Copied to
clipboard

G = C8:D10order 160 = 25·5

1st semidirect product of C8 and D10 acting via D10/C5=C22

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

C1C20 — C8:D10
C1C5C10C20D20C2xD20 — C8:D10
C5C10C20 — C8:D10
C1C2C2xC4M4(2)

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

Smallest permutation representation of C8:D10
On 40 points
Generators in S40
(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 2A2B2C2D2E4A4B4C5A5B8A8B10A10B10C10D20A20B20C20D20E20F40A···40H
order12222244455881010101020202020202040···40
size1122020202220224422442222444···4

31 irreducible representations

dim111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D20D20C8:C22C8:D10
kernelC8:D10C40:C2D40C5xM4(2)C2xD20C4oD20C20C2xC10M4(2)C8C2xC4C4C22C5C1
# reps122111112424414

Matrix representation of C8:D10 in GL4(F41) generated by

931390
1222039
0233210
8192919
,
6600
35100
38173535
316640
,
6600
13500
3962516
249216
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

׿
x
:
Z
F
o
wr
Q
<