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G = D10.D4order 160 = 25·5

1st non-split extension by D10 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.1D4, (C2×C4)⋊F5, (C2×C20)⋊1C4, C51(C23⋊C4), C22⋊F51C2, (C2×D20).2C2, (C22×D5)⋊2C4, C22.2(C2×F5), C2.4(C22⋊F5), C10.1(C22⋊C4), (C22×D5).14C22, (C2×C10).2(C2×C4), SmallGroup(160,74)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D10.D4
C1C5C10D10C22×D5C22⋊F5 — D10.D4
C5C10C2×C10 — D10.D4
C1C2C22C2×C4

Generators and relations for D10.D4
 G = < a,b,c,d | a10=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc-1 >

2C2
10C2
10C2
20C2
2C4
5C22
5C22
10C22
20C4
20C4
20C22
20C22
2D5
2D5
2C10
4D5
5C23
5C23
10C2×C4
10D4
10D4
10C2×C4
2C20
2D10
4D10
4F5
4F5
4D10
5C2×D4
5C22⋊C4
5C22⋊C4
2C2×F5
2C2×F5
2D20
2D20
5C23⋊C4

Character table of D10.D4

 class 12A2B2C2D2E4A4B4C4D4E510A10B10C20A20B20C20D
 size 11210102042020202044444444
ρ11111111111111111111    trivial
ρ211111-1-1-111-11111-1-1-1-1    linear of order 2
ρ311111-1-11-1-111111-1-1-1-1    linear of order 2
ρ41111111-1-1-1-111111111    linear of order 2
ρ5111-1-11-1-i-iii1111-1-1-1-1    linear of order 4
ρ6111-1-11-1ii-i-i1111-1-1-1-1    linear of order 4
ρ7111-1-1-11-ii-ii11111111    linear of order 4
ρ8111-1-1-11i-ii-i11111111    linear of order 4
ρ922-22-200000022-2-20000    orthogonal lifted from D4
ρ1022-2-2200000022-2-20000    orthogonal lifted from D4
ρ11444000-40000-1-1-1-11111    orthogonal lifted from C2×F5
ρ124-40000000004-4000000    orthogonal lifted from C23⋊C4
ρ1344400040000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444-400000000-1-11155-5-5    orthogonal lifted from C22⋊F5
ρ1544-400000000-1-111-5-555    orthogonal lifted from C22⋊F5
ρ164-4000000000-115-54ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543    orthogonal faithful
ρ174-4000000000-11-5543ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ54+2ζ4ζ5244ζ53+2ζ4ζ54    orthogonal faithful
ρ184-4000000000-11-5543ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ53+2ζ4ζ544ζ54+2ζ4ζ524    orthogonal faithful
ρ194-4000000000-115-54ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ52+2ζ43ζ54343ζ54+2ζ43ζ5343    orthogonal faithful

Smallest permutation representation of D10.D4
On 40 points
Generators in S40
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)
(1 24 12 36)(2 21 11 39)(3 28 20 32)(4 25 19 35)(5 22 18 38)(6 29 17 31)(7 26 16 34)(8 23 15 37)(9 30 14 40)(10 27 13 33)
(1 31 17 29)(2 38 16 22)(3 35 15 25)(4 32 14 28)(5 39 13 21)(6 36 12 24)(7 33 11 27)(8 40 20 30)(9 37 19 23)(10 34 18 26)

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

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

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

D10.D4 is a maximal subgroup of
C42⋊F5  C422F5  (C2×D4)⋊F5  (C2×Q8)⋊F5  C23⋊F55C2  (C2×D4)⋊7F5  (C2×Q8)⋊7F5  D10.4D12  (C2×C60)⋊C4
D10.D4 is a maximal quotient of
C42⋊F5  C422F5  C42.F5  C42.2F5  C5⋊C2≀C4  C22⋊C4⋊F5  D10.1D8  D10.1Q16  (C2×C20)⋊1C8  C5⋊(C23⋊C8)  C22⋊F5⋊C4  D10.4D12  (C2×C60)⋊C4

Matrix representation of D10.D4 in GL4(𝔽41) generated by

6600
35100
00040
2727135
,
251600
21600
32272839
1430213
,
40133715
739264
416149
651130
,
340390
11039
52770
30204040
G:=sub<GL(4,GF(41))| [6,35,0,27,6,1,0,27,0,0,0,1,0,0,40,35],[25,2,32,14,16,16,27,30,0,0,28,2,0,0,39,13],[40,7,4,6,13,39,16,5,37,26,14,11,15,4,9,30],[34,1,5,30,0,1,27,20,39,0,7,40,0,39,0,40] >;

D10.D4 in GAP, Magma, Sage, TeX

D_{10}.D_4
% in TeX

G:=Group("D10.D4");
// GroupNames label

G:=SmallGroup(160,74);
// by ID

G=gap.SmallGroup(160,74);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,579,2309,1169]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^-1>;
// generators/relations

Export

Subgroup lattice of D10.D4 in TeX
Character table of D10.D4 in TeX

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