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G = D46D10order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46D10, C232D10, D208C22, C10.7C24, C512+ 1+4, C20.21C23, D10.3C23, Dic108C22, Dic5.4C23, (D4×D5)⋊4C2, (C2×D4)⋊7D5, (C2×C4)⋊3D10, C4○D205C2, (D4×C10)⋊7C2, D42D54C2, (C2×C20)⋊3C22, (C5×D4)⋊7C22, (C4×D5)⋊1C22, C5⋊D43C22, 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

C1C10 — D46D10
C1C5C10D10C22×D5D4×D5 — D46D10
C5C10 — D46D10
C1C2C2×D4

Generators and relations for D46D10
 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 [×9], C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], D5 [×4], C10, C10 [×5], C2×D4, C2×D4 [×8], C4○D4 [×6], Dic5 [×4], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×4], C2×C10 [×2], 2+ 1+4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C4○D20 [×2], D4×D5 [×4], D42D5 [×4], C2×C5⋊D4 [×4], D4×C10, D46D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, C22×D5 [×7], C23×D5, D46D10

Smallest permutation representation of D46D10
On 40 points
Generators in S40
(1 33 7 38)(2 39 8 34)(3 35 9 40)(4 31 10 36)(5 37 6 32)(11 26 16 21)(12 22 17 27)(13 28 18 23)(14 24 19 29)(15 30 20 25)
(1 25)(2 21)(3 27)(4 23)(5 29)(6 24)(7 30)(8 26)(9 22)(10 28)(11 34)(12 40)(13 36)(14 32)(15 38)(16 39)(17 35)(18 31)(19 37)(20 33)
(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 2)(3 5)(6 9)(7 8)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(32 40)(33 39)(34 38)(35 37)

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

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

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

D46D10 is a maximal subgroup of
C23⋊D20  C23.5D20  D20.1D4  D201D4  C242D10  C22⋊C4⋊D10  C425D10  D205D4  D813D10  D20.29D4  D85D10  D86D10  C10.C25  D5×2+ 1+4  D20.37C23  D2026D6  D2013D6  D1214D10  C15⋊2+ 1+4  D46D30
D46D10 is a maximal quotient of
C232Dic10  C24.24D10  C24.27D10  C233D20  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  D45Dic10  C42.104D10  C4211D10  C42.108D10  D45D20  C4216D10  C42.113D10  C42.114D10  C4217D10  C42.115D10  C42.116D10  C42.118D10  C24.32D10  C243D10  C244D10  C24.33D10  C24.34D10  C24.35D10  C245D10  C24.36D10  C10.682- 1+4  Dic1020D4  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  D2020D4  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  C4220D10  C4221D10  C4222D10  C42.145D10  C42.166D10  C4226D10  D2011D4  Dic1011D4  C42.168D10  C4228D10  Dic109Q8  C42.174D10  D209Q8  C42.178D10  C42.179D10  C42.180D10  C24.38D10  D4×C5⋊D4  C248D10  C24.41D10  C24.42D10  D2026D6  D2013D6  D1214D10  C15⋊2+ 1+4  D46D30

37 conjugacy classes

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

37 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D5D10D10D102+ 1+4D46D10
kernelD46D10C4○D20D4×D5D42D5C2×C5⋊D4D4×C10C2×D4C2×C4D4C23C5C1
# reps124441228414

Matrix representation of D46D10 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] >;

D46D10 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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