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G = C5xC4.D4order 160 = 25·5

Direct product of C5 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5xC4.D4, C23.C20, C20.58D4, M4(2):3C10, C4.9(C5xD4), (D4xC10).8C2, (C2xD4).2C10, (C5xM4(2)):9C2, C22.3(C2xC20), (C22xC10).1C4, (C2xC20).59C22, C10.33(C22:C4), (C2xC4).1(C2xC10), C2.4(C5xC22:C4), (C2xC10).40(C2xC4), SmallGroup(160,50)

Series: Derived Chief Lower central Upper central

C1C22 — C5xC4.D4
C1C2C4C2xC4C2xC20C5xM4(2) — C5xC4.D4
C1C2C22 — C5xC4.D4
C1C10C2xC20 — C5xC4.D4

Generators and relations for C5xC4.D4
 G = < a,b,c,d | a5=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 84 in 46 conjugacy classes, 24 normal (12 characteristic)
Quotients: C1, C2, C4, C22, C5, C2xC4, D4, C10, C22:C4, C20, C2xC10, C4.D4, C2xC20, C5xD4, C5xC22:C4, C5xC4.D4
2C2
4C2
4C2
2C22
2C22
4C22
4C22
2C10
4C10
4C10
2D4
2C8
2D4
2C8
2C2xC10
2C2xC10
4C2xC10
4C2xC10
2C5xD4
2C40
2C40
2C5xD4

Smallest permutation representation of C5xC4.D4
On 40 points
Generators in S40
(1 19 16 37 25)(2 20 9 38 26)(3 21 10 39 27)(4 22 11 40 28)(5 23 12 33 29)(6 24 13 34 30)(7 17 14 35 31)(8 18 15 36 32)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)
(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 4 3 2 5 8 7 6)(9 12 15 14 13 16 11 10)(17 24 19 22 21 20 23 18)(25 28 27 26 29 32 31 30)(33 36 35 34 37 40 39 38)

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

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

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

C5xC4.D4 is a maximal subgroup of   C23.3D20  C23.4D20  M4(2).19D10  D20.1D4  D20:1D4  D20.2D4  D20.3D4

55 conjugacy classes

class 1 2A2B2C2D4A4B5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H10I···10P20A···20H40A···40P
order122224455558888101010101010101010···1020···2040···40
size112442211114444111122224···42···24···4

55 irreducible representations

dim111111112244
type+++++
imageC1C2C2C4C5C10C10C20D4C5xD4C4.D4C5xC4.D4
kernelC5xC4.D4C5xM4(2)D4xC10C22xC10C4.D4M4(2)C2xD4C23C20C4C5C1
# reps1214484162814

Matrix representation of C5xC4.D4 in GL4(F41) generated by

18000
01800
00180
00018
,
403900
1100
03201
99400
,
320039
0011
10032
0109
,
90390
0011
01320
1090
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[40,1,0,9,39,1,32,9,0,0,0,40,0,0,1,0],[32,0,1,0,0,0,0,1,0,1,0,0,39,1,32,9],[9,0,0,1,0,0,1,0,39,1,32,9,0,1,0,0] >;

C5xC4.D4 in GAP, Magma, Sage, TeX

C_5\times C_4.D_4
% in TeX

G:=Group("C5xC4.D4");
// GroupNames label

G:=SmallGroup(160,50);
// by ID

G=gap.SmallGroup(160,50);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1810,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

Export

Subgroup lattice of C5xC4.D4 in TeX

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