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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, (D4×D5)⋊4C2, (C2×D4)⋊7D5, (C2×C4)⋊3D10, C4○D20⋊5C2, (D4×C10)⋊7C2, D4⋊2D5⋊4C2, (C2×C20)⋊3C22, (C5×D4)⋊7C22, (C4×D5)⋊1C22, C5⋊D4⋊3C22, C2.8(C23×D5), (C2×C10).2C23, C4.21(C22×D5), (C22×C10)⋊5C22, (C2×Dic5)⋊4C22, (C22×D5)⋊3C22, C22.6(C22×D5), (C2×C5⋊D4)⋊11C2, SmallGroup(160,219)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D4⋊6D10
Chief series
C1 — C5 — C10 — D10 — C22×D5 — D4×D5 — D4⋊6D10
Lower central
C5 — C10 — D4⋊6D10
Upper central
C1 — C2 — C2×D4

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, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D4⋊6D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C23×D5, 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  D5×2+ 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  D4×C5⋊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⋊6D10C4○D20D4×D5D4⋊2D5C2×C5⋊D4D4×C10C2×D4C2×C4D4C23C5C1
# 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
 

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