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## G = C33⋊3SD16order 432 = 24·33

### 3rd semidirect product of C33 and SD16 acting faithfully

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

 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 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
12C3×S3
12C3×S3
12C3×S3
24C3×S3
27SD16
9D12
6S32
12S32
3F9

Character table of C333SD16

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 8A 8B 12A 12B 12C size 1 9 36 2 8 16 18 36 18 72 54 54 36 36 36 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 -1 2 -1 2 -2 -1 0 0 0 1 1 -1 orthogonal lifted from D6 ρ6 2 2 0 -1 2 -1 2 2 -1 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 2 2 2 -2 0 2 0 0 0 0 0 -2 orthogonal lifted from D4 ρ8 2 2 0 -1 2 -1 -2 0 -1 0 0 0 √-3 -√-3 1 complex lifted from C3⋊D4 ρ9 2 2 0 -1 2 -1 -2 0 -1 0 0 0 -√-3 √-3 1 complex lifted from C3⋊D4 ρ10 2 -2 0 2 2 2 0 0 -2 0 -√-2 √-2 0 0 0 complex lifted from SD16 ρ11 2 -2 0 2 2 2 0 0 -2 0 √-2 -√-2 0 0 0 complex lifted from SD16 ρ12 4 -4 0 -2 4 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ13 8 0 2 8 -1 -1 0 0 0 -1 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ14 8 0 -2 8 -1 -1 0 0 0 1 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ15 16 0 0 -8 -2 1 0 0 0 0 0 0 0 0 0 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)

 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 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 72 72 72 72 72 72 72 72 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 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 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 72 72 72 72 72 72 72
,
 72 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 72 72 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 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 0 67 0 0 0 0 0 0 0 0 0 6 6 6 6 0 0 0 0 0 0 0 0 6 0 67 0 0 0 0 0 0 0 0 0 67 67 6 6 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 72 72 72 72 72 72 72 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 72 72 72 72 72 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

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

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