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## G = D4⋊Dic5order 160 = 25·5

### 1st semidirect product of D4 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — D4⋊Dic5
 Lower central C5 — C10 — C20 — D4⋊Dic5
 Upper central C1 — C22 — C2×C4 — C2×D4

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
Generators in S80
(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)]])

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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