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## G = C33⋊7SD16order 432 = 24·33

### 3rd semidirect product of C33 and SD16 acting via SD16/C2=D4

Aliases: C337SD16, C6.17S3≀C2, D6⋊S3.S3, C334C83C2, C335Q82C2, C3⋊Dic3.12D6, (C32×C6).11D4, C324(D4.S3), C2.6(C33⋊D4), C32(C322SD16), (C3×D6⋊S3).1C2, (C3×C6).17(C3⋊D4), (C3×C3⋊Dic3).9C22, SmallGroup(432,584)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊Dic3 — C33⋊7SD16
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊5Q8 — C33⋊7SD16
 Lower central C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊7SD16
 Upper central C1 — C2

Generators and relations for C337SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 508 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3, C6, C6 [×8], C8, D4, Q8, C32, C32 [×4], Dic3 [×5], C12 [×2], D6, C2×C6 [×3], SD16, C3×S3 [×4], C3×C6, C3×C6 [×5], C3⋊C8, Dic6 [×2], C3⋊D4, C3×D4, C33, C3×Dic3 [×5], C3⋊Dic3, C3⋊Dic3, S3×C6 [×3], C62, D4.S3, S3×C32, C32×C6, C322C8, D6⋊S3, C322Q8 [×2], C3×C3⋊D4, C3×C3⋊Dic3, C3×C3⋊Dic3, S3×C3×C6, C322SD16, C334C8, C3×D6⋊S3, C335Q8, C337SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, C3⋊D4, D4.S3, S3≀C2, C322SD16, C33⋊D4, C337SD16

Character table of C337SD16

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 12A 12B 12C size 1 1 12 2 4 4 4 4 8 18 36 2 4 4 4 4 8 12 12 12 12 12 12 12 12 54 54 36 36 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 2 2 2 2 2 2 -2 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2 0 orthogonal lifted from D4 ρ6 2 2 2 -1 -1 2 -1 2 -1 2 0 -1 -1 2 2 -1 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 0 -1 0 orthogonal lifted from S3 ρ7 2 2 -2 -1 -1 2 -1 2 -1 2 0 -1 -1 2 2 -1 -1 1 -2 1 1 -2 1 1 1 0 0 0 -1 0 orthogonal lifted from D6 ρ8 2 2 0 -1 -1 2 -1 2 -1 -2 0 -1 -1 2 2 -1 -1 -√-3 0 √-3 √-3 0 √-3 -√-3 -√-3 0 0 0 1 0 complex lifted from C3⋊D4 ρ9 2 2 0 -1 -1 2 -1 2 -1 -2 0 -1 -1 2 2 -1 -1 √-3 0 -√-3 -√-3 0 -√-3 √-3 √-3 0 0 0 1 0 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 0 0 0 0 0 0 -√-2 √-2 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 0 0 0 0 0 0 √-2 -√-2 0 0 0 complex lifted from SD16 ρ12 4 4 2 4 1 -2 1 1 -2 0 0 4 1 -2 1 1 -2 -1 -1 -1 2 -1 -1 -1 2 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 4 -2 1 -2 -2 1 0 -2 4 -2 1 -2 -2 1 0 0 0 0 0 0 0 0 0 0 1 0 1 orthogonal lifted from S3≀C2 ρ14 4 4 0 4 -2 1 -2 -2 1 0 2 4 -2 1 -2 -2 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ15 4 4 -2 4 1 -2 1 1 -2 0 0 4 1 -2 1 1 -2 1 1 1 -2 1 1 1 -2 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 -4 0 -2 -2 4 -2 4 -2 0 0 2 2 -4 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ17 4 -4 0 4 -2 1 -2 -2 1 0 0 -4 2 -1 2 2 -1 0 0 0 0 0 0 0 0 0 0 -√3 0 √3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ18 4 -4 0 4 -2 1 -2 -2 1 0 0 -4 2 -1 2 2 -1 0 0 0 0 0 0 0 0 0 0 √3 0 -√3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ19 4 -4 0 4 1 -2 1 1 -2 0 0 -4 -1 2 -1 -1 2 -√-3 -√-3 -√-3 0 √-3 √-3 √-3 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ20 4 4 -2 -2 -1-3√-3/2 -2 -1+3√-3/2 1 1 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 ζ3 1 ζ32 1+√-3 1 ζ32 ζ3 1-√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ21 4 -4 0 -2 -1-3√-3/2 -2 -1+3√-3/2 1 1 0 0 2 1-3√-3/2 2 -1 1+3√-3/2 -1 3+√-3/2 -√-3 -3+√-3/2 0 √-3 3-√-3/2 -3-√-3/2 0 0 0 0 0 0 complex faithful ρ22 4 4 2 -2 -1-3√-3/2 -2 -1+3√-3/2 1 1 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 ζ65 -1 ζ6 -1-√-3 -1 ζ6 ζ65 -1+√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ23 4 -4 0 -2 -1-3√-3/2 -2 -1+3√-3/2 1 1 0 0 2 1-3√-3/2 2 -1 1+3√-3/2 -1 -3-√-3/2 √-3 3-√-3/2 0 -√-3 -3+√-3/2 3+√-3/2 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 -2 -1+3√-3/2 -2 -1-3√-3/2 1 1 0 0 2 1+3√-3/2 2 -1 1-3√-3/2 -1 -3+√-3/2 -√-3 3+√-3/2 0 √-3 -3-√-3/2 3-√-3/2 0 0 0 0 0 0 complex faithful ρ25 4 4 -2 -2 -1+3√-3/2 -2 -1-3√-3/2 1 1 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 ζ32 1 ζ3 1-√-3 1 ζ3 ζ32 1+√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ26 4 -4 0 -2 -1+3√-3/2 -2 -1-3√-3/2 1 1 0 0 2 1+3√-3/2 2 -1 1-3√-3/2 -1 3-√-3/2 √-3 -3-√-3/2 0 -√-3 3+√-3/2 -3+√-3/2 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 4 1 -2 1 1 -2 0 0 -4 -1 2 -1 -1 2 √-3 √-3 √-3 0 -√-3 -√-3 -√-3 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ28 4 4 2 -2 -1+3√-3/2 -2 -1-3√-3/2 1 1 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 ζ6 -1 ζ65 -1+√-3 -1 ζ65 ζ6 -1-√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ29 8 8 0 -4 2 2 2 -4 -1 0 0 -4 2 2 -4 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊D4 ρ30 8 -8 0 -4 2 2 2 -4 -1 0 0 4 -2 -2 4 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C337SD16
On 24 points - transitive group 24T1291
Generators in S24
(2 23 12)(4 14 17)(6 19 16)(8 10 21)
(1 11 22)(3 24 13)(5 15 18)(7 20 9)
(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)| (2,23,12)(4,14,17)(6,19,16)(8,10,21), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (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( (2,23,12)(4,14,17)(6,19,16)(8,10,21), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (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([(2,23,12),(4,14,17),(6,19,16),(8,10,21)], [(1,11,22),(3,24,13),(5,15,18),(7,20,9)], [(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(24,1291);

Matrix representation of C337SD16 in GL4(𝔽7) generated by

 1 0 4 0 5 6 1 4 4 4 0 6 0 0 0 1
,
 5 3 5 3 3 5 2 3 0 0 1 0 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 0 6 3 0 0 2 4 5 3 4 2 5 2 2 6 3
,
 1 0 0 5 5 6 0 2 4 4 1 6 0 0 0 6
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[0,0,3,2,6,2,4,2,3,4,2,6,0,5,5,3],[1,5,4,0,0,6,4,0,0,0,1,0,5,2,6,6] >;

C337SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("C3^3:7SD16");
// GroupNames label

G:=SmallGroup(432,584);
// by ID

G=gap.SmallGroup(432,584);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,254,135,58,1684,571,298,677,1027,14118]);
// 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=b^-1,e*a*e=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

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