metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D10, C20.15D4, D20⋊6C22, C20.12C23, Dic10⋊5C22, D4⋊D5⋊5C2, (C2×D4)⋊2D5, C4○D20⋊3C2, (D4×C10)⋊2C2, C5⋊4(C8⋊C22), D4.D5⋊5C2, C5⋊2C8⋊3C22, C10.45(C2×D4), (C2×C4).17D10, (C2×C10).39D4, C4.Dic5⋊6C2, C4.16(C5⋊D4), (C5×D4).6C22, C4.12(C22×D5), (C2×C20).30C22, C22.10(C5⋊D4), C2.9(C2×C5⋊D4), SmallGroup(160,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c9 >
Subgroups: 208 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C8⋊C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×C10, C4.Dic5, D4⋊D5, D4.D5, C4○D20, D4×C10, D4.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, C5⋊D4, C22×D5, C2×C5⋊D4, D4.D10
►On 40 points
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(1 16)(2 7)(3 18)(4 9)(5 20)(6 11)(8 13)(10 15)(12 17)(14 19)(22 32)(24 34)(26 36)(28 38)(30 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 21 11 31)(2 30 12 40)(3 39 13 29)(4 28 14 38)(5 37 15 27)(6 26 16 36)(7 35 17 25)(8 24 18 34)(9 33 19 23)(10 22 20 32)
G:=sub<Sym(40)| (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,16)(2,7)(3,18)(4,9)(5,20)(6,11)(8,13)(10,15)(12,17)(14,19)(22,32)(24,34)(26,36)(28,38)(30,40), (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), (1,21,11,31)(2,30,12,40)(3,39,13,29)(4,28,14,38)(5,37,15,27)(6,26,16,36)(7,35,17,25)(8,24,18,34)(9,33,19,23)(10,22,20,32)>;
G:=Group( (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,16)(2,7)(3,18)(4,9)(5,20)(6,11)(8,13)(10,15)(12,17)(14,19)(22,32)(24,34)(26,36)(28,38)(30,40), (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), (1,21,11,31)(2,30,12,40)(3,39,13,29)(4,28,14,38)(5,37,15,27)(6,26,16,36)(7,35,17,25)(8,24,18,34)(9,33,19,23)(10,22,20,32) );
G=PermutationGroup([[(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30)], [(1,16),(2,7),(3,18),(4,9),(5,20),(6,11),(8,13),(10,15),(12,17),(14,19),(22,32),(24,34),(26,36),(28,38),(30,40)], [(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)], [(1,21,11,31),(2,30,12,40),(3,39,13,29),(4,28,14,38),(5,37,15,27),(6,26,16,36),(7,35,17,25),(8,24,18,34),(9,33,19,23),(10,22,20,32)]])
D4.D10 is a maximal subgroup of
D20.2D4 D20.3D4 D20.14D4 D20⋊5D4 C40.23D4 C40.44D4 D20⋊18D4 D20.38D4 D8⋊13D10 D20.29D4 D5×C8⋊C22 SD16⋊D10 C20.C24 D20.32C23 D20.33C23 D20⋊21D6 D60⋊36C22 D20⋊10D6 D12.9D10 D4.D30
D4.D10 is a maximal quotient of
C20.47(C4⋊C4) C4○D20⋊9C4 C4⋊C4.228D10 C4⋊C4.230D10 D4.3Dic10 C42.48D10 D4.1D20 C42.51D10 (C2×D4).D10 D20⋊17D4 C4⋊D4⋊D5 C4.(D4×D5) C42.72D10 C20⋊2D8 C42.74D10 Dic10⋊9D4 C42.76D10 D20⋊5Q8 C42.82D10 Dic10⋊5Q8 (D4×C10)⋊18C4 (C2×C10)⋊8D8 (C5×D4).31D4 D20⋊21D6 D60⋊36C22 D20⋊10D6 D12.9D10 D4.D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 20 | 2 | 2 | 20 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4.D10 |
| kernel | D4.D10 | C4.Dic5 | D4⋊D5 | D4.D5 | C4○D20 | D4×C10 | C20 | C2×C10 | C2×D4 | C2×C4 | D4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 4 |
Matrix representation of D4.D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 38 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 13 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 38 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 38 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 38 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 9 | 16 | 0 | 0 | 0 | 0 |
| 36 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 38 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 40 | 38 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,1,13,0,0,0,0,3,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,0,1],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,40,28,0,0,0,0,38,1,0,0] >;
D4.D10 in GAP, Magma, Sage, TeX
D_4.D_{10} % in TeX
G:=Group("D4.D10"); // GroupNames label
G:=SmallGroup(160,153);
// by ID
G=gap.SmallGroup(160,153);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,579,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^9>;
// generators/relations