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G = D10.4D10order 400 = 24·52

4th non-split extension by D10 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial

Aliases: D10.4D10, Dic5.4D10, C102.8C22, C5⋊D41D5, C22.2D52, (D5×Dic5)⋊2C2, (C2×C10).4D10, C527(C4○D4), C522D43C2, C54(D42D5), C522Q85C2, (C5×C10).12C23, (D5×C10).4C22, C10.12(C22×D5), (C5×Dic5).5C22, C526C4.12C22, C2.13(C2×D52), (C5×C5⋊D4)⋊2C2, (C2×C526C4)⋊5C2, SmallGroup(400,174)

Series: Derived Chief Lower central Upper central

C1C5×C10 — D10.4D10
C1C5C52C5×C10D5×C10D5×Dic5 — D10.4D10
C52C5×C10 — D10.4D10
C1C2C22

Generators and relations for D10.4D10
 G = < a,b,c,d | a10=b2=1, c10=d2=a5, bab=cac-1=dad-1=a-1, cbc-1=a3b, dbd-1=a8b, dcd-1=c9 >

Subgroups: 508 in 96 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×2], C5 [×2], C5 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10 [×2], C10 [×10], C4○D4, Dic5 [×2], Dic5 [×8], C20 [×2], D10 [×2], C2×C10 [×2], C2×C10 [×4], C52, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×2], C5×D4 [×2], C5×D5 [×2], C5×C10, C5×C10, D42D5 [×2], C5×Dic5 [×2], C526C4 [×2], D5×C10 [×2], C102, D5×Dic5 [×2], C522D4, C522Q8, C5×C5⋊D4 [×2], C2×C526C4, D10.4D10
Quotients: C1, C2 [×7], C22 [×7], C23, D5 [×2], C4○D4, D10 [×6], C22×D5 [×2], D42D5 [×2], D52, C2×D52, D10.4D10

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

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

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

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

46 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D5E5F5G5H10A10B10C10D10E···10T10U10V10W10X20A20B20C20D
order1222244444555555551010101010···101010101020202020
size112101010102525502222444422224···42020202020202020

46 irreducible representations

dim111111222224444
type++++++++++-++-
imageC1C2C2C2C2C2D5C4○D4D10D10D10D42D5D52C2×D52D10.4D10
kernelD10.4D10D5×Dic5C522D4C522Q8C5×C5⋊D4C2×C526C4C5⋊D4C52Dic5D10C2×C10C5C22C2C1
# reps121121424444448

Matrix representation of D10.4D10 in GL6(𝔽41)

4000000
0400000
00403400
007700
000010
000001
,
0400000
4000000
001700
0004000
000010
000001
,
010000
4000000
001000
00344000
00004034
000077
,
3200000
0320000
0040000
007100
00004034
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,34,1] >;

D10.4D10 in GAP, Magma, Sage, TeX

D_{10}._4D_{10}
% in TeX

G:=Group("D10.4D10");
// GroupNames label

G:=SmallGroup(400,174);
// by ID

G=gap.SmallGroup(400,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=1,c^10=d^2=a^5,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^8*b,d*c*d^-1=c^9>;
// generators/relations

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