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G = C33⋊3SD16  order 432 = 24·33

3rd semidirect product of C33 and SD16 acting faithfully

non-abelian, soluble, monomial

Aliases: C33⋊3SD16, PSU3(𝔽2)⋊2S3, C3⋊3AΓL1(𝔽9), C3⋊F9⋊2C2, C32⋊C4.2D6, C32⋊(Q8⋊2S3), C32⋊2D12.2C2, (C3×PSU3(𝔽2))⋊1C2, (C3×C3⋊S3).3D4, C3⋊S3.2(C3⋊D4), (C3×C32⋊C4).3C22, SmallGroup(432,739)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C32⋊C4 — C33⋊3SD16
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C32⋊2D12 — C33⋊3SD16
Lower central
C33 — C3×C3⋊S3 — C3×C32⋊C4 — C33⋊3SD16
Upper central
C1

Generators and relations for C33⋊3SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b, ae=ea, bc=cb, dbd-1=ab-1, ebe=a-1b-1, dcd-1=ece=c-1, ede=d3 >

C1
9C2
36C2
C3
4C3
8C3
9C4
18C4
54C22
9C6
12S3
12S3
12S3
24S3
36C6
C32
4C32
8C32
9Q8
27D4
27C8
9C12
18D6
18C12
36D6
C3⋊S3
4C3⋊S3
12C3×S3
12C3×S3
12C3×S3
24C3×S3
C33
27SD16
9C3⋊C8
9D12
9C3×Q8
C32⋊C4
2C32⋊C4
6S32
12S32
C3×C3⋊S3
4C3×C3⋊S3
9Q8⋊2S3
PSU3(𝔽2)
3F9
3S3≀C2
C3×C32⋊C4
2C32⋊4D6
2C3×C32⋊C4
3AΓL1(𝔽9)
C3⋊F9
C3×PSU3(𝔽2)
C32⋊2D12
C33⋊3SD16

Character table of C33⋊3SD16

 class 12A2B3A3B3C4A4B6A6B8A8B12A12B12C
 size 19362816183618725454363636
ρ1111111111111111    trivial
ρ211-11111-11-111-1-11    linear of order 2
ρ311-1111111-1-1-1111    linear of order 2
ρ41111111-111-1-1-1-11    linear of order 2
ρ5220-12-12-2-100011-1    orthogonal lifted from D6
ρ6220-12-122-1000-1-1-1    orthogonal lifted from S3
ρ7220222-20200000-2    orthogonal lifted from D4
ρ8220-12-1-20-1000√-3-√-31    complex lifted from C3⋊D4
ρ9220-12-1-20-1000-√-3√-31    complex lifted from C3⋊D4
ρ102-2022200-20-√-2√-2000    complex lifted from SD16
ρ112-2022200-20√-2-√-2000    complex lifted from SD16
ρ124-40-24-2002000000    orthogonal lifted from Q8⋊2S3
ρ138028-1-1000-100000    orthogonal lifted from AΓL1(𝔽9)
ρ1480-28-1-1000100000    orthogonal lifted from AΓL1(𝔽9)
ρ151600-8-21000000000    orthogonal faithful

Permutation representations of C33⋊3SD16
►On 24 points - transitive group 24T1332
Generators in S24
(2 22 16)(3 9 23)(4 10 24)(6 12 18)(7 19 13)(8 20 14)

(1 21 15)(2 16 22)(3 9 23)(5 11 17)(6 18 12)(7 19 13)

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

(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 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
 
G:=sub<Sym(24)| (2,22,16)(3,9,23)(4,10,24)(6,12,18)(7,19,13)(8,20,14), (1,21,15)(2,16,22)(3,9,23)(5,11,17)(6,18,12)(7,19,13), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (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)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
 
G:=Group( (2,22,16)(3,9,23)(4,10,24)(6,12,18)(7,19,13)(8,20,14), (1,21,15)(2,16,22)(3,9,23)(5,11,17)(6,18,12)(7,19,13), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (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)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
 
G=PermutationGroup([[(2,22,16),(3,9,23),(4,10,24),(6,12,18),(7,19,13),(8,20,14)], [(1,21,15),(2,16,22),(3,9,23),(5,11,17),(6,18,12),(7,19,13)], [(1,15,21),(2,22,16),(3,9,23),(4,24,10),(5,11,17),(6,18,12),(7,13,19),(8,20,14)], [(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),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
 
G:=TransitiveGroup(24,1332);
 

►On 27 points - transitive group 27T140
Generators in S27
(1 17 13)(2 7 11)(3 23 27)(4 10 21)(5 20 26)(6 8 25)(9 22 24)(12 15 14)(16 18 19)

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

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

(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)

(2 3)(4 22)(5 25)(6 20)(7 23)(8 26)(9 21)(10 24)(11 27)(12 18)(14 16)(15 19)
 
G:=sub<Sym(27)| (1,17,13)(2,7,11)(3,23,27)(4,10,21)(5,20,26)(6,8,25)(9,22,24)(12,15,14)(16,18,19), (1,16,12)(2,22,26)(3,6,10)(4,27,25)(5,7,24)(8,21,23)(9,20,11)(13,19,14)(15,17,18), (1,3,2)(4,20,14)(5,15,21)(6,22,16)(7,17,23)(8,24,18)(9,19,25)(10,26,12)(11,13,27), (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), (2,3)(4,22)(5,25)(6,20)(7,23)(8,26)(9,21)(10,24)(11,27)(12,18)(14,16)(15,19)>;
 
G:=Group( (1,17,13)(2,7,11)(3,23,27)(4,10,21)(5,20,26)(6,8,25)(9,22,24)(12,15,14)(16,18,19), (1,16,12)(2,22,26)(3,6,10)(4,27,25)(5,7,24)(8,21,23)(9,20,11)(13,19,14)(15,17,18), (1,3,2)(4,20,14)(5,15,21)(6,22,16)(7,17,23)(8,24,18)(9,19,25)(10,26,12)(11,13,27), (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), (2,3)(4,22)(5,25)(6,20)(7,23)(8,26)(9,21)(10,24)(11,27)(12,18)(14,16)(15,19) );
 
G=PermutationGroup([[(1,17,13),(2,7,11),(3,23,27),(4,10,21),(5,20,26),(6,8,25),(9,22,24),(12,15,14),(16,18,19)], [(1,16,12),(2,22,26),(3,6,10),(4,27,25),(5,7,24),(8,21,23),(9,20,11),(13,19,14),(15,17,18)], [(1,3,2),(4,20,14),(5,15,21),(6,22,16),(7,17,23),(8,24,18),(9,19,25),(10,26,12),(11,13,27)], [(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)], [(2,3),(4,22),(5,25),(6,20),(7,23),(8,26),(9,21),(10,24),(11,27),(12,18),(14,16),(15,19)]])
 
G:=TransitiveGroup(27,140);
 

Matrix representation of C33⋊3SD16 ►in GL12(𝔽73)

100000000000
010000000000
001000000000
000100000000
000000000100
000010000000
00007272727272727272
000000000001
000000010000
000001000000
000000100000
000000001000
,
100000000000
010000000000
001000000000
000100000000
000000010000
000000001000
000010000000
000000100000
000000000010
000000000001
000001000000
00007272727272727272
,
72720000000000
100000000000
00727200000000
001000000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
67067000000000
666600000000
6067000000000
67676600000000
000010000000
000000100000
000000001000
00007272727272727272
000000000001
000000010000
000001000000
000000000100
,
100000000000
72720000000000
0072000000000
001100000000
000010000000
000001000000
000000000001
000000000010
00007272727272727272
000000000100
000000010000
000000100000

G:=sub<GL(12,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,1,0,72],[72,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[67,6,6,67,0,0,0,0,0,0,0,0,0,6,0,67,0,0,0,0,0,0,0,0,67,6,67,6,0,0,0,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0],[1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,1,0,72,0,0,0] >;
 

C33⋊3SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_3{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:3SD16");
 
// GroupNames label
 
G:=SmallGroup(432,739);
 
// by ID
 
G=gap.SmallGroup(432,739);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,254,135,58,1691,998,165,5381,348,1363,530,14118]);
 
// 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,a*e=e*a,b*c=c*b,d*b*d^-1=a*b^-1,e*b*e=a^-1*b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

Export

Subgroup lattice of C33⋊3SD16 in TeX
Character table of C33⋊3SD16 in TeX

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