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## G = He3⋊SD16order 432 = 24·33

### The semidirect product of He3 and SD16 acting faithfully

Aliases: He3⋊SD16, SU3(𝔽2)⋊C2, C3.AΓL1(𝔽9), He3⋊C8⋊C2, He3⋊D4.C2, He3⋊C2.D4, He3⋊C4.C22, SmallGroup(432,520)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C4 — He3⋊SD16
 Chief series C1 — C3 — He3 — He3⋊C2 — He3⋊C4 — He3⋊C8 — He3⋊SD16
 Lower central He3 — He3⋊C2 — He3⋊C4 — He3⋊SD16
 Upper central C1

Generators and relations for He3⋊SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=c, eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=ab-1c, ece=a-1b-1c, ede=d3 >

9C2
36C2
12C3
9C4
18C4
54C22
9C6
12S3
12S3
36C6
36S3
4C32
9Q8
27C8
27D4
9C12
18C12
18D6
36D6
12C3×S3
12C3×S3
27SD16
9D12
12S32

Character table of He3⋊SD16

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 8A 8B 12A 12B 12C size 1 9 36 2 24 18 36 18 72 54 54 36 36 36 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 2 2 -2 0 2 0 0 0 -2 0 0 orthogonal lifted from D4 ρ6 2 -2 0 2 2 0 0 -2 0 -√-2 √-2 0 0 0 complex lifted from SD16 ρ7 2 -2 0 2 2 0 0 -2 0 √-2 -√-2 0 0 0 complex lifted from SD16 ρ8 6 -2 0 -3 0 2 -2 1 0 0 0 -1 1 1 orthogonal faithful ρ9 6 -2 0 -3 0 2 2 1 0 0 0 -1 -1 -1 orthogonal faithful ρ10 6 -2 0 -3 0 -2 0 1 0 0 0 1 -√-3 √-3 complex faithful ρ11 6 -2 0 -3 0 -2 0 1 0 0 0 1 √-3 -√-3 complex faithful ρ12 8 0 -2 8 -1 0 0 0 1 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ13 8 0 2 8 -1 0 0 0 -1 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ14 12 4 0 -6 0 0 0 -2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of He3⋊SD16
On 27 points - transitive group 27T141
Generators in S27
(1 18 11)(2 26 14)(3 7 22)(4 12 10)(5 8 25)(6 24 19)(9 13 15)(16 27 17)(20 21 23)
(1 3 2)(4 15 23)(5 24 16)(6 17 25)(7 26 18)(8 19 27)(9 20 12)(10 13 21)(11 22 14)
(1 17 10)(2 6 21)(3 25 13)(4 7 24)(5 23 18)(8 12 14)(9 11 19)(15 26 16)(20 22 27)
(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 13)(5 16)(6 19)(7 14)(8 17)(9 12)(10 15)(11 18)(21 23)(22 26)(25 27)

G:=sub<Sym(27)| (1,18,11)(2,26,14)(3,7,22)(4,12,10)(5,8,25)(6,24,19)(9,13,15)(16,27,17)(20,21,23), (1,3,2)(4,15,23)(5,24,16)(6,17,25)(7,26,18)(8,19,27)(9,20,12)(10,13,21)(11,22,14), (1,17,10)(2,6,21)(3,25,13)(4,7,24)(5,23,18)(8,12,14)(9,11,19)(15,26,16)(20,22,27), (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)(4,13)(5,16)(6,19)(7,14)(8,17)(9,12)(10,15)(11,18)(21,23)(22,26)(25,27)>;

G:=Group( (1,18,11)(2,26,14)(3,7,22)(4,12,10)(5,8,25)(6,24,19)(9,13,15)(16,27,17)(20,21,23), (1,3,2)(4,15,23)(5,24,16)(6,17,25)(7,26,18)(8,19,27)(9,20,12)(10,13,21)(11,22,14), (1,17,10)(2,6,21)(3,25,13)(4,7,24)(5,23,18)(8,12,14)(9,11,19)(15,26,16)(20,22,27), (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)(4,13)(5,16)(6,19)(7,14)(8,17)(9,12)(10,15)(11,18)(21,23)(22,26)(25,27) );

G=PermutationGroup([[(1,18,11),(2,26,14),(3,7,22),(4,12,10),(5,8,25),(6,24,19),(9,13,15),(16,27,17),(20,21,23)], [(1,3,2),(4,15,23),(5,24,16),(6,17,25),(7,26,18),(8,19,27),(9,20,12),(10,13,21),(11,22,14)], [(1,17,10),(2,6,21),(3,25,13),(4,7,24),(5,23,18),(8,12,14),(9,11,19),(15,26,16),(20,22,27)], [(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),(4,13),(5,16),(6,19),(7,14),(8,17),(9,12),(10,15),(11,18),(21,23),(22,26),(25,27)]])

G:=TransitiveGroup(27,141);

Matrix representation of He3⋊SD16 in GL6(ℤ)

 0 1 1 0 -1 1 -2 0 -1 0 -2 -1 2 0 1 1 1 2 1 0 0 0 -1 1 -1 -1 -1 -1 0 -2 0 0 0 0 2 -1
,
 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 -1 0 -1 0 -1 -1 0 1 0 1 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 -1 -2 -2 -1 -1 -2 1 -1 0 -1 1 -1 0 1 1 0 0 1 -1 1 1 1 0 1
,
 0 0 -1 1 0 0 0 0 0 1 0 0 0 0 0 -1 1 -1 -1 -1 -2 -1 -1 -2 -1 0 0 0 -1 0 1 1 2 0 0 2
,
 1 1 2 0 2 1 0 0 0 -1 1 -1 2 0 1 1 1 2 1 0 1 1 2 1 -1 0 -1 0 -1 -1 -2 -1 -2 -1 -3 -2

G:=sub<GL(6,Integers())| [0,-2,2,1,-1,0,1,0,0,0,-1,0,1,-1,1,0,-1,0,0,0,1,0,-1,0,-1,-2,1,-1,0,2,1,-1,2,1,-2,-1],[0,1,0,0,-1,0,-1,-1,0,0,0,1,0,0,0,1,-1,0,0,0,-1,-1,0,1,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,-1,1,0,-1,0,1,-2,-1,1,1,0,0,-2,0,1,1,0,0,-1,-1,0,1,0,0,-1,1,0,0,0,0,-2,-1,1,1],[0,0,0,-1,-1,1,0,0,0,-1,0,1,-1,0,0,-2,0,2,1,1,-1,-1,0,0,0,0,1,-1,-1,0,0,0,-1,-2,0,2],[1,0,2,1,-1,-2,1,0,0,0,0,-1,2,0,1,1,-1,-2,0,-1,1,1,0,-1,2,1,1,2,-1,-3,1,-1,2,1,-1,-2] >;

He3⋊SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes {\rm SD}_{16}
% in TeX

G:=Group("He3:SD16");
// GroupNames label

G:=SmallGroup(432,520);
// by ID

G=gap.SmallGroup(432,520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,197,135,58,1684,4491,998,1425,4709,2028,2875,1286,537,14118,7069]);
// Polycyclic

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

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