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G = D42Dic5order 160 = 25·5

2nd semidirect product of D4 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42Dic5, Q82Dic5, C20.56D4, C55C4≀C2, (C5×D4)⋊5C4, (C5×Q8)⋊5C4, C4○D4.1D5, (C2×C10).3D4, C20.30(C2×C4), (C4×Dic5)⋊2C2, (C2×C4).41D10, C4.Dic54C2, C4.3(C2×Dic5), C4.31(C5⋊D4), (C2×C20).20C22, C22.3(C5⋊D4), C2.8(C23.D5), C10.29(C22⋊C4), (C5×C4○D4).1C2, SmallGroup(160,44)

Series: Derived Chief Lower central Upper central

C1C20 — D42Dic5
C1C5C10C20C2×C20C4.Dic5 — D42Dic5
C5C10C20 — D42Dic5
C1C4C2×C4C4○D4

Generators and relations for D42Dic5
 G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

2C2
4C2
2C4
2C22
10C4
10C4
2C10
4C10
2D4
2C2×C4
10C2×C4
10C8
2C2×C10
2Dic5
2C20
2Dic5
5C42
5M4(2)
2C52C8
2C2×C20
2C5×D4
2C2×Dic5
5C4≀C2

Smallest permutation representation of D42Dic5
On 40 points
Generators in S40
(1 20 14 6)(2 16 15 7)(3 17 11 8)(4 18 12 9)(5 19 13 10)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 31 14 36)(2 37 15 32)(3 33 11 38)(4 39 12 34)(5 35 13 40)(6 26 20 21)(7 22 16 27)(8 28 17 23)(9 24 18 29)(10 30 19 25)
(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)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 18)(7 17)(8 16)(9 20)(10 19)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)

G:=sub<Sym(40)| (1,20,14,6)(2,16,15,7)(3,17,11,8)(4,18,12,9)(5,19,13,10)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,31,14,36)(2,37,15,32)(3,33,11,38)(4,39,12,34)(5,35,13,40)(6,26,20,21)(7,22,16,27)(8,28,17,23)(9,24,18,29)(10,30,19,25), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,18)(7,17)(8,16)(9,20)(10,19)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)>;

G:=Group( (1,20,14,6)(2,16,15,7)(3,17,11,8)(4,18,12,9)(5,19,13,10)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,31,14,36)(2,37,15,32)(3,33,11,38)(4,39,12,34)(5,35,13,40)(6,26,20,21)(7,22,16,27)(8,28,17,23)(9,24,18,29)(10,30,19,25), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,18)(7,17)(8,16)(9,20)(10,19)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40) );

G=PermutationGroup([[(1,20,14,6),(2,16,15,7),(3,17,11,8),(4,18,12,9),(5,19,13,10),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,31,14,36),(2,37,15,32),(3,33,11,38),(4,39,12,34),(5,35,13,40),(6,26,20,21),(7,22,16,27),(8,28,17,23),(9,24,18,29),(10,30,19,25)], [(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,18),(7,17),(8,16),(9,20),(10,19),(21,39,26,34),(22,38,27,33),(23,37,28,32),(24,36,29,31),(25,35,30,40)]])

D42Dic5 is a maximal subgroup of
D5×C4≀C2  C42⋊D10  C40.93D4  C40.50D4  D85Dic5  D84Dic5  D2018D4  D20.38D4  D20.39D4  D20.40D4  (D4×C10)⋊21C4  2+ 1+4⋊D5  2+ 1+4.D5  2- 1+42D5  2- 1+4.2D5  C60.98D4  C60.99D4  Q83Dic15  C52U2(𝔽3)
D42Dic5 is a maximal quotient of
C20.32C42  C20.57D8  C20.26Q16  C4⋊C4⋊Dic5  C10.29C4≀C2  C42.7D10  C42.8D10  C60.98D4  C60.99D4  Q83Dic15

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B10A10B10C···10H20A20B20C20D20E···20J
order1222444444445588101010···102020202020···20
size1124112410101010222020224···422224···4

34 irreducible representations

dim1111112222222224
type++++++++--
imageC1C2C2C2C4C4D4D4D5D10Dic5Dic5C4≀C2C5⋊D4C5⋊D4D42Dic5
kernelD42Dic5C4.Dic5C4×Dic5C5×C4○D4C5×D4C5×Q8C20C2×C10C4○D4C2×C4D4Q8C5C4C22C1
# reps1111221122224444

Matrix representation of D42Dic5 in GL4(𝔽41) generated by

40000
04000
00320
0009
,
17100
402400
00032
00320
,
34100
40000
00400
0001
,
73400
13400
00320
00040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[17,40,0,0,1,24,0,0,0,0,0,32,0,0,32,0],[34,40,0,0,1,0,0,0,0,0,40,0,0,0,0,1],[7,1,0,0,34,34,0,0,0,0,32,0,0,0,0,40] >;

D42Dic5 in GAP, Magma, Sage, TeX

D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("D4:2Dic5");
// GroupNames label

G:=SmallGroup(160,44);
// by ID

G=gap.SmallGroup(160,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D42Dic5 in TeX

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