direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.D4, C23.C20, C20.58D4, M4(2)⋊3C10, C4.9(C5×D4), (D4×C10).8C2, (C2×D4).2C10, (C5×M4(2))⋊9C2, C22.3(C2×C20), (C22×C10).1C4, (C2×C20).59C22, C10.33(C22⋊C4), (C2×C4).1(C2×C10), C2.4(C5×C22⋊C4), (C2×C10).40(C2×C4), SmallGroup(160,50)
Series: Derived ►Chief ►Lower central ►Upper central
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
(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)]])
C5×C4.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 | 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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