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G = D20⋊C4order 160 = 25·5

1st semidirect product of D20 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D41F5, D201C4, D5.2D8, D10.18D4, Dic5.2D4, D5.2SD16, C4⋊F51C2, D5⋊C81C2, C5⋊(D4⋊C4), (C5×D4)⋊1C4, C4.1(C2×F5), C20.1(C2×C4), (D4×D5).2C2, (C4×D5).7C22, C2.6(C22⋊F5), C10.5(C22⋊C4), SmallGroup(160,82)

Series: Derived Chief Lower central Upper central

C1C20 — D20⋊C4
C1C5C10D10C4×D5C4⋊F5 — D20⋊C4
C5C10C20 — D20⋊C4
C1C2C4D4

Generators and relations for D20⋊C4
 G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a17b >

4C2
5C2
5C2
20C2
2C22
5C4
5C22
10C22
20C22
20C22
20C4
4D5
4C10
5D4
5C2×C4
10D4
10C23
10C2×C4
10C8
2C2×C10
2D10
4D10
4F5
4D10
5C4⋊C4
5C2×D4
5C2×C8
2C5⋊C8
2C5⋊D4
2C2×F5
2C22×D5
5D4⋊C4

Character table of D20⋊C4

 class 12A2B2C2D2E4A4B4C4D58A8B8C8D10A10B10C20
 size 114552021020204101010104888
ρ11111111111111111111    trivial
ρ211-111-111111-1-1-1-11-1-11    linear of order 2
ρ311-111-111-1-1111111-1-11    linear of order 2
ρ411111111-1-11-1-1-1-11111    linear of order 2
ρ5111-1-1-11-1i-i1i-i-ii1111    linear of order 4
ρ611-1-1-111-1i-i1-iii-i1-1-11    linear of order 4
ρ7111-1-1-11-1-ii1-iii-i1111    linear of order 4
ρ811-1-1-111-1-ii1i-i-ii1-1-11    linear of order 4
ρ9220-2-20-220020000200-2    orthogonal lifted from D4
ρ10220220-2-20020000200-2    orthogonal lifted from D4
ρ112-202-20000022-22-2-2000    orthogonal lifted from D8
ρ122-202-2000002-22-22-2000    orthogonal lifted from D8
ρ132-20-22000002--2--2-2-2-2000    complex lifted from SD16
ρ142-20-22000002-2-2--2--2-2000    complex lifted from SD16
ρ154440004000-10000-1-1-1-1    orthogonal lifted from F5
ρ1644-40004000-10000-111-1    orthogonal lifted from C2×F5
ρ17440000-4000-10000-15-51    orthogonal lifted from C22⋊F5
ρ18440000-4000-10000-1-551    orthogonal lifted from C22⋊F5
ρ198-800000000-200002000    orthogonal faithful, Schur index 2

Smallest permutation representation of D20⋊C4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)
(1 28)(2 35 10 31)(3 22 19 34)(4 29 8 37)(5 36 17 40)(6 23)(7 30 15 26)(9 24 13 32)(11 38)(12 25 20 21)(14 39 18 27)(16 33)

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

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

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

D20⋊C4 is a maximal subgroup of
D8×F5  D40⋊C4  SD16×F5  SD16⋊F5  (D4×C10)⋊C4  C4○D4⋊F5  C4○D20⋊C4  D60⋊C4  D12⋊F5  D20⋊Dic3
D20⋊C4 is a maximal quotient of
D10.1D8  D20⋊C8  Dic5.D8  D10.18D8  D10.D8  D5.D16  D8.F5  D40.C4  D401C4  D5.Q32  Q16.F5  Dic20.C4  D10.SD16  Dic5.23D8  Dic5.SD16  D60⋊C4  D12⋊F5  D20⋊Dic3

Matrix representation of D20⋊C4 in GL6(𝔽41)

1390000
1400000
0064000
0036100
000007
00003535
,
4020000
010000
001100
0004000
000010
0000540
,
0300000
1500000
000020
000002
0032400
00183800

G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,35,0,0,0,0,7,35],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[0,15,0,0,0,0,30,0,0,0,0,0,0,0,0,0,3,18,0,0,0,0,24,38,0,0,2,0,0,0,0,0,0,2,0,0] >;

D20⋊C4 in GAP, Magma, Sage, TeX

D_{20}\rtimes C_4
% in TeX

G:=Group("D20:C4");
// GroupNames label

G:=SmallGroup(160,82);
// by ID

G=gap.SmallGroup(160,82);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,579,297,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;
// generators/relations

Export

Subgroup lattice of D20⋊C4 in TeX
Character table of D20⋊C4 in TeX

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