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G = C20.D4  order 160 = 25·5

8th non-split extension by C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.8D4, C23.Dic5, (C2×D4).2D5, (C2×C4).3D10, (D4×C10).2C2, C5⋊3(C4.D4), C4.Dic5⋊3C2, C4.13(C5⋊D4), (C22×C10).2C4, (C2×C20).17C22, C2.4(C23.D5), C22.2(C2×Dic5), C10.25(C22⋊C4), (C2×C10).48(C2×C4), SmallGroup(160,40)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C20.D4
 G = < a,b,c | a20=1, b4=a10, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a15b3 >

C1
C2
2C2
4C2
4C2
C5
C4
C4
C22
2C22
2C22
4C22
4C22
C10
2C10
4C10
4C10
C23
C23
C2×C4
2D4
2D4
10C8
10C8
C2×C10
C20
C20
2C2×C10
2C2×C10
4C2×C10
4C2×C10
C2×D4
5M4(2)
5M4(2)
C22×C10
C22×C10
C2×C20
2C5×D4
2C5⋊2C8
2C5⋊2C8
2C5×D4
5C4.D4
D4×C10
C4.Dic5
C4.Dic5
C20.D4

Smallest permutation representation of C20.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 28 6 23 11 38 16 33)(2 27 7 22 12 37 17 32)(3 26 8 21 13 36 18 31)(4 25 9 40 14 35 19 30)(5 24 10 39 15 34 20 29)

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

C20.D4 is a maximal subgroup of
 C5⋊3C2≀C4  (C2×C20).D4  C24⋊2Dic5  (C22×C20)⋊C4  D5×C4.D4  M4(2).19D10  C42⋊5D10  D20⋊5D4  C40.23D4  C40.44D4  M4(2).D10  M4(2).13D10  (D4×C10).29C4  2+ 1+4⋊D5  2+ 1+4.D5  C20.5D12  C60.8D4
C20.D4 is a maximal quotient of
 C24.Dic5  (C2×C20).Q8  C42.7D10  C20.9D8  C20.5Q16  C20.5D12  C60.8D4

31 conjugacy classes

class 1 2A2B2C2D4A4B5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D
order122224455888810···1010···1020202020
size112442222202020202···24···44444

31 irreducible representations

dim11112222244
type++++++-+
imageC1C2C2C4D4D5D10Dic5C5⋊D4C4.D4C20.D4
kernelC20.D4C4.Dic5D4×C10C22×C10C20C2×D4C2×C4C23C4C5C1
# reps12142224814

Matrix representation of C20.D4 ►in GL6(𝔽41)

2410000
1240000
000100
0040000
000001
0000400
,
9110000
30320000
000010
0000040
000100
001000
,
1190000
32300000
000010
000001
000100
0040000

G:=sub<GL(6,GF(41))| [24,1,0,0,0,0,1,24,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[9,30,0,0,0,0,11,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0],[11,32,0,0,0,0,9,30,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C20.D4 in GAP, Magma, Sage, TeX 

C_{20}.D_4
 
% in TeX
 
G:=Group("C20.D4");
 
// GroupNames label
 
G:=SmallGroup(160,40);
 
// by ID
 
G=gap.SmallGroup(160,40);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,86,579,4613]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C20.D4 in TeX

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