metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.7D4, D4⋊1Dic5, C10.12D8, C10.6SD16, (C5×D4)⋊4C4, (C2×D4).1D5, C5⋊4(D4⋊C4), C20.28(C2×C4), C4⋊Dic5⋊10C2, C2.3(D4⋊D5), (D4×C10).1C2, (C2×C10).33D4, (C2×C4).39D10, C4.1(C2×Dic5), C4.12(C5⋊D4), C2.3(D4.D5), (C2×C20).16C22, C2.3(C23.D5), C10.24(C22⋊C4), C22.17(C5⋊D4), (C2×C5⋊2C8)⋊2C2, SmallGroup(160,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊Dic5
G = < a,b,c,d | a4=b2=c10=1, d2=c5, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4⋊Dic5
►On 80 points
(1 45 39 14)(2 46 40 15)(3 47 31 16)(4 48 32 17)(5 49 33 18)(6 50 34 19)(7 41 35 20)(8 42 36 11)(9 43 37 12)(10 44 38 13)(21 56 65 76)(22 57 66 77)(23 58 67 78)(24 59 68 79)(25 60 69 80)(26 51 70 71)(27 52 61 72)(28 53 62 73)(29 54 63 74)(30 55 64 75)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 11)(9 12)(10 13)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(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 56 6 51)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 68 16 63)(12 67 17 62)(13 66 18 61)(14 65 19 70)(15 64 20 69)(21 50 26 45)(22 49 27 44)(23 48 28 43)(24 47 29 42)(25 46 30 41)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
G:=sub<Sym(80)| (1,45,39,14)(2,46,40,15)(3,47,31,16)(4,48,32,17)(5,49,33,18)(6,50,34,19)(7,41,35,20)(8,42,36,11)(9,43,37,12)(10,44,38,13)(21,56,65,76)(22,57,66,77)(23,58,67,78)(24,59,68,79)(25,60,69,80)(26,51,70,71)(27,52,61,72)(28,53,62,73)(29,54,63,74)(30,55,64,75), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,11)(9,12)(10,13)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,68,16,63)(12,67,17,62)(13,66,18,61)(14,65,19,70)(15,64,20,69)(21,50,26,45)(22,49,27,44)(23,48,28,43)(24,47,29,42)(25,46,30,41)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)>;
G:=Group( (1,45,39,14)(2,46,40,15)(3,47,31,16)(4,48,32,17)(5,49,33,18)(6,50,34,19)(7,41,35,20)(8,42,36,11)(9,43,37,12)(10,44,38,13)(21,56,65,76)(22,57,66,77)(23,58,67,78)(24,59,68,79)(25,60,69,80)(26,51,70,71)(27,52,61,72)(28,53,62,73)(29,54,63,74)(30,55,64,75), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,11)(9,12)(10,13)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,68,16,63)(12,67,17,62)(13,66,18,61)(14,65,19,70)(15,64,20,69)(21,50,26,45)(22,49,27,44)(23,48,28,43)(24,47,29,42)(25,46,30,41)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75) );
G=PermutationGroup([[(1,45,39,14),(2,46,40,15),(3,47,31,16),(4,48,32,17),(5,49,33,18),(6,50,34,19),(7,41,35,20),(8,42,36,11),(9,43,37,12),(10,44,38,13),(21,56,65,76),(22,57,66,77),(23,58,67,78),(24,59,68,79),(25,60,69,80),(26,51,70,71),(27,52,61,72),(28,53,62,73),(29,54,63,74),(30,55,64,75)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,11),(9,12),(10,13),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,56,6,51),(2,55,7,60),(3,54,8,59),(4,53,9,58),(5,52,10,57),(11,68,16,63),(12,67,17,62),(13,66,18,61),(14,65,19,70),(15,64,20,69),(21,50,26,45),(22,49,27,44),(23,48,28,43),(24,47,29,42),(25,46,30,41),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75)]])
D4⋊Dic5 is a maximal subgroup of
Dic5.14D8 Dic5.5D8 D4⋊Dic10 D4.Dic10 C4⋊C4.D10 C20⋊Q8⋊C2 D4.2Dic10 (C8×Dic5)⋊C2 D5×D4⋊C4 (D4×D5)⋊C4 D4⋊(C4×D5) D4⋊2D5⋊C4 D10.12D8 D10.16SD16 C40⋊6C4⋊C2 C40⋊5C4⋊C2 C20.50D8 C20.38SD16 D4.3Dic10 C4×D4⋊D5 C42.48D10 C4×D4.D5 C42.51D10 (C2×C10).D8 C4⋊D4.D5 (C2×D4).D10 (C2×C10)⋊D8 C4⋊D4⋊D5 C5⋊2C8⋊23D4 C4.(D4×D5) C42.61D10 C42.62D10 C42.213D10 D20.23D4 C20.16D8 C42.72D10 C20⋊2D8 Dic10⋊9D4 D8×Dic5 Dic5⋊D8 D8⋊Dic5 (C2×D8).D5 D20⋊D4 C40⋊6D4 Dic10⋊D4 C40⋊12D4 SD16×Dic5 Dic5⋊5SD16 SD16⋊Dic5 (C5×Q8).D4 D10⋊6SD16 C40⋊14D4 Dic10.16D4 C40⋊8D4 (D4×C10)⋊18C4 (C2×C10)⋊8D8 (C5×D4).31D4 C4○D4⋊Dic5 C20.(C2×D4) (C5×D4)⋊14D4 (C5×D4).32D4 D12⋊Dic5 C10.D24 D4⋊Dic15
D4⋊Dic5 is a maximal quotient of
C20.31C42 C20.57D8 C4⋊C4⋊Dic5 C20.9D8 C20.10D8 C10.D16 D8.Dic5 C40.15D4 Q16.Dic5 D8⋊2Dic5 C20.58D8 D12⋊Dic5 C10.D24 D4⋊Dic15
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 20 | 20 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | Dic5 | C5⋊D4 | C5⋊D4 | D4⋊D5 | D4.D5 |
| kernel | D4⋊Dic5 | C2×C5⋊2C8 | C4⋊Dic5 | D4×C10 | C5×D4 | C20 | C2×C10 | C2×D4 | C10 | C10 | C2×C4 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of D4⋊Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 32 |
| 0 | 0 | 23 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 32 |
| 0 | 0 | 0 | 40 |
| 25 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 17 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,23,0,0,32,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,32,40],[25,0,0,0,0,23,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,0,17,0,0,12,0] >;
D4⋊Dic5 in GAP, Magma, Sage, TeX
D_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("D4:Dic5"); // GroupNames label
G:=SmallGroup(160,39);
// by ID
G=gap.SmallGroup(160,39);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,579,297,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^10=1,d^2=c^5,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
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