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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(D4×D5), C52⋊3(C2×D4), (C5×D20)⋊7C2, (C5×C20)⋊2C22, C52⋊2D4⋊2C2, (D5×C10)⋊2C22, (C5×C10).9C23, C10.9(C22×D5), C52⋊6C4⋊3C22, (C2×D52)⋊2C2, (C4×C5⋊D5)⋊2C2, C2.11(C2×D52), (C2×C5⋊D5).16C22, SmallGroup(400,171)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — C20⋊D10
Chief series
C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — C20⋊D10
Lower central
C52 — C5×C10 — 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, C2×C4, D4, C23, D5, C10, C10, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C52, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C5×D5, C5⋊D5, C5×C10, D4×D5, C52⋊6C4, C5×C20, D52, D5×C10, C2×C5⋊D5, C52⋊2D4, C5×D20, C4×C5⋊D5, C2×D52, C20⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, D4×D5, D52, C2×D52, 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++++++++++++
imageC1C2C2C2C2D4D5D10D10D4×D5D52C2×D52C20⋊D10
kernelC20⋊D10C52⋊2D4C5×D20C4×C5⋊D5C2×D52C5⋊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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