5, C20, C20, D10, C2×C10, C52, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×C20, C5×Q8, C5×D5, C5×C10, C2×Dic10, Q8×D5, C5×Dic5, C5×Dic5, C52⋊6C4, C5×C20, D5×C10, D5×Dic5, C52⋊2Q8, C5×Dic10, D5×C20, C52⋊4Q8, D5×Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, Dic10, C22×D5, C2×Dic10, Q8×D5, D52, C2×D52, D5×Dic10
►On 80 points
Generators in S80
(1 13 5 17 9)(2 14 6 18 10)(3 15 7 19 11)(4 16 8 20 12)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)(41 49 57 45 53)(42 50 58 46 54)(43 51 59 47 55)(44 52 60 48 56)(61 73 65 77 69)(62 74 66 78 70)(63 75 67 79 71)(64 76 68 80 72)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)
(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)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 40 11 30)(2 39 12 29)(3 38 13 28)(4 37 14 27)(5 36 15 26)(6 35 16 25)(7 34 17 24)(8 33 18 23)(9 32 19 22)(10 31 20 21)(41 79 51 69)(42 78 52 68)(43 77 53 67)(44 76 54 66)(45 75 55 65)(46 74 56 64)(47 73 57 63)(48 72 58 62)(49 71 59 61)(50 70 60 80)
G:=sub<Sym(80)| (1,13,5,17,9)(2,14,6,18,10)(3,15,7,19,11)(4,16,8,20,12)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36)(41,49,57,45,53)(42,50,58,46,54)(43,51,59,47,55)(44,52,60,48,56)(61,73,65,77,69)(62,74,66,78,70)(63,75,67,79,71)(64,76,68,80,72), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,40,11,30)(2,39,12,29)(3,38,13,28)(4,37,14,27)(5,36,15,26)(6,35,16,25)(7,34,17,24)(8,33,18,23)(9,32,19,22)(10,31,20,21)(41,79,51,69)(42,78,52,68)(43,77,53,67)(44,76,54,66)(45,75,55,65)(46,74,56,64)(47,73,57,63)(48,72,58,62)(49,71,59,61)(50,70,60,80)>;
G:=Group( (1,13,5,17,9)(2,14,6,18,10)(3,15,7,19,11)(4,16,8,20,12)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36)(41,49,57,45,53)(42,50,58,46,54)(43,51,59,47,55)(44,52,60,48,56)(61,73,65,77,69)(62,74,66,78,70)(63,75,67,79,71)(64,76,68,80,72), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,40,11,30)(2,39,12,29)(3,38,13,28)(4,37,14,27)(5,36,15,26)(6,35,16,25)(7,34,17,24)(8,33,18,23)(9,32,19,22)(10,31,20,21)(41,79,51,69)(42,78,52,68)(43,77,53,67)(44,76,54,66)(45,75,55,65)(46,74,56,64)(47,73,57,63)(48,72,58,62)(49,71,59,61)(50,70,60,80) );
G=PermutationGroup([[(1,13,5,17,9),(2,14,6,18,10),(3,15,7,19,11),(4,16,8,20,12),(21,29,37,25,33),(22,30,38,26,34),(23,31,39,27,35),(24,32,40,28,36),(41,49,57,45,53),(42,50,58,46,54),(43,51,59,47,55),(44,52,60,48,56),(61,73,65,77,69),(62,74,66,78,70),(63,75,67,79,71),(64,76,68,80,72)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,40,11,30),(2,39,12,29),(3,38,13,28),(4,37,14,27),(5,36,15,26),(6,35,16,25),(7,34,17,24),(8,33,18,23),(9,32,19,22),(10,31,20,21),(41,79,51,69),(42,78,52,68),(43,77,53,67),(44,76,54,66),(45,75,55,65),(46,74,56,64),(47,73,57,63),(48,72,58,62),(49,71,59,61),(50,70,60,80)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R | 20S | 20T | 20U | 20V |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 10 | 10 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D5 | D10 | D10 | D10 | Dic10 | Q8×D5 | D52 | C2×D52 | D5×Dic10 |
| kernel | D5×Dic10 | D5×Dic5 | C52⋊2Q8 | C5×Dic10 | D5×C20 | C52⋊4Q8 | C5×D5 | Dic10 | C4×D5 | Dic5 | C20 | D10 | D5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 2 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of D5×Dic10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 35 | 35 |
| 32 | 11 | 0 | 0 |
| 30 | 27 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 27 | 28 | 0 | 0 |
| 12 | 14 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,6,35,0,0,40,35],[32,30,0,0,11,27,0,0,0,0,40,0,0,0,0,40],[27,12,0,0,28,14,0,0,0,0,40,0,0,0,0,40] >;
D5×Dic10 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_{10} % in TeX
G:=Group("D5xDic10"); // GroupNames label
G:=SmallGroup(400,163);
// by ID
G=gap.SmallGroup(400,163);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations