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G = D4.D10order 160 = 25·5

1st non-split extension by D4 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.6D10, C20.15D4, D206C22, C20.12C23, Dic105C22, D4⋊D55C2, (C2×D4)⋊2D5, C4○D203C2, (D4×C10)⋊2C2, C54(C8⋊C22), D4.D55C2, C52C83C22, C10.45(C2×D4), (C2×C4).17D10, (C2×C10).39D4, C4.Dic56C2, C4.16(C5⋊D4), (C5×D4).6C22, C4.12(C22×D5), (C2×C20).30C22, C22.10(C5⋊D4), C2.9(C2×C5⋊D4), SmallGroup(160,153)

Series: Derived Chief Lower central Upper central

C1C20 — D4.D10
C1C5C10C20D20C4○D20 — D4.D10
C5C10C20 — D4.D10
C1C2C2×C4C2×D4

Generators and relations for D4.D10
 G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c9 >

Subgroups: 208 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, D5, C10, C10 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, D4.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C5⋊D4, D4.D10

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

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

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

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

D4.D10 is a maximal subgroup of
D20.2D4  D20.3D4  D20.14D4  D205D4  C40.23D4  C40.44D4  D2018D4  D20.38D4  D813D10  D20.29D4  D5×C8⋊C22  SD16⋊D10  C20.C24  D20.32C23  D20.33C23  D2021D6  D6036C22  D2010D6  D12.9D10  D4.D30
D4.D10 is a maximal quotient of
C20.47(C4⋊C4)  C4○D209C4  C4⋊C4.228D10  C4⋊C4.230D10  D4.3Dic10  C42.48D10  D4.1D20  C42.51D10  (C2×D4).D10  D2017D4  C4⋊D4⋊D5  C4.(D4×D5)  C42.72D10  C202D8  C42.74D10  Dic109D4  C42.76D10  D205Q8  C42.82D10  Dic105Q8  (D4×C10)⋊18C4  (C2×C10)⋊8D8  (C5×D4).31D4  D2021D6  D6036C22  D2010D6  D12.9D10  D4.D30

31 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B10A···10F10G···10N20A20B20C20D
order122222444558810···1010···1020202020
size112442022202220202···24···44444

31 irreducible representations

dim111111222222244
type++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4C8⋊C22D4.D10
kernelD4.D10C4.Dic5D4⋊D5D4.D5C4○D20D4×C10C20C2×C10C2×D4C2×C4D4C4C22C5C1
# reps112211112244414

Matrix representation of D4.D10 in GL6(𝔽41)

100000
010000
00403800
0028100
000013
00001340
,
4000000
0400000
00403800
000100
0000400
0000281
,
060000
3470000
00403800
0028100
00004038
0000281
,
9160000
36320000
00004038
0000281
00403800
0028100

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,1,13,0,0,0,0,3,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,0,1],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,40,28,0,0,0,0,38,1,0,0] >;

D4.D10 in GAP, Magma, Sage, TeX

D_4.D_{10}
% in TeX

G:=Group("D4.D10");
// GroupNames label

G:=SmallGroup(160,153);
// by ID

G=gap.SmallGroup(160,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,579,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^9>;
// generators/relations

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