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G = He3⋊SD16  order 432 = 24·33

The semidirect product of He3 and SD16 acting faithfully

non-abelian, soluble

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 >

C1
9C2
36C2
C3
12C3
9C4
18C4
54C22
9C6
12S3
12S3
36C6
36S3
4C32
9Q8
27C8
27D4
9C12
18C12
18D6
36D6
4C3⋊S3
12C3×S3
12C3×S3
He3
27SD16
9C3×Q8
9C3⋊C8
9D12
12S32
He3⋊C2
4C32⋊C6
9Q8⋊2S3
He3⋊C4
2C32⋊D6
2He3⋊C4
He3⋊C8
SU3(𝔽2)
He3⋊D4
He3⋊SD16

Character table of He3⋊SD16

 class 12A2B3A3B4A4B6A6B8A8B12A12B12C
 size 1936224183618725454363636
ρ111111111111111    trivial
ρ211-1111-11-1111-1-1    linear of order 2
ρ311-111111-1-1-1111    linear of order 2
ρ4111111-111-1-11-1-1    linear of order 2
ρ522022-202000-200    orthogonal lifted from D4
ρ62-202200-20-√-2√-2000    complex lifted from SD16
ρ72-202200-20√-2-√-2000    complex lifted from SD16
ρ86-20-302-21000-111    orthogonal faithful
ρ96-20-30221000-1-1-1    orthogonal faithful
ρ106-20-30-2010001-√-3√-3    complex faithful
ρ116-20-30-2010001√-3-√-3    complex faithful
ρ1280-28-1000100000    orthogonal lifted from AΓL1(𝔽9)
ρ138028-1000-100000    orthogonal lifted from AΓL1(𝔽9)
ρ141240-6000-2000000    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)

0110-11
-20-10-2-1
201112
1000-11
-1-1-1-10-2
00002-1
,
0-10000
1-10000
000-100
001-100
-10-10-1-1
010110
,
100000
010000
-1-2-2-1-1-2
1-10-11-1
011001
-111101
,
00-1100
000100
000-11-1
-1-1-2-1-1-2
-1000-10
112002
,
112021
000-11-1
201112
101121
-10-10-1-1
-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
 

Export

Subgroup lattice of He3⋊SD16 in TeX
Character table of He3⋊SD16 in TeX

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