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G = C32:D20order 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

C1C32C5xC3:S3 — C32:D20
C1C5C3xC15C5xC3:S3D15:S3 — C32:D20
C3xC15C5xC3:S3 — C32:D20
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
9C2
30C2
30C2
2C3
2C3
9C4
45C22
45C22
6S3
6S3
10S3
10S3
30C6
30C6
6D5
6D5
9C10
2C15
2C15
45D4
30D6
30D6
10C3xS3
10C3xS3
9D10
9D10
9C20
2D15
2D15
6C3xD5
6C3xD5
6C5xS3
6C5xS3
5S32
5S32
9D20
6S3xD5
6S3xD5
2C3xD15
2C3xD15
5S3wrC2

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+5/243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ112-200220-1-5/2-1+5/2001+5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/2ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    orthogonal lifted from D20
ρ122-200220-1-5/2-1+5/2001+5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/24ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    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+5/2ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    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
ρ188000-420-2-25-2+250000-1+5/21-51+5-1-5/20000    orthogonal faithful
ρ1980002-40-2-25-2+2500001-5-1+5/2-1-5/21+50000    orthogonal faithful
ρ208000-420-2+25-2-250000-1-5/21+51-5-1+5/20000    orthogonal faithful
ρ2180002-40-2+25-2-2500001+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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