Copied to
clipboard

## G = C33⋊SD16order 432 = 24·33

### 2nd semidirect product of C33 and SD16 acting faithfully

Aliases: C332SD16, C32AΓL1(𝔽9), S3≀C2.S3, C3⋊F91C2, C32⋊C4.1D6, C32⋊(D4.S3), C33⋊Q82C2, (C3×C3⋊S3).2D4, (C3×S3≀C2).1C2, C3⋊S3.1(C3⋊D4), (C3×C32⋊C4).2C22, SmallGroup(432,738)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊C4 — C33⋊SD16
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C3⋊F9 — C33⋊SD16
 Lower central C33 — C3×C3⋊S3 — C3×C32⋊C4 — C33⋊SD16
 Upper central C1

Generators and relations for C33⋊SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b, eae=dbd-1=ab-1, bc=cb, ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

9C2
12C2
4C3
8C3
9C4
18C22
54C4
4S3
9C6
12C6
12C6
12S3
24C6
4C32
8C32
9D4
27C8
27Q8
9C12
12D6
18Dic3
18C2×C6
12C3×C6
12C3×S3
27SD16
9Dic6
2S32
12S3×C6
3F9

Character table of C33⋊SD16

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 12 size 1 9 12 2 8 8 8 18 108 12 12 18 24 24 24 54 54 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 linear of order 2 ρ5 2 2 2 -1 -1 -1 2 2 0 -1 -1 -1 -1 2 -1 0 0 -1 orthogonal lifted from S3 ρ6 2 2 -2 -1 -1 -1 2 2 0 1 1 -1 1 -2 1 0 0 -1 orthogonal lifted from D6 ρ7 2 2 0 2 2 2 2 -2 0 0 0 2 0 0 0 0 0 -2 orthogonal lifted from D4 ρ8 2 2 0 -1 -1 -1 2 -2 0 √-3 -√-3 -1 √-3 0 -√-3 0 0 1 complex lifted from C3⋊D4 ρ9 2 2 0 -1 -1 -1 2 -2 0 -√-3 √-3 -1 -√-3 0 √-3 0 0 1 complex lifted from C3⋊D4 ρ10 2 -2 0 2 2 2 2 0 0 0 0 -2 0 0 0 √-2 -√-2 0 complex lifted from SD16 ρ11 2 -2 0 2 2 2 2 0 0 0 0 -2 0 0 0 -√-2 √-2 0 complex lifted from SD16 ρ12 4 -4 0 -2 -2 -2 4 0 0 0 0 2 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ13 8 0 -2 8 -1 -1 -1 0 0 -2 -2 0 1 1 1 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ14 8 0 2 8 -1 -1 -1 0 0 2 2 0 -1 -1 -1 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ15 8 0 -2 -4 1+3√-3/2 1-3√-3/2 -1 0 0 1-√-3 1+√-3 0 ζ3 1 ζ32 0 0 0 complex faithful ρ16 8 0 2 -4 1+3√-3/2 1-3√-3/2 -1 0 0 -1+√-3 -1-√-3 0 ζ65 -1 ζ6 0 0 0 complex faithful ρ17 8 0 -2 -4 1-3√-3/2 1+3√-3/2 -1 0 0 1+√-3 1-√-3 0 ζ32 1 ζ3 0 0 0 complex faithful ρ18 8 0 2 -4 1-3√-3/2 1+3√-3/2 -1 0 0 -1-√-3 -1+√-3 0 ζ6 -1 ζ65 0 0 0 complex faithful

Permutation representations of C33⋊SD16
On 24 points - transitive group 24T1331
Generators in S24
(1 11 22)(3 13 24)(4 17 14)(5 18 15)(7 20 9)(8 10 21)
(2 12 23)(3 24 13)(4 17 14)(6 19 16)(7 9 20)(8 10 21)
(1 22 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)

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

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

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

G:=TransitiveGroup(24,1331);

On 27 points - transitive group 27T136
Generators in S27
(1 26 22)(2 11 7)(3 17 13)(4 19 10)(5 16 18)(6 15 8)(9 14 12)(20 25 27)(21 24 23)
(1 25 21)(2 16 12)(3 10 6)(4 15 17)(5 14 7)(8 13 19)(9 11 18)(20 23 22)(24 26 27)
(1 3 2)(4 18 27)(5 20 19)(6 12 21)(7 22 13)(8 14 23)(9 24 15)(10 16 25)(11 26 17)
(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)
(5 7)(6 10)(9 11)(12 16)(13 19)(15 17)(20 22)(21 25)(24 26)

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

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

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

G:=TransitiveGroup(27,136);

Matrix representation of C33⋊SD16 in GL8(𝔽73)

 8 8 0 0 0 0 72 72 0 64 0 0 0 0 0 0 0 0 8 8 0 0 72 72 0 0 0 64 0 0 0 0 0 0 0 0 8 8 1 1 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 9 0 1 0 0 0 0 0 0 0 0 64 65 0 0 65 0 0 0 0 8 0 0 0 0 0 0 0 0 8 8 0 65 0 0 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 64
,
 64 0 0 0 0 0 65 65 0 64 0 0 0 0 0 0 0 0 64 0 0 0 65 65 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8
,
 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 72 0 1 0 0 0 0 0 0 0 7 72 0 0 72 72 7 72 0 0 0 0 72 72 0 1 0 0 0 0 0 0
,
 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 7 72 0 0 72 72 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 72 1 1

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,8,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,72,0,72,0,1,0,1,0,72,0,72,0,1,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,65,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,1,0,65,0,0,0,8,0,9,0,0,0,65,0,0,64],[64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,65,0,65,0,0,0,8,0,65,0,65,0,0,0,0,8],[1,0,1,0,72,0,7,0,0,0,0,0,0,0,72,1,0,0,0,0,1,7,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,72,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,72,0,0,0,0,1,0,1,0,72,0,0,7,0,0,0,0,0,0,1,72,0,0,0,72,0,1,0,1,0,0,0,72,0,0,0,1] >;

C33⋊SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C3^3:SD16");
// GroupNames label

G:=SmallGroup(432,738);
// by ID

G=gap.SmallGroup(432,738);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,84,85,135,58,2244,1971,998,165,677,2028,1363,530,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b,e*a*e=d*b*d^-1=a*b^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

Export

׿
×
𝔽