Copied to
clipboard

G = D4:6D10  order 160 = 25·5

2nd semidirect product of D4 and D10 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:6D10, C23:2D10, D20:8C22, C10.7C24, C5:12+ 1+4, C20.21C23, D10.3C23, Dic10:8C22, Dic5.4C23, (D4xD5):4C2, (C2xD4):7D5, (C2xC4):3D10, C4oD20:5C2, (D4xC10):7C2, D4:2D5:4C2, (C2xC20):3C22, (C5xD4):7C22, (C4xD5):1C22, C5:D4:3C22, C2.8(C23xD5), (C2xC10).2C23, C4.21(C22xD5), (C22xC10):5C22, (C2xDic5):4C22, (C22xD5):3C22, C22.6(C22xD5), (C2xC5:D4):11C2, SmallGroup(160,219)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D4:6D10
Chief series
C1 — C5 — C10 — D10 — C22xD5 — D4xD5 — D4:6D10
Lower central
C5 — C10 — D4:6D10
Upper central
C1 — C2 — C2xD4

Generators and relations for D4:6D10
 G = < a,b,c,d | a4=b2=c10=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 520 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, D4, Q8, C23, C23, D5, C10, C10, C2xD4, C2xD4, C4oD4, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, 2+ 1+4, Dic10, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xC10, C4oD20, D4xD5, D4:2D5, C2xC5:D4, D4xC10, D4:6D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22xD5, C23xD5, D4:6D10

Smallest permutation representation of D4:6D10
►On 40 points
Generators in S40
(1 37 10 32)(2 33 6 38)(3 39 7 34)(4 35 8 40)(5 31 9 36)(11 24 16 29)(12 30 17 25)(13 26 18 21)(14 22 19 27)(15 28 20 23)

(1 23)(2 29)(3 25)(4 21)(5 27)(6 24)(7 30)(8 26)(9 22)(10 28)(11 38)(12 34)(13 40)(14 36)(15 32)(16 33)(17 39)(18 35)(19 31)(20 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)

(2 5)(3 4)(6 9)(7 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 30)(22 29)(23 28)(24 27)(25 26)(31 33)(34 40)(35 39)(36 38)
 
G:=sub<Sym(40)| (1,37,10,32)(2,33,6,38)(3,39,7,34)(4,35,8,40)(5,31,9,36)(11,24,16,29)(12,30,17,25)(13,26,18,21)(14,22,19,27)(15,28,20,23), (1,23)(2,29)(3,25)(4,21)(5,27)(6,24)(7,30)(8,26)(9,22)(10,28)(11,38)(12,34)(13,40)(14,36)(15,32)(16,33)(17,39)(18,35)(19,31)(20,37), (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), (2,5)(3,4)(6,9)(7,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,30)(22,29)(23,28)(24,27)(25,26)(31,33)(34,40)(35,39)(36,38)>;
 
G:=Group( (1,37,10,32)(2,33,6,38)(3,39,7,34)(4,35,8,40)(5,31,9,36)(11,24,16,29)(12,30,17,25)(13,26,18,21)(14,22,19,27)(15,28,20,23), (1,23)(2,29)(3,25)(4,21)(5,27)(6,24)(7,30)(8,26)(9,22)(10,28)(11,38)(12,34)(13,40)(14,36)(15,32)(16,33)(17,39)(18,35)(19,31)(20,37), (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), (2,5)(3,4)(6,9)(7,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,30)(22,29)(23,28)(24,27)(25,26)(31,33)(34,40)(35,39)(36,38) );
 
G=PermutationGroup([[(1,37,10,32),(2,33,6,38),(3,39,7,34),(4,35,8,40),(5,31,9,36),(11,24,16,29),(12,30,17,25),(13,26,18,21),(14,22,19,27),(15,28,20,23)], [(1,23),(2,29),(3,25),(4,21),(5,27),(6,24),(7,30),(8,26),(9,22),(10,28),(11,38),(12,34),(13,40),(14,36),(15,32),(16,33),(17,39),(18,35),(19,31),(20,37)], [(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)], [(2,5),(3,4),(6,9),(7,8),(11,19),(12,18),(13,17),(14,16),(15,20),(21,30),(22,29),(23,28),(24,27),(25,26),(31,33),(34,40),(35,39),(36,38)]])
 

D4:6D10 is a maximal subgroup of
 C23:D20  C23.5D20  D20.1D4  D20:1D4  C24:2D10  C22:C4:D10  C42:5D10  D20:5D4  D8:13D10  D20.29D4  D8:5D10  D8:6D10  C10.C25  D5x2+ 1+4  D20.37C23  D20:26D6  D20:13D6  D12:14D10  C15:2+ 1+4  D4:6D30
D4:6D10 is a maximal quotient of
 C23:2Dic10  C24.24D10  C24.27D10  C23:3D20  C24.30D10  C24.31D10  C10.12- 1+4  C10.82+ 1+4  C10.2+ 1+4  C10.102+ 1+4  C10.112+ 1+4  C10.62- 1+4  D4:5Dic10  C42.104D10  C42:11D10  C42.108D10  D4:5D20  C42:16D10  C42.113D10  C42.114D10  C42:17D10  C42.115D10  C42.116D10  C42.118D10  C24.32D10  C24:3D10  C24:4D10  C24.33D10  C24.34D10  C24.35D10  C24:5D10  C24.36D10  C10.682- 1+4  Dic10:20D4  C10.342+ 1+4  C10.352+ 1+4  C10.362+ 1+4  C10.372+ 1+4  C10.382+ 1+4  C10.392+ 1+4  C10.402+ 1+4  D20:20D4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C10.452+ 1+4  C10.462+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.742- 1+4  C10.502+ 1+4  C10.512+ 1+4  C10.522+ 1+4  C10.532+ 1+4  C10.202- 1+4  C10.222- 1+4  C10.562+ 1+4  C10.572+ 1+4  C10.582+ 1+4  C10.262- 1+4  C10.812- 1+4  C10.612+ 1+4  C10.622+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.842- 1+4  C10.662+ 1+4  C10.672+ 1+4  C10.682+ 1+4  C10.692+ 1+4  C42.137D10  C42.138D10  C42.140D10  C42:20D10  C42:21D10  C42:22D10  C42.145D10  C42.166D10  C42:26D10  D20:11D4  Dic10:11D4  C42.168D10  C42:28D10  Dic10:9Q8  C42.174D10  D20:9Q8  C42.178D10  C42.179D10  C42.180D10  C24.38D10  D4xC5:D4  C24:8D10  C24.41D10  C24.42D10  D20:26D6  D20:13D6  D12:14D10  C15:2+ 1+4  D4:6D30

37 conjugacy classes

class 1 2A2B···2F2G2H2I2J4A4B4C4D4E4F5A5B10A···10F10G···10N20A20B20C20D
order122···222224444445510···1010···1020202020
size112···2101010102210101010222···24···44444

37 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D5D10D10D102+ 1+4D4:6D10
kernelD4:6D10C4oD20D4xD5D4:2D5C2xC5:D4D4xC10C2xD4C2xC4D4C23C5C1
# reps124441228414

Matrix representation of D4:6D10 ►in GL4(F41) generated by

40403925
1717025
040181
173537
,
00341
113940
00400
10340
,
40700
34700
28353534
253360
,
40000
34100
2223640
31383535
G:=sub<GL(4,GF(41))| [40,17,0,17,40,17,40,35,39,0,18,3,25,25,1,7],[0,1,0,1,0,1,0,0,34,39,40,34,1,40,0,0],[40,34,28,25,7,7,35,33,0,0,35,6,0,0,34,0],[40,34,22,31,0,1,23,38,0,0,6,35,0,0,40,35] >;
 

D4:6D10 in GAP, Magma, Sage, TeX 

D_4\rtimes_6D_{10}
 
% in TeX
 
G:=Group("D4:6D10");
 
// GroupNames label
 
G:=SmallGroup(160,219);
 
// by ID
 
G=gap.SmallGroup(160,219);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,579,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^10=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<