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G = S32:D4  order 288 = 25·32

The semidirect product of S32 and D4 acting via D4/C4=C2

non-abelian, soluble, monomial

Aliases: S32:1D4, C4:2S3wrC2, (C3xC12):1D4, C3:Dic3:5D4, D6:D6:8C2, C32:1(C4:D4), C6.D6.12C22, S32:C4:4C2, (C4xS32):1C2, (C2xS3wrC2):1C2, C3:S3.2(C2xD4), C2.13(C2xS3wrC2), C4:(C32:C4):4C2, (C2xS32).7C22, (C3xC6).11(C2xD4), C3:S3.4(C4oD4), (C2xC3:S3).5C23, (C4xC3:S3).35C22, (C2xC32:C4).5C22, SmallGroup(288,878)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2xC3:S3 — S32:D4
Chief series
C1 — C32 — C3:S3 — C2xC3:S3 — C2xS32 — C2xS3wrC2 — S32:D4
Lower central
C32 — C2xC3:S3 — S32:D4
Upper central
C1 — C2 — C4

Generators and relations for S32:D4
 G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 872 in 148 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2xC4, D4, C23, C32, Dic3, C12, D6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C4:D4, C3xDic3, C3:Dic3, C3xC12, C32:C4, S32, S32, S3xC6, C2xC3:S3, S3xC2xC4, S3xD4, S3xDic3, C6.D6, D6:S3, S3xC12, C3xD12, C4xC3:S3, S3wrC2, C2xC32:C4, C2xS32, C2xS32, S32:C4, C4:(C32:C4), C4xS32, D6:D6, C2xS3wrC2, S32:D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C4:D4, S3wrC2, C2xS3wrC2, S32:D4

Character table of S32:D4

 class 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D12E
 size 11669912124426618363644121224244481212
ρ1111111111111111111111111111    trivial
ρ21111111-111-1-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ3111111-1111-1-1-1-11-11111-11-1-1-1-1-1    linear of order 2
ρ4111111-1-1111111-1-11111-1-111111    linear of order 2
ρ511-1-111-1111-111-1-1111-1-1-11-1-1-111    linear of order 2
ρ611-1-111-1-1111-1-111111-1-1-1-1111-1-1    linear of order 2
ρ711-1-1111-111-111-11-111-1-11-1-1-1-111    linear of order 2
ρ811-1-11111111-1-11-1-111-1-111111-1-1    linear of order 2
ρ92200-2-20022-200200220000-2-2-200    orthogonal lifted from D4
ρ102-22-22-20022000000-2-22-20000000    orthogonal lifted from D4
ρ112-2-222-20022000000-2-2-220000000    orthogonal lifted from D4
ρ122200-2-20022200-20022000022200    orthogonal lifted from D4
ρ132-200-2200220-2i2i000-2-200000002i-2i    complex lifted from C4oD4
ρ142-200-22002202i-2i000-2-20000000-2i2i    complex lifted from C4oD4
ρ15442200001-2-4-2-2000-21-1-100-1-1211    orthogonal lifted from C2xS3wrC2
ρ164400002-2-21-4000001-200-1122-100    orthogonal lifted from C2xS3wrC2
ρ1744000022-214000001-200-1-1-2-2100    orthogonal lifted from S3wrC2
ρ18440000-22-21-4000001-2001-122-100    orthogonal lifted from C2xS3wrC2
ρ19440000-2-2-214000001-20011-2-2100    orthogonal lifted from S3wrC2
ρ2044-2-200001-2-422000-211100-1-12-1-1    orthogonal lifted from C2xS3wrC2
ρ21442200001-2422000-21-1-10011-2-1-1    orthogonal lifted from S3wrC2
ρ2244-2-200001-24-2-2000-21110011-211    orthogonal lifted from S3wrC2
ρ234-42-200001-20-2i2i0002-1-11003i-3i0-ii    complex faithful
ρ244-4-2200001-202i-2i0002-11-1003i-3i0i-i    complex faithful
ρ254-4-2200001-20-2i2i0002-11-100-3i3i0-ii    complex faithful
ρ264-42-200001-202i-2i0002-1-1100-3i3i0i-i    complex faithful
ρ278-8000000-42000000-24000000000    orthogonal faithful

Permutation representations of S32:D4
►On 24 points - transitive group 24T644
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)

(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)

(1 23 3 21)(2 22 4 24)(5 14 12 17)(6 13 9 20)(7 16 10 19)(8 15 11 18)

(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
 
G:=sub<Sym(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), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,23,3,21)(2,22,4,24)(5,14,12,17)(6,13,9,20)(7,16,10,19)(8,15,11,18), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)>;
 
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), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,23,3,21)(2,22,4,24)(5,14,12,17)(6,13,9,20)(7,16,10,19)(8,15,11,18), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20) );
 
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)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,23,3,21),(2,22,4,24),(5,14,12,17),(6,13,9,20),(7,16,10,19),(8,15,11,18)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)]])
 
G:=TransitiveGroup(24,644);
 

►On 24 points - transitive group 24T649
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)

(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)

(1 22)(2 21)(3 24)(4 23)(5 15 10 20)(6 14 11 19)(7 13 12 18)(8 16 9 17)

(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
 
G:=sub<Sym(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), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,21)(3,24)(4,23)(5,15,10,20)(6,14,11,19)(7,13,12,18)(8,16,9,17), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)>;
 
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), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,21)(3,24)(4,23)(5,15,10,20)(6,14,11,19)(7,13,12,18)(8,16,9,17), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20) );
 
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)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,22),(2,21),(3,24),(4,23),(5,15,10,20),(6,14,11,19),(7,13,12,18),(8,16,9,17)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)]])
 
G:=TransitiveGroup(24,649);
 

►On 24 points - transitive group 24T652
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(5 10 21)(6 11 22)(7 12 23)(8 9 24)

(1 19 14)(2 20 15)(3 17 16)(4 18 13)

(1 24)(2 23)(3 22)(4 21)(5 13 10 18)(6 16 11 17)(7 15 12 20)(8 14 9 19)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
 
G:=sub<Sym(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), (5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,24)(2,23)(3,22)(4,21)(5,13,10,18)(6,16,11,17)(7,15,12,20)(8,14,9,19), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)>;
 
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), (5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,24)(2,23)(3,22)(4,21)(5,13,10,18)(6,16,11,17)(7,15,12,20)(8,14,9,19), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24) );
 
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)], [(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13)], [(1,24),(2,23),(3,22),(4,21),(5,13,10,18),(6,16,11,17),(7,15,12,20),(8,14,9,19)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)]])
 
G:=TransitiveGroup(24,652);
 

►On 24 points - transitive group 24T656
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(5 10 21)(6 11 22)(7 12 23)(8 9 24)

(1 19 14)(2 20 15)(3 17 16)(4 18 13)

(1 22 3 24)(2 21 4 23)(5 13 12 20)(6 16 9 19)(7 15 10 18)(8 14 11 17)

(5 12)(6 9)(7 10)(8 11)(21 23)(22 24)
 
G:=sub<Sym(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), (5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,22,3,24)(2,21,4,23)(5,13,12,20)(6,16,9,19)(7,15,10,18)(8,14,11,17), (5,12)(6,9)(7,10)(8,11)(21,23)(22,24)>;
 
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), (5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,22,3,24)(2,21,4,23)(5,13,12,20)(6,16,9,19)(7,15,10,18)(8,14,11,17), (5,12)(6,9)(7,10)(8,11)(21,23)(22,24) );
 
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)], [(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13)], [(1,22,3,24),(2,21,4,23),(5,13,12,20),(6,16,9,19),(7,15,10,18),(8,14,11,17)], [(5,12),(6,9),(7,10),(8,11),(21,23),(22,24)]])
 
G:=TransitiveGroup(24,656);
 

Matrix representation of S32:D4 ►in GL4(F5) generated by

0400
1000
0001
0040
,
3112
4324
1303
3420
,
3442
1321
4302
3130
,
1012
0431
0002
0020
,
1012
0124
0003
0020
G:=sub<GL(4,GF(5))| [0,1,0,0,4,0,0,0,0,0,0,4,0,0,1,0],[3,4,1,3,1,3,3,4,1,2,0,2,2,4,3,0],[3,1,4,3,4,3,3,1,4,2,0,3,2,1,2,0],[1,0,0,0,0,4,0,0,1,3,0,2,2,1,2,0],[1,0,0,0,0,1,0,0,1,2,0,2,2,4,3,0] >;
 

S32:D4 in GAP, Magma, Sage, TeX 

S_3^2\rtimes D_4
 
% in TeX
 
G:=Group("S3^2:D4");
 
// GroupNames label
 
G:=SmallGroup(288,878);
 
// by ID
 
G=gap.SmallGroup(288,878);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,100,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,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

Character table of S32:D4 in TeX

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