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G = S32:D5order 360 = 23·32·5

The semidirect product of S32 and D5 acting via D5/C5=C2

non-abelian, soluble, monomial

Aliases: S32:D5, C5:2S3wrC2, (C3xC15):1D4, C32:(C5:D4), D15:S3:1C2, C3:S3.1D10, C32:Dic5:2C2, (C5xS32):3C2, (C5xC3:S3).2C22, SmallGroup(360,133)

Series: Derived Chief Lower central Upper central

C1C32C5xC3:S3 — S32:D5
C1C5C3xC15C5xC3:S3D15:S3 — S32:D5
C3xC15C5xC3:S3 — S32:D5
C1

Generators and relations for S32:D5
 G = < a,b,c,d,e | a5=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 432 in 52 conjugacy classes, 11 normal (all characteristic)
Quotients: C1, C2, C22, D4, D5, D10, C5:D4, S3wrC2, S32:D5
6C2
9C2
30C2
2C3
2C3
9C22
45C22
45C4
2S3
6S3
6C6
6S3
10S3
30C6
6D5
6C10
9C10
2C15
2C15
45D4
6D6
30D6
2C3xS3
10C3xS3
9D10
9Dic5
9C2xC10
2C5xS3
2D15
6C3xD5
6C30
6C5xS3
6C5xS3
5S32
5C32:C4
9C5:D4
6S3xC10
6S3xD5
2C3xD15
2S3xC15
5S3wrC2

Character table of S32:D5

 class 12A2B2C3A3B45A5B6A6B10A10B10C10D10E10F15A15B15C15D15E15F30A30B30C30D
 size 1693044902212606666181844448812121212
ρ1111111111111111111111111111    trivial
ρ21-11111-111-11-1-1-1-111111111-1-1-1-1    linear of order 2
ρ31-11-111111-1-1-1-1-1-111111111-1-1-1-1    linear of order 2
ρ4111-111-1111-11111111111111111    linear of order 2
ρ520-2022022000000-2-22222220000    orthogonal lifted from D4
ρ62-220220-1-5/2-1+5/2-201-5/21-5/21+5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ72220220-1+5/2-1-5/220-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ82220220-1-5/2-1+5/220-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ92-220220-1+5/2-1-5/2-201+5/21+5/21-5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1020-20220-1+5/2-1-5/200ζ53525352ζ5455451-5/21+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2ζ545545ζ53525352    complex lifted from C5:D4
ρ1120-20220-1+5/2-1-5/2005352ζ5352545ζ5451-5/21+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2545ζ5455352ζ5352    complex lifted from C5:D4
ρ1220-20220-1-5/2-1+5/200545ζ545ζ535253521+5/21-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2ζ53525352545ζ545    complex lifted from C5:D4
ρ1320-20220-1-5/2-1+5/200ζ5455455352ζ53521+5/21-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/25352ζ5352ζ545545    complex lifted from C5:D4
ρ14400-21-204401000000-2-2-2-2110000    orthogonal lifted from S3wrC2
ρ154-200-2104410-2-2-2-2001111-2-21111    orthogonal lifted from S3wrC2
ρ1640021-20440-1000000-2-2-2-2110000    orthogonal lifted from S3wrC2
ρ174200-21044-102222001111-2-2-1-1-1-1    orthogonal lifted from S3wrC2
ρ184-200-210-1+5-1-510-2ζ53-2ζ52-2ζ54-2ζ500535254553+2ζ5254+2ζ51+5/21-5/2ζ54ζ5ζ53ζ52    complex faithful
ρ194200-210-1-5-1+5-1054552530054553+2ζ5254+2ζ553521-5/21+5/25253545    complex faithful
ρ204200-210-1-5-1+5-1055453520054+2ζ5535254553+2ζ521-5/21+5/25352554    complex faithful
ρ214200-210-1+5-1-5-10535254500535254553+2ζ5254+2ζ51+5/21-5/25455352    complex faithful
ρ224-200-210-1-5-1+510-2ζ54-2ζ5-2ζ52-2ζ530054553+2ζ5254+2ζ553521-5/21+5/2ζ52ζ53ζ54ζ5    complex faithful
ρ234-200-210-1-5-1+510-2ζ5-2ζ54-2ζ53-2ζ520054+2ζ5535254553+2ζ521-5/21+5/2ζ53ζ52ζ5ζ54    complex faithful
ρ244-200-210-1+5-1-510-2ζ52-2ζ53-2ζ5-2ζ540053+2ζ5254+2ζ553525451+5/21-5/2ζ5ζ54ζ52ζ53    complex faithful
ρ254200-210-1+5-1-5-1052535540053+2ζ5254+2ζ553525451+5/21-5/25545253    complex faithful
ρ2680002-40-2+25-2-25000000001+51-51+51-5-1-5/2-1+5/20000    orthogonal faithful
ρ2780002-40-2-25-2+25000000001-51+51-51+5-1+5/2-1-5/20000    orthogonal faithful

Permutation representations of S32:D5
On 30 points - transitive group 30T96
Generators in S30
(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)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)
(1 19)(2 18)(3 17)(4 16)(5 20)(6 22 11 27)(7 21 12 26)(8 25 13 30)(9 24 14 29)(10 23 15 28)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 27)(22 26)(23 30)(24 29)(25 28)

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

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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (1,19)(2,18)(3,17)(4,16)(5,20)(6,22,11,27)(7,21,12,26)(8,25,13,30)(9,24,14,29)(10,23,15,28), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28) );

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)], [(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13)], [(1,19),(2,18),(3,17),(4,16),(5,20),(6,22,11,27),(7,21,12,26),(8,25,13,30),(9,24,14,29),(10,23,15,28)], [(2,5),(3,4),(6,7),(8,10),(11,12),(13,15),(16,17),(18,20),(21,27),(22,26),(23,30),(24,29),(25,28)]])

G:=TransitiveGroup(30,96);

On 30 points - transitive group 30T100
Generators in S30
(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 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 29 11 24)(7 28 12 23)(8 27 13 22)(9 26 14 21)(10 30 15 25)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(9 21)(10 25)(11 29)(12 28)(13 27)(14 26)(15 30)

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

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), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,29,11,24)(7,28,12,23)(8,27,13,22)(9,26,14,21)(10,30,15,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,21)(10,25)(11,29)(12,28)(13,27)(14,26)(15,30) );

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)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,29,11,24),(7,28,12,23),(8,27,13,22),(9,26,14,21),(10,30,15,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(9,21),(10,25),(11,29),(12,28),(13,27),(14,26),(15,30)]])

G:=TransitiveGroup(30,100);

Matrix representation of S32:D5 in GL6(F61)

3400000
090000
001000
000100
000010
000001
,
100000
010000
001000
000100
00005915
0000121
,
100000
010000
00591500
0012100
000010
000001
,
0600000
100000
0000600
0000121
0060000
0006000
,
0600000
6000000
0060000
0006000
0000600
0000121

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,60,12,0,0,0,0,0,1,0,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,12,0,0,0,0,0,1] >;

S32:D5 in GAP, Magma, Sage, TeX

S_3^2\rtimes D_5
% in TeX

G:=Group("S3^2:D5");
// GroupNames label

G:=SmallGroup(360,133);
// by ID

G=gap.SmallGroup(360,133);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,73,579,201,111,244,376,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of S32:D5 in TeX
Character table of S32:D5 in TeX

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