metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic5, D4⋊2Dic5, C20.56D4, C5⋊5C4≀C2, (C5×D4)⋊5C4, (C5×Q8)⋊5C4, C4○D4.1D5, (C2×C10).3D4, C20.30(C2×C4), (C4×Dic5)⋊2C2, (C2×C4).41D10, C4.Dic5⋊4C2, C4.3(C2×Dic5), C4.31(C5⋊D4), (C2×C20).20C22, C22.3(C5⋊D4), C2.8(C23.D5), C10.29(C22⋊C4), (C5×C4○D4).1C2, SmallGroup(160,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2Dic5
G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Smallest permutation representation of Q8⋊2Dic5
►On 40 points
Generators in S40
(1 18 13 8)(2 19 14 9)(3 20 15 10)(4 16 11 6)(5 17 12 7)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 28 13 23)(2 24 14 29)(3 30 15 25)(4 26 11 21)(5 22 12 27)(6 31 16 36)(7 37 17 32)(8 33 18 38)(9 39 19 34)(10 35 20 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 12)(2 11)(3 15)(4 14)(5 13)(6 19)(7 18)(8 17)(9 16)(10 20)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)
G:=sub<Sym(40)| (1,18,13,8)(2,19,14,9)(3,20,15,10)(4,16,11,6)(5,17,12,7)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,28,13,23)(2,24,14,29)(3,30,15,25)(4,26,11,21)(5,22,12,27)(6,31,16,36)(7,37,17,32)(8,33,18,38)(9,39,19,34)(10,35,20,40), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,19)(7,18)(8,17)(9,16)(10,20)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35)>;
G:=Group( (1,18,13,8)(2,19,14,9)(3,20,15,10)(4,16,11,6)(5,17,12,7)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,28,13,23)(2,24,14,29)(3,30,15,25)(4,26,11,21)(5,22,12,27)(6,31,16,36)(7,37,17,32)(8,33,18,38)(9,39,19,34)(10,35,20,40), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,19)(7,18)(8,17)(9,16)(10,20)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35) );
G=PermutationGroup([(1,18,13,8),(2,19,14,9),(3,20,15,10),(4,16,11,6),(5,17,12,7),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,28,13,23),(2,24,14,29),(3,30,15,25),(4,26,11,21),(5,22,12,27),(6,31,16,36),(7,37,17,32),(8,33,18,38),(9,39,19,34),(10,35,20,40)], [(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,19),(7,18),(8,17),(9,16),(10,20),(21,34,26,39),(22,33,27,38),(23,32,28,37),(24,31,29,36),(25,40,30,35)])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | Dic5 | Dic5 | C4≀C2 | C5⋊D4 | C5⋊D4 | Q8⋊2Dic5 |
| kernel | Q8⋊2Dic5 | C4.Dic5 | C4×Dic5 | C5×C4○D4 | C5×D4 | C5×Q8 | C20 | C2×C10 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of Q8⋊2Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 17 | 1 | 0 | 0 |
| 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 34 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[17,40,0,0,1,24,0,0,0,0,0,32,0,0,32,0],[34,40,0,0,1,0,0,0,0,0,40,0,0,0,0,1],[7,1,0,0,34,34,0,0,0,0,32,0,0,0,0,40] >;
Q8⋊2Dic5 in GAP, Magma, Sage, TeX
Q_8\rtimes_2{\rm Dic}_5 % in TeX
G:=Group("Q8:2Dic5"); // GroupNames label
G:=SmallGroup(160,44);
// by ID
G=gap.SmallGroup(160,44);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,579,297,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations