metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D10, D40⋊2C2, C40⋊1C22, C4.14D20, C20.12D4, D20⋊4C22, M4(2)⋊1D5, C22.5D20, C20.32C23, Dic10⋊4C22, C4○D20⋊2C2, (C2×D20)⋊7C2, C40⋊C2⋊1C2, C5⋊1(C8⋊C22), (C2×C10).5D4, C10.13(C2×D4), C2.15(C2×D20), (C2×C4).15D10, (C5×M4(2))⋊1C2, C4.30(C22×D5), (C2×C20).27C22, SmallGroup(160,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D10
G = < a,b,c | a8=b10=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 304 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C8⋊C22, C40, Dic10, C4×D5, D20, D20, D20, C5⋊D4, C2×C20, C22×D5, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, C8⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, D20, C22×D5, C2×D20, C8⋊D10
►On 40 points
(1 21 8 38 20 26 11 33)(2 27 9 34 16 22 12 39)(3 23 10 40 17 28 13 35)(4 29 6 36 18 24 14 31)(5 25 7 32 19 30 15 37)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 17)(12 16)(13 20)(14 19)(15 18)(21 23)(24 30)(25 29)(26 28)(31 32)(33 40)(34 39)(35 38)(36 37)
G:=sub<Sym(40)| (1,21,8,38,20,26,11,33)(2,27,9,34,16,22,12,39)(3,23,10,40,17,28,13,35)(4,29,6,36,18,24,14,31)(5,25,7,32,19,30,15,37), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,17)(12,16)(13,20)(14,19)(15,18)(21,23)(24,30)(25,29)(26,28)(31,32)(33,40)(34,39)(35,38)(36,37)>;
G:=Group( (1,21,8,38,20,26,11,33)(2,27,9,34,16,22,12,39)(3,23,10,40,17,28,13,35)(4,29,6,36,18,24,14,31)(5,25,7,32,19,30,15,37), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,17)(12,16)(13,20)(14,19)(15,18)(21,23)(24,30)(25,29)(26,28)(31,32)(33,40)(34,39)(35,38)(36,37) );
G=PermutationGroup([[(1,21,8,38,20,26,11,33),(2,27,9,34,16,22,12,39),(3,23,10,40,17,28,13,35),(4,29,6,36,18,24,14,31),(5,25,7,32,19,30,15,37)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,17),(12,16),(13,20),(14,19),(15,18),(21,23),(24,30),(25,29),(26,28),(31,32),(33,40),(34,39),(35,38),(36,37)]])
C8⋊D10 is a maximal subgroup of
D20⋊1D4 D20.3D4 D20.5D4 D20.6D4 D4⋊4D20 D4.10D20 C8.21D20 C8.24D20 C40.9C23 D4.11D20 D4.12D20 D5×C8⋊C22 D8⋊5D10 D40⋊C22 C40.C23 C40⋊1D6 D40⋊S3 D20⋊19D6 D60⋊30C22 C8⋊D30
C8⋊D10 is a maximal quotient of
C8⋊Dic10 C42.16D10 D40⋊9C4 C8⋊D20 C42.19D10 C42.20D10 C23.35D20 D20.31D4 D20⋊13D4 D20⋊14D4 C23.38D20 C23.13D20 D20⋊3Q8 C4⋊D40 D20.19D4 D20.3Q8 Dic10⋊8D4 C20.7Q16 C23.47D20 C23.48D20 C23.49D20 C40⋊2D4 C40⋊3D4 C40⋊1D6 D40⋊S3 D20⋊19D6 D60⋊30C22 C8⋊D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 20 | 2 | 2 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | D20 | D20 | C8⋊C22 | C8⋊D10 |
| kernel | C8⋊D10 | C40⋊C2 | D40 | C5×M4(2) | C2×D20 | C4○D20 | C20 | C2×C10 | M4(2) | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 1 | 4 |
Matrix representation of C8⋊D10 ►in GL4(𝔽41) generated by
| 9 | 31 | 39 | 0 |
| 12 | 22 | 0 | 39 |
| 0 | 23 | 32 | 10 |
| 8 | 19 | 29 | 19 |
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 38 | 17 | 35 | 35 |
| 31 | 6 | 6 | 40 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 39 | 6 | 25 | 16 |
| 24 | 9 | 2 | 16 |
G:=sub<GL(4,GF(41))| [9,12,0,8,31,22,23,19,39,0,32,29,0,39,10,19],[6,35,38,31,6,1,17,6,0,0,35,6,0,0,35,40],[6,1,39,24,6,35,6,9,0,0,25,2,0,0,16,16] >;
C8⋊D10 in GAP, Magma, Sage, TeX
C_8\rtimes D_{10} % in TeX
G:=Group("C8:D10"); // GroupNames label
G:=SmallGroup(160,129);
// by ID
G=gap.SmallGroup(160,129);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^8=b^10=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations