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G = C20:D10  order 400 = 24·52

2nd semidirect product of C20 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial

Aliases: D20:4D5, C20:2D10, D10:2D10, C4:2D52, C5:D5:2D4, C5:2(D4xD5), C52:3(C2xD4), (C5xD20):7C2, (C5xC20):2C22, C52:2D4:2C2, (D5xC10):2C22, (C5xC10).9C23, C10.9(C22xD5), C52:6C4:3C22, (C2xD52):2C2, (C4xC5:D5):2C2, C2.11(C2xD52), (C2xC5:D5).16C22, SmallGroup(400,171)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5xC10 — C20:D10
Chief series
C1 — C5 — C52 — C5xC10 — D5xC10 — C2xD52 — C20:D10
Lower central
C52 — C5xC10 — C20:D10
Upper central
C1 — C2 — C4

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

Subgroups: 940 in 124 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2xC4, D4, C23, D5, C10, C10, C2xD4, Dic5, C20, C20, D10, D10, C2xC10, C52, C4xD5, D20, C5:D4, C5xD4, C22xD5, C5xD5, C5:D5, C5xC10, D4xD5, C52:6C4, C5xC20, D52, D5xC10, C2xC5:D5, C52:2D4, C5xD20, C4xC5:D5, C2xD52, C20:D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22xD5, D4xD5, D52, C2xD52, C20:D10

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

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

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D5E5F5G5H10A10B10C10D10E10F10G10H10I···10P20A···20L
order122222224455555555101010101010101010···1020···20
size11101010102525250222244442222444420···204···4

46 irreducible representations

dim1111122224444
type++++++++++++
imageC1C2C2C2C2D4D5D10D10D4xD5D52C2xD52C20:D10
kernelC20:D10C52:2D4C5xD20C4xC5:D5C2xD52C5:D5D20C20D10C5C4C2C1
# reps1221224484448

Matrix representation of C20:D10 ►in GL6(F41)

100000
010000
001200
00404000
0000040
0000135
,
1350000
660000
001200
0004000
0000400
0000351
,
4060000
010000
0040000
0004000
000010
0000640

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1],[40,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
 

C20:D10 in GAP, Magma, Sage, TeX 

C_{20}\rtimes D_{10}
 
% in TeX
 
G:=Group("C20:D10");
 
// GroupNames label
 
G:=SmallGroup(400,171);
 
// by ID
 
G=gap.SmallGroup(400,171);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,116,50,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=b^10=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
 
// generators/relations
 

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