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G = C33⋊6SD16order 432 = 24·33

2nd semidirect product of C33 and SD16 acting via SD16/C2=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊Dic3 — C33⋊6SD16
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C33⋊6SD16
 Lower central C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊6SD16
 Upper central C1 — C2

Generators and relations for C336SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=b, bc=cb, dbd-1=a-1, ebe=a, dcd-1=ece=c-1, ede=d3 >

Subgroups: 588 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3 [×4], C6, C6 [×5], C8, D4, Q8, C32, C32 [×4], Dic3 [×3], C12 [×4], D6 [×3], C2×C6, SD16, C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×4], C3⋊C8, Dic6, D12, C3⋊D4, C3×Q8, C33, C3×Dic3 [×5], C3⋊Dic3, C3×C12, S3×C6 [×3], C2×C3⋊S3, Q82S3, C3×C3⋊S3, C32×C6, C322C8, D6⋊S3, C3⋊D12, C322Q8, C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C322SD16, C334C8, C3×C322Q8, C339D4, C336SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, C3⋊D4, Q82S3, S3≀C2, C322SD16, C33⋊D4, C336SD16

Character table of C336SD16

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I size 1 1 36 2 4 4 4 4 8 12 18 2 4 4 4 4 8 36 36 54 54 12 12 12 12 12 12 12 12 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 -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 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 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -1 2 2 -1 -1 -1 2 2 -1 2 -1 -1 2 -1 0 0 0 0 -1 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 2 2 2 2 2 0 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ7 2 2 0 -1 2 2 -1 -1 -1 -2 2 -1 2 -1 -1 2 -1 0 0 0 0 1 -2 1 1 1 1 -2 1 -1 orthogonal lifted from D6 ρ8 2 2 0 -1 2 2 -1 -1 -1 0 -2 -1 2 -1 -1 2 -1 0 0 0 0 -√-3 0 √-3 -√-3 √-3 -√-3 0 √-3 1 complex lifted from C3⋊D4 ρ9 2 2 0 -1 2 2 -1 -1 -1 0 -2 -1 2 -1 -1 2 -1 0 0 0 0 √-3 0 -√-3 √-3 -√-3 √-3 0 -√-3 1 complex lifted from C3⋊D4 ρ10 2 -2 0 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ11 2 -2 0 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ12 4 4 2 4 1 -2 -2 -2 1 0 0 4 -2 -2 -2 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 4 -2 1 1 1 -2 -2 0 4 1 1 1 -2 -2 0 0 0 0 1 1 1 1 -2 -2 1 1 0 orthogonal lifted from S3≀C2 ρ14 4 -4 0 -2 4 4 -2 -2 -2 0 0 2 -4 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ15 4 4 0 4 -2 1 1 1 -2 2 0 4 1 1 1 -2 -2 0 0 0 0 -1 -1 -1 -1 2 2 -1 -1 0 orthogonal lifted from S3≀C2 ρ16 4 4 -2 4 1 -2 -2 -2 1 0 0 4 -2 -2 -2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 -4 0 4 -2 1 1 1 -2 0 0 -4 -1 -1 -1 2 2 0 0 0 0 √3 -√3 -√3 -√3 0 0 √3 √3 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ18 4 -4 0 4 -2 1 1 1 -2 0 0 -4 -1 -1 -1 2 2 0 0 0 0 -√3 √3 √3 √3 0 0 -√3 -√3 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ19 4 4 0 -2 -2 1 -1+3√-3/2 -1-3√-3/2 1 -2 0 -2 1 -1+3√-3/2 -1-3√-3/2 -2 1 0 0 0 0 ζ32 1 ζ3 ζ32 1-√-3 1+√-3 1 ζ3 0 complex lifted from C33⋊D4 ρ20 4 -4 0 4 1 -2 -2 -2 1 0 0 -4 2 2 2 -1 -1 √-3 -√-3 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ21 4 4 0 -2 -2 1 -1-3√-3/2 -1+3√-3/2 1 -2 0 -2 1 -1-3√-3/2 -1+3√-3/2 -2 1 0 0 0 0 ζ3 1 ζ32 ζ3 1+√-3 1-√-3 1 ζ32 0 complex lifted from C33⋊D4 ρ22 4 -4 0 4 1 -2 -2 -2 1 0 0 -4 2 2 2 -1 -1 -√-3 √-3 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ23 4 4 0 -2 -2 1 -1+3√-3/2 -1-3√-3/2 1 2 0 -2 1 -1+3√-3/2 -1-3√-3/2 -2 1 0 0 0 0 ζ6 -1 ζ65 ζ6 -1+√-3 -1-√-3 -1 ζ65 0 complex lifted from C33⋊D4 ρ24 4 -4 0 -2 -2 1 -1+3√-3/2 -1-3√-3/2 1 0 0 2 -1 1-3√-3/2 1+3√-3/2 2 -1 0 0 0 0 ζ4ζ32+2ζ4 √3 ζ4ζ3+2ζ4 ζ43ζ32+2ζ43 0 0 -√3 ζ43ζ3+2ζ43 0 complex faithful ρ25 4 -4 0 -2 -2 1 -1-3√-3/2 -1+3√-3/2 1 0 0 2 -1 1+3√-3/2 1-3√-3/2 2 -1 0 0 0 0 ζ43ζ3+2ζ43 √3 ζ43ζ32+2ζ43 ζ4ζ3+2ζ4 0 0 -√3 ζ4ζ32+2ζ4 0 complex faithful ρ26 4 -4 0 -2 -2 1 -1+3√-3/2 -1-3√-3/2 1 0 0 2 -1 1-3√-3/2 1+3√-3/2 2 -1 0 0 0 0 ζ43ζ32+2ζ43 -√3 ζ43ζ3+2ζ43 ζ4ζ32+2ζ4 0 0 √3 ζ4ζ3+2ζ4 0 complex faithful ρ27 4 4 0 -2 -2 1 -1-3√-3/2 -1+3√-3/2 1 2 0 -2 1 -1-3√-3/2 -1+3√-3/2 -2 1 0 0 0 0 ζ65 -1 ζ6 ζ65 -1-√-3 -1+√-3 -1 ζ6 0 complex lifted from C33⋊D4 ρ28 4 -4 0 -2 -2 1 -1-3√-3/2 -1+3√-3/2 1 0 0 2 -1 1+3√-3/2 1-3√-3/2 2 -1 0 0 0 0 ζ4ζ3+2ζ4 -√3 ζ4ζ32+2ζ4 ζ43ζ3+2ζ43 0 0 √3 ζ43ζ32+2ζ43 0 complex faithful ρ29 8 -8 0 -4 2 -4 2 2 -1 0 0 4 4 -2 -2 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ30 8 8 0 -4 2 -4 2 2 -1 0 0 -4 -4 2 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊D4

Permutation representations of C336SD16
On 24 points - transitive group 24T1315
Generators in S24
(1 21 13)(2 14 22)(3 15 23)(4 24 16)(5 17 9)(6 10 18)(7 11 19)(8 20 12)
(1 13 21)(2 14 22)(3 23 15)(4 24 16)(5 9 17)(6 10 18)(7 19 11)(8 20 12)
(1 13 21)(2 22 14)(3 15 23)(4 24 16)(5 9 17)(6 18 10)(7 11 19)(8 20 12)
(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 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)

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

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

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

G:=TransitiveGroup(24,1315);

Matrix representation of C336SD16 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 61 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 13 43 0 0 0 0 0 0 30 60 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 1 0 0 0 0 72 0 72 0
,
 1 0 0 0 0 0 0 0 1 72 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,6,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,13,30,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C336SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("C3^3:6SD16");
// GroupNames label

G:=SmallGroup(432,583);
// by ID

G=gap.SmallGroup(432,583);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,254,135,58,1684,571,298,677,1027,14118]);
// 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=e*a*e=b,b*c=c*b,d*b*d^-1=a^-1,e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

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