metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2Dic5, Q8⋊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 D4⋊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 D4⋊2Dic5
►On 40 points
(1 20 14 6)(2 16 15 7)(3 17 11 8)(4 18 12 9)(5 19 13 10)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 31 14 36)(2 37 15 32)(3 33 11 38)(4 39 12 34)(5 35 13 40)(6 26 20 21)(7 22 16 27)(8 28 17 23)(9 24 18 29)(10 30 19 25)
(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 18)(7 17)(8 16)(9 20)(10 19)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)
G:=sub<Sym(40)| (1,20,14,6)(2,16,15,7)(3,17,11,8)(4,18,12,9)(5,19,13,10)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,31,14,36)(2,37,15,32)(3,33,11,38)(4,39,12,34)(5,35,13,40)(6,26,20,21)(7,22,16,27)(8,28,17,23)(9,24,18,29)(10,30,19,25), (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)(6,18)(7,17)(8,16)(9,20)(10,19)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)>;
G:=Group( (1,20,14,6)(2,16,15,7)(3,17,11,8)(4,18,12,9)(5,19,13,10)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,31,14,36)(2,37,15,32)(3,33,11,38)(4,39,12,34)(5,35,13,40)(6,26,20,21)(7,22,16,27)(8,28,17,23)(9,24,18,29)(10,30,19,25), (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)(6,18)(7,17)(8,16)(9,20)(10,19)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40) );
G=PermutationGroup([[(1,20,14,6),(2,16,15,7),(3,17,11,8),(4,18,12,9),(5,19,13,10),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,31,14,36),(2,37,15,32),(3,33,11,38),(4,39,12,34),(5,35,13,40),(6,26,20,21),(7,22,16,27),(8,28,17,23),(9,24,18,29),(10,30,19,25)], [(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),(6,18),(7,17),(8,16),(9,20),(10,19),(21,39,26,34),(22,38,27,33),(23,37,28,32),(24,36,29,31),(25,35,30,40)]])
D4⋊2Dic5 is a maximal subgroup of
D5×C4≀C2 C42⋊D10 C40.93D4 C40.50D4 D8⋊5Dic5 D8⋊4Dic5 D20⋊18D4 D20.38D4 D20.39D4 D20.40D4 (D4×C10)⋊21C4 2+ 1+4⋊D5 2+ 1+4.D5 2- 1+4⋊2D5 2- 1+4.2D5 C60.98D4 C60.99D4 Q8⋊3Dic15 C5⋊2U2(𝔽3)
D4⋊2Dic5 is a maximal quotient of
C20.32C42 C20.57D8 C20.26Q16 C4⋊C4⋊Dic5 C10.29C4≀C2 C42.7D10 C42.8D10 C60.98D4 C60.99D4 Q8⋊3Dic15
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 | D4⋊2Dic5 |
| kernel | D4⋊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 D4⋊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] >;
D4⋊2Dic5 in GAP, Magma, Sage, TeX
D_4\rtimes_2{\rm Dic}_5 % in TeX
G:=Group("D4:2Dic5"); // 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
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