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, C4oD20:2C2, (C2xD20):7C2, C40:C2:1C2, C5:1(C8:C22), (C2xC10).5D4, C10.13(C2xD4), C2.15(C2xD20), (C2xC4).15D10, (C5xM4(2)):1C2, C4.30(C22xD5), (C2xC20).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, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, M4(2), D8, SD16, C2xD4, C4oD4, Dic5, C20, D10, C2xC10, C8:C22, C40, Dic10, C4xD5, D20, D20, D20, C5:D4, C2xC20, C22xD5, C40:C2, D40, C5xM4(2), C2xD20, C4oD20, C8:D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C8:C22, D20, C22xD5, C2xD20, C8:D10
(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 D5xC8: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 | C5xM4(2) | C2xD20 | C4oD20 | C20 | C2xC10 | M4(2) | C8 | C2xC4 | 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