Copied to
clipboard

G = C333SD16order 432 = 24·33

3rd semidirect product of C33 and SD16 acting faithfully

non-abelian, soluble, monomial

Aliases: C333SD16, PSU3(𝔽2)⋊2S3, C33AΓL1(𝔽9), C3⋊F92C2, C32⋊C4.2D6, C32⋊(Q82S3), C322D12.2C2, (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

C1C32C3×C32⋊C4 — C333SD16
C1C3C33C3×C3⋊S3C3×C32⋊C4C322D12 — C333SD16
C33C3×C3⋊S3C3×C32⋊C4 — C333SD16
C1

Generators and relations for C333SD16
 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 >

9C2
36C2
4C3
8C3
9C4
18C4
54C22
9C6
12S3
12S3
12S3
24S3
36C6
4C32
8C32
9Q8
27D4
27C8
9C12
18D6
18C12
36D6
4C3⋊S3
12C3×S3
12C3×S3
12C3×S3
24C3×S3
27SD16
9C3⋊C8
9D12
9C3×Q8
2C32⋊C4
6S32
12S32
4C3×C3⋊S3
9Q82S3
3F9
3S3≀C2
2C324D6
2C3×C32⋊C4
3AΓL1(𝔽9)

Character table of C333SD16

 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 Q82S3
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 C333SD16
On 24 points - transitive group 24T1332
Generators in S24
(2 22 11)(3 12 23)(4 13 24)(6 15 18)(7 19 16)(8 20 9)
(1 21 10)(2 11 22)(3 12 23)(5 14 17)(6 18 15)(7 19 16)
(1 10 21)(2 22 11)(3 12 23)(4 24 13)(5 14 17)(6 18 15)(7 16 19)(8 20 9)
(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 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)

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

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

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

G:=TransitiveGroup(24,1332);

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

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

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

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

G:=TransitiveGroup(27,140);

Matrix representation of C333SD16 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] >;

C333SD16 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 C333SD16 in TeX
Character table of C333SD16 in TeX

׿
×
𝔽