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## G = C32⋊5SD16order 144 = 24·32

### 3rd semidirect product of C32 and SD16 acting via SD16/C4=C22

Aliases: Dic61S3, C6.14D12, C12.13D6, C325SD16, C3⋊C83S3, C4.3S32, (C3×C6).10D4, C32(C24⋊C2), (C3×Dic6)⋊2C2, C6.3(C3⋊D4), C12⋊S3.2C2, C31(Q82S3), (C3×C12).5C22, C2.6(C3⋊D12), (C3×C3⋊C8)⋊3C2, SmallGroup(144,60)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊5SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊5SD16
 Lower central C32 — C3×C6 — C3×C12 — C32⋊5SD16
 Upper central C1 — C2 — C4

Generators and relations for C325SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=b-1, dcd=c3 >

Character table of C325SD16

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D size 1 1 36 2 2 4 2 12 2 2 4 6 6 2 2 4 4 4 12 12 6 6 6 6 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 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 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 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 linear of order 2 ρ5 2 2 0 -1 2 -1 2 -2 -1 2 -1 0 0 2 2 -1 -1 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 2 -1 -1 2 0 2 -1 -1 -2 -2 -1 -1 2 -1 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ7 2 2 0 -1 2 -1 2 2 -1 2 -1 0 0 2 2 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 -1 -1 2 0 2 -1 -1 2 2 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 2 0 2 2 2 -2 0 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 2 -1 -1 -2 0 2 -1 -1 0 0 1 1 -2 1 1 0 0 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ11 2 2 0 2 -1 -1 -2 0 2 -1 -1 0 0 1 1 -2 1 1 0 0 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ12 2 -2 0 2 2 2 0 0 -2 -2 -2 -√-2 √-2 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ13 2 -2 0 2 2 2 0 0 -2 -2 -2 √-2 -√-2 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ14 2 2 0 -1 2 -1 -2 0 -1 2 -1 0 0 -2 -2 1 1 1 -√-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ15 2 2 0 -1 2 -1 -2 0 -1 2 -1 0 0 -2 -2 1 1 1 √-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ16 2 -2 0 2 -1 -1 0 0 -2 1 1 -√-2 √-2 -√3 √3 0 √3 -√3 0 0 -ζ87ζ32+ζ85ζ32+ζ85 -ζ87ζ3+ζ85ζ3+ζ85 -ζ83ζ3+ζ8ζ3+ζ8 -ζ83ζ32+ζ8ζ32+ζ8 complex lifted from C24⋊C2 ρ17 2 -2 0 2 -1 -1 0 0 -2 1 1 √-2 -√-2 √3 -√3 0 -√3 √3 0 0 -ζ83ζ3+ζ8ζ3+ζ8 -ζ83ζ32+ζ8ζ32+ζ8 -ζ87ζ32+ζ85ζ32+ζ85 -ζ87ζ3+ζ85ζ3+ζ85 complex lifted from C24⋊C2 ρ18 2 -2 0 2 -1 -1 0 0 -2 1 1 √-2 -√-2 -√3 √3 0 √3 -√3 0 0 -ζ83ζ32+ζ8ζ32+ζ8 -ζ83ζ3+ζ8ζ3+ζ8 -ζ87ζ3+ζ85ζ3+ζ85 -ζ87ζ32+ζ85ζ32+ζ85 complex lifted from C24⋊C2 ρ19 2 -2 0 2 -1 -1 0 0 -2 1 1 -√-2 √-2 √3 -√3 0 -√3 √3 0 0 -ζ87ζ3+ζ85ζ3+ζ85 -ζ87ζ32+ζ85ζ32+ζ85 -ζ83ζ32+ζ8ζ32+ζ8 -ζ83ζ3+ζ8ζ3+ζ8 complex lifted from C24⋊C2 ρ20 4 -4 0 -2 4 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ21 4 4 0 -2 -2 1 -4 0 -2 -2 1 0 0 2 2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ22 4 4 0 -2 -2 1 4 0 -2 -2 1 0 0 -2 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ23 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 -2√3 2√3 0 -√3 √3 0 0 0 0 0 0 orthogonal faithful ρ24 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 2√3 -2√3 0 √3 -√3 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C325SD16
On 24 points - transitive group 24T233
Generators in S24
(1 13 24)(2 17 14)(3 15 18)(4 19 16)(5 9 20)(6 21 10)(7 11 22)(8 23 12)
(1 13 24)(2 14 17)(3 15 18)(4 16 19)(5 9 20)(6 10 21)(7 11 22)(8 12 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 3)(2 6)(5 7)(9 22)(10 17)(11 20)(12 23)(13 18)(14 21)(15 24)(16 19)

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

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

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

G:=TransitiveGroup(24,233);

Matrix representation of C325SD16 in GL6(𝔽73)

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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C325SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C3^2:5SD16");
// GroupNames label

G:=SmallGroup(144,60);
// by ID

G=gap.SmallGroup(144,60);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,31,218,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

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