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G = C20.29D10  order 400 = 24·52

3rd non-split extension by C20 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, A-group

Aliases: C20.29D10, C5⋊D5⋊3C8, C5⋊3(C8×D5), C4.14D52, C5⋊2C8⋊6D5, C52⋊10(C2×C8), C10.8(C4×D5), C52⋊6C4.4C4, (C5×C20).28C22, C2.1(Dic5⋊2D5), (C5×C5⋊2C8)⋊5C2, (C4×C5⋊D5).3C2, (C2×C5⋊D5).4C4, (C5×C10).42(C2×C4), SmallGroup(400,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C20.29D10
Chief series
C1 — C5 — C52 — C5×C10 — C5×C20 — C5×C5⋊2C8 — C20.29D10
Lower central
C52 — C20.29D10
Upper central
C1 — C4

Generators and relations for C20.29D10
 G = < a,b,c | a20=1, b10=a5, c2=a10, bab-1=cac-1=a9, cbc-1=b9 >

Subgroups: 364 in 60 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, C2×C8, Dic5, C20, C20, D10, C52, C5⋊2C8, C40, C4×D5, C5⋊D5, C5×C10, C8×D5, C52⋊6C4, C5×C20, C2×C5⋊D5, C5×C5⋊2C8, C4×C5⋊D5, C20.29D10
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C2×C8, D10, C4×D5, C8×D5, D52, Dic5⋊2D5, C20.29D10

Smallest permutation representation of C20.29D10
►On 40 points
Generators in S40
(1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39)(2 20 38 16 34 12 30 8 26 4 22 40 18 36 14 32 10 28 6 24)

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

64 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D5E5F5G5H8A···8H10A10B10C10D10E10F10G10H20A···20H20I···20P40A···40P
order12224444555555558···8101010101010101020···2020···2040···40
size112525112525222244445···5222244442···24···410···10

64 irreducible representations

dim1111112222444
type+++++++
imageC1C2C2C4C4C8D5D10C4×D5C8×D5D52Dic5⋊2D5C20.29D10
kernelC20.29D10C5×C5⋊2C8C4×C5⋊D5C52⋊6C4C2×C5⋊D5C5⋊D5C5⋊2C8C20C10C5C4C2C1
# reps12122844816448

Matrix representation of C20.29D10 ►in GL4(𝔽41) generated by

03200
92200
0010
0001
,
14000
252700
00406
003535
,
9000
193200
00406
0001
G:=sub<GL(4,GF(41))| [0,9,0,0,32,22,0,0,0,0,1,0,0,0,0,1],[14,25,0,0,0,27,0,0,0,0,40,35,0,0,6,35],[9,19,0,0,0,32,0,0,0,0,40,0,0,0,6,1] >;
 

C20.29D10 in GAP, Magma, Sage, TeX 

C_{20}._{29}D_{10}
 
% in TeX
 
G:=Group("C20.29D10");
 
// GroupNames label
 
G:=SmallGroup(400,61);
 
// by ID
 
G=gap.SmallGroup(400,61);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,31,50,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=1,b^10=a^5,c^2=a^10,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=b^9>;
 
// generators/relations
 

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