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## G = C33⋊8SD16order 432 = 24·33

### 4th 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⋊8SD16
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C33⋊8SD16
 Lower central C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8SD16
 Upper central C1 — C2

Generators and relations for C338SD16
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, cd=dc, ece=c-1, ede=d3 >

Subgroups: 640 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 [×5], C12 [×2], D6 [×3], C2×C6, SD16, C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×4], C24, Dic6 [×2], D12, C3⋊D4, C33, C3×Dic3 [×5], C3⋊Dic3, C3⋊Dic3, S3×C6 [×3], C2×C3⋊S3, C24⋊C2, C3×C3⋊S3, C32×C6, C322C8, D6⋊S3, C3⋊D12, C322Q8 [×2], C3×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C322SD16, C3×C322C8, C339D4, C335Q8, C338SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, D12, C24⋊C2, S3≀C2, C322SD16, C322D12, C338SD16

Character table of C338SD16

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 36 2 4 4 8 8 18 36 2 4 4 8 8 36 36 18 18 18 18 36 36 18 18 18 18 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 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 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 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 linear of order 2 ρ5 2 2 0 -1 2 2 -1 -1 2 0 -1 2 2 -1 -1 0 0 -2 -2 -1 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ6 2 2 0 2 2 2 2 2 -2 0 2 2 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ7 2 2 0 -1 2 2 -1 -1 2 0 -1 2 2 -1 -1 0 0 2 2 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 -1 2 2 -1 -1 -2 0 -1 2 2 -1 -1 0 0 0 0 1 1 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ9 2 2 0 -1 2 2 -1 -1 -2 0 -1 2 2 -1 -1 0 0 0 0 1 1 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ10 2 -2 0 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ11 2 -2 0 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ12 2 -2 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 0 0 -√-2 √-2 √3 -√3 0 0 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 complex lifted from C24⋊C2 ρ13 2 -2 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 0 0 √-2 -√-2 √3 -√3 0 0 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 complex lifted from C24⋊C2 ρ14 2 -2 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 0 0 √-2 -√-2 -√3 √3 0 0 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 complex lifted from C24⋊C2 ρ15 2 -2 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 0 0 -√-2 √-2 -√3 √3 0 0 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 complex lifted from C24⋊C2 ρ16 4 4 -2 4 -2 1 -2 1 0 0 4 -2 1 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 4 2 4 -2 1 -2 1 0 0 4 -2 1 -2 1 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 4 1 -2 1 -2 0 -2 4 1 -2 1 -2 0 0 0 0 0 0 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ19 4 4 0 4 1 -2 1 -2 0 2 4 1 -2 1 -2 0 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ20 4 -4 0 4 1 -2 1 -2 0 0 -4 -1 2 -1 2 0 0 0 0 0 0 √3 -√3 0 0 0 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ21 4 -4 0 4 1 -2 1 -2 0 0 -4 -1 2 -1 2 0 0 0 0 0 0 -√3 √3 0 0 0 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ22 4 -4 0 4 -2 1 -2 1 0 0 -4 2 -1 2 -1 √-3 -√-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ23 4 -4 0 4 -2 1 -2 1 0 0 -4 2 -1 2 -1 -√-3 √-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ24 8 -8 0 -4 -4 2 2 -1 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 8 8 0 -4 -4 2 2 -1 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ26 8 8 0 -4 2 -4 -1 2 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ27 8 -8 0 -4 2 -4 -1 2 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C338SD16
On 24 points - transitive group 24T1307
Generators in S24
(1 13 21)(2 22 14)(3 23 15)(4 16 24)(5 9 17)(6 18 10)(7 19 11)(8 12 20)
(1 21 13)(2 22 14)(3 15 23)(4 16 24)(5 17 9)(6 18 10)(7 11 19)(8 12 20)
(1 21 13)(2 22 14)(3 23 15)(4 24 16)(5 17 9)(6 18 10)(7 19 11)(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)| (1,13,21)(2,22,14)(3,23,15)(4,16,24)(5,9,17)(6,18,10)(7,19,11)(8,12,20), (1,21,13)(2,22,14)(3,15,23)(4,16,24)(5,17,9)(6,18,10)(7,11,19)(8,12,20), (1,21,13)(2,22,14)(3,23,15)(4,24,16)(5,17,9)(6,18,10)(7,19,11)(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( (1,13,21)(2,22,14)(3,23,15)(4,16,24)(5,9,17)(6,18,10)(7,19,11)(8,12,20), (1,21,13)(2,22,14)(3,15,23)(4,16,24)(5,17,9)(6,18,10)(7,11,19)(8,12,20), (1,21,13)(2,22,14)(3,23,15)(4,24,16)(5,17,9)(6,18,10)(7,19,11)(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([(1,13,21),(2,22,14),(3,23,15),(4,16,24),(5,9,17),(6,18,10),(7,19,11),(8,12,20)], [(1,21,13),(2,22,14),(3,15,23),(4,16,24),(5,17,9),(6,18,10),(7,11,19),(8,12,20)], [(1,21,13),(2,22,14),(3,23,15),(4,24,16),(5,17,9),(6,18,10),(7,19,11),(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(24,1307);

Matrix representation of C338SD16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 1 72 0 0 0 1 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 72 1 0 0 0 0 72 0 0 0 0 1 0 0 72
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 62 0 0 0 0 11 25 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72,72,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,11,0,0,0,0,62,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C338SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("C3^3:8SD16");
// GroupNames label

G:=SmallGroup(432,589);
// by ID

G=gap.SmallGroup(432,589);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,36,254,58,1684,1691,298,677,348,1027,14118]);
// 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,c*d=d*c,e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

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