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G = D4⋊Dic5order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.7D4, D41Dic5, C10.12D8, C10.6SD16, (C5×D4)⋊4C4, (C2×D4).1D5, C54(D4⋊C4), C20.28(C2×C4), C4⋊Dic510C2, 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×C52C8)⋊2C2, SmallGroup(160,39)

Series: Derived Chief Lower central Upper central

C1C20 — D4⋊Dic5
C1C5C10C2×C10C2×C20C4⋊Dic5 — D4⋊Dic5
C5C10C20 — D4⋊Dic5
C1C22C2×C4C2×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 >

4C2
4C2
2C22
2C22
4C22
4C22
20C4
4C10
4C10
2D4
2C23
10C2×C4
10C8
2C2×C10
2C2×C10
4C2×C10
4Dic5
4C2×C10
5C4⋊C4
5C2×C8
2C52C8
2C5×D4
2C2×Dic5
2C22×C10
5D4⋊C4

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)]])

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)  D42D5⋊C4  D10.12D8  D10.16SD16  C406C4⋊C2  C405C4⋊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  C52C823D4  C4.(D4×D5)  C42.61D10  C42.62D10  C42.213D10  D20.23D4  C20.16D8  C42.72D10  C202D8  Dic109D4  D8×Dic5  Dic5⋊D8  D8⋊Dic5  (C2×D8).D5  D20⋊D4  C406D4  Dic10⋊D4  C4012D4  SD16×Dic5  Dic55SD16  SD16⋊Dic5  (C5×Q8).D4  D106SD16  C4014D4  Dic10.16D4  C408D4  (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  D82Dic5  C20.58D8  D12⋊Dic5  C10.D24  D4⋊Dic15

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D
order122222444455888810···1010···1020202020
size11114422202022101010102···24···44444

34 irreducible representations

dim1111122222222244
type+++++++++-+-
imageC1C2C2C2C4D4D4D5D8SD16D10Dic5C5⋊D4C5⋊D4D4⋊D5D4.D5
kernelD4⋊Dic5C2×C52C8C4⋊Dic5D4×C10C5×D4C20C2×C10C2×D4C10C10C2×C4D4C4C22C2C2
# reps1111411222244422

Matrix representation of D4⋊Dic5 in GL4(𝔽41) generated by

40000
04000
00132
002340
,
40000
0100
00132
00040
,
25000
02300
00400
00040
,
0100
40000
00012
00170
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

Export

Subgroup lattice of D4⋊Dic5 in TeX

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