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G = C32:D20  order 360 = 23·32·5

The semidirect product of C32 and D20 acting via D20/C5=D4

non-abelian, soluble, monomial

Aliases: C32:D20, C5:1S3wrC2, C32:C4:D5, (C3xC15):2D4, D15:S3:2C2, C3:S3.2D10, (C5xC32:C4):1C2, (C5xC3:S3).3C22, SmallGroup(360,134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C5xC3:S3 — C32:D20
Chief series
C1 — C5 — C3xC15 — C5xC3:S3 — D15:S3 — C32:D20
Lower central
C3xC15 — C5xC3:S3 — C32:D20
Upper central
C1

Generators and relations for C32:D20
 G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

Subgroups: 520 in 52 conjugacy classes, 11 normal (9 characteristic)
Quotients: C1, C2, C22, D4, D5, D10, D20, S3wrC2, C32:D20
C1
9C2
30C2
30C2
2C3
2C3
C5
9C4
45C22
45C22
6S3
6S3
10S3
10S3
30C6
30C6
C32
6D5
6D5
9C10
2C15
2C15
45D4
30D6
30D6
C3:S3
10C3xS3
10C3xS3
9D10
9D10
9C20
2D15
2D15
6C3xD5
6C3xD5
6C5xS3
6C5xS3
C3xC15
C32:C4
5S32
5S32
9D20
6S3xD5
6S3xD5
C5xC3:S3
2C3xD15
2C3xD15
5S3wrC2
D15:S3
D15:S3
C5xC32:C4
C32:D20

Character table of C32:D20

 class 12A2B2C3A3B45A5B6A6B10A10B15A15B15C15D20A20B20C20D
 size 19303044182260601818888818181818
ρ1111111111111111111111    trivial
ρ211-1111-111-11111111-1-1-1-1    linear of order 2
ρ3111-111-1111-1111111-1-1-1-1    linear of order 2
ρ411-1-111111-1-11111111111    linear of order 2
ρ52-2002202200-2-222220000    orthogonal lifted from D4
ρ6220022-2-1-√5/2-1+√5/200-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ72200222-1-√5/2-1+√5/200-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ82200222-1+√5/2-1-√5/200-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ9220022-2-1+√5/2-1-√5/200-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ102-200220-1+√5/2-1-√5/2001-√5/21+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ112-200220-1-√5/2-1+√5/2001+√5/21-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ122-200220-1-√5/2-1+√5/2001+√5/21-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ132-200220-1+√5/2-1-√5/2001-√5/21+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ144020-21044-10001-2-210000    orthogonal lifted from S3wrC2
ρ15400-21-20440100-211-20000    orthogonal lifted from S3wrC2
ρ1640-20-2104410001-2-210000    orthogonal lifted from S3wrC2
ρ1740021-20440-100-211-20000    orthogonal lifted from S3wrC2
ρ188000-420-2-2√5-2+2√50000-1+√5/21-√51+√5-1-√5/20000    orthogonal faithful
ρ1980002-40-2-2√5-2+2√500001-√5-1+√5/2-1-√5/21+√50000    orthogonal faithful
ρ208000-420-2+2√5-2-2√50000-1-√5/21+√51-√5-1+√5/20000    orthogonal faithful
ρ2180002-40-2+2√5-2-2√500001+√5-1-√5/2-1+√5/21-√50000    orthogonal faithful

Permutation representations of C32:D20
►On 30 points - transitive group 30T95
Generators in S30
(2 27 17)(4 19 29)(6 11 21)(8 23 13)(10 15 25)

(1 26 16)(3 18 28)(5 30 20)(7 22 12)(9 14 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)

(1 9)(2 8)(3 7)(4 6)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)
 
G:=sub<Sym(30)| (2,27,17)(4,19,29)(6,11,21)(8,23,13)(10,15,25), (1,26,16)(3,18,28)(5,30,20)(7,22,12)(9,14,24), (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), (1,9)(2,8)(3,7)(4,6)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)>;
 
G:=Group( (2,27,17)(4,19,29)(6,11,21)(8,23,13)(10,15,25), (1,26,16)(3,18,28)(5,30,20)(7,22,12)(9,14,24), (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), (1,9)(2,8)(3,7)(4,6)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21) );
 
G=PermutationGroup([[(2,27,17),(4,19,29),(6,11,21),(8,23,13),(10,15,25)], [(1,26,16),(3,18,28),(5,30,20),(7,22,12),(9,14,24)], [(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)], [(1,9),(2,8),(3,7),(4,6),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21)]])
 
G:=TransitiveGroup(30,95);
 

Matrix representation of C32:D20 ►in GL6(F61)

100000
010000
0060100
0060000
0000060
0000160
,
100000
010000
0060100
0060000
0000601
0000600
,
1590000
52150000
000010
000001
000100
001000
,
9460000
46520000
0000600
0000060
0060000
0006000

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[15,52,0,0,0,0,9,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[9,46,0,0,0,0,46,52,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,60,0,0,0,0,0,0,60,0,0] >;
 

C32:D20 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_{20}
 
% in TeX
 
G:=Group("C3^2:D20");
 
// GroupNames label
 
G:=SmallGroup(360,134);
 
// by ID
 
G=gap.SmallGroup(360,134);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-5,73,31,579,585,111,244,130,376,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32:D20 in TeX
Character table of C32:D20 in TeX

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