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## G = C5×C4.D4order 160 = 25·5

### Direct product of C5 and C4.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4.D4
 Lower central C1 — C2 — C22 — C5×C4.D4
 Upper central C1 — C10 — C2×C20 — C5×C4.D4

Generators and relations for C5×C4.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 >

Smallest permutation representation of C5×C4.D4
On 40 points
Generators in S40
(1 19 16 37 27)(2 20 9 38 28)(3 21 10 39 29)(4 22 11 40 30)(5 23 12 33 31)(6 24 13 34 32)(7 17 14 35 25)(8 18 15 36 26)
(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 32 27 30 29 28 31 26)(33 36 35 34 37 40 39 38)

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

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

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

C5×C4.D4 is a maximal subgroup of   C23.3D20  C23.4D20  M4(2).19D10  D20.1D4  D201D4  D20.2D4  D20.3D4

55 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20H 40A ··· 40P order 1 2 2 2 2 4 4 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 2 4 4 2 2 1 1 1 1 4 4 4 4 1 1 1 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C4 C5 C10 C10 C20 D4 C5×D4 C4.D4 C5×C4.D4 kernel C5×C4.D4 C5×M4(2) D4×C10 C22×C10 C4.D4 M4(2) C2×D4 C23 C20 C4 C5 C1 # reps 1 2 1 4 4 8 4 16 2 8 1 4

Matrix representation of C5×C4.D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 40 39 0 0 1 1 0 0 0 32 0 1 9 9 40 0
,
 32 0 0 39 0 0 1 1 1 0 0 32 0 1 0 9
,
 9 0 39 0 0 0 1 1 0 1 32 0 1 0 9 0
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] >;

C5×C4.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

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