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G = C33⋊SD16  order 432 = 24·33

2nd semidirect product of C33 and SD16 acting faithfully

non-abelian, soluble, monomial

Aliases: C33⋊2SD16, C3⋊2AΓL1(𝔽9), S3≀C2.S3, C3⋊F9⋊1C2, C32⋊C4.1D6, C32⋊(D4.S3), C33⋊Q8⋊2C2, (C3×C3⋊S3).2D4, (C3×S3≀C2).1C2, C3⋊S3.1(C3⋊D4), (C3×C32⋊C4).2C22, SmallGroup(432,738)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C32⋊C4 — C33⋊SD16
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C3⋊F9 — C33⋊SD16
Lower central
C33 — C3×C3⋊S3 — C3×C32⋊C4 — C33⋊SD16
Upper central
C1

Generators and relations for C33⋊SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b, eae=dbd-1=ab-1, bc=cb, ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

C1
9C2
12C2
C3
4C3
8C3
9C4
18C22
54C4
4S3
9C6
12C6
12C6
12S3
24C6
C32
4C32
8C32
9D4
27C8
27Q8
9C12
12D6
18Dic3
18C2×C6
C3⋊S3
4C3×S3
4C3×S3
8C3×S3
12C3×C6
12C3×S3
C33
27SD16
9C3⋊C8
9Dic6
9C3×D4
C32⋊C4
2S32
6C32⋊C4
12S3×C6
C3×C3⋊S3
4S3×C32
9D4.S3
S3≀C2
3PSU3(𝔽2)
3F9
C3×C32⋊C4
2C33⋊C4
2C3×S32
3AΓL1(𝔽9)
C33⋊Q8
C3×S3≀C2
C3⋊F9
C33⋊SD16

Character table of C33⋊SD16

 class 12A2B3A3B3C3D4A4B6A6B6C6D6E6F8A8B12
 size 1912288818108121218242424545436
ρ1111111111111111111    trivial
ρ211-1111111-1-11-1-1-1-1-11    linear of order 2
ρ311111111-1111111-1-11    linear of order 2
ρ411-111111-1-1-11-1-1-1111    linear of order 2
ρ5222-1-1-1220-1-1-1-12-100-1    orthogonal lifted from S3
ρ622-2-1-1-122011-11-2100-1    orthogonal lifted from D6
ρ72202222-2000200000-2    orthogonal lifted from D4
ρ8220-1-1-12-20√-3-√-3-1√-30-√-3001    complex lifted from C3⋊D4
ρ9220-1-1-12-20-√-3√-3-1-√-30√-3001    complex lifted from C3⋊D4
ρ102-2022220000-2000√-2-√-20    complex lifted from SD16
ρ112-2022220000-2000-√-2√-20    complex lifted from SD16
ρ124-40-2-2-2400002000000    symplectic lifted from D4.S3, Schur index 2
ρ1380-28-1-1-100-2-20111000    orthogonal lifted from AΓL1(𝔽9)
ρ148028-1-1-100220-1-1-1000    orthogonal lifted from AΓL1(𝔽9)
ρ1580-2-41+3√-3/21-3√-3/2-1001-√-31+√-30ζ31ζ32000    complex faithful
ρ16802-41+3√-3/21-3√-3/2-100-1+√-3-1-√-30ζ65-1ζ6000    complex faithful
ρ1780-2-41-3√-3/21+3√-3/2-1001+√-31-√-30ζ321ζ3000    complex faithful
ρ18802-41-3√-3/21+3√-3/2-100-1-√-3-1+√-30ζ6-1ζ65000    complex faithful

Permutation representations of C33⋊SD16
►On 24 points - transitive group 24T1331
Generators in S24
(1 9 22)(3 11 24)(4 17 12)(5 18 13)(7 20 15)(8 16 21)

(2 10 23)(3 24 11)(4 17 12)(6 19 14)(7 15 20)(8 16 21)

(1 22 9)(2 10 23)(3 24 11)(4 12 17)(5 18 13)(6 14 19)(7 20 15)(8 16 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)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)
 
G:=sub<Sym(24)| (1,9,22)(3,11,24)(4,17,12)(5,18,13)(7,20,15)(8,16,21), (2,10,23)(3,24,11)(4,17,12)(6,19,14)(7,15,20)(8,16,21), (1,22,9)(2,10,23)(3,24,11)(4,12,17)(5,18,13)(6,14,19)(7,20,15)(8,16,21), (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)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)>;
 
G:=Group( (1,9,22)(3,11,24)(4,17,12)(5,18,13)(7,20,15)(8,16,21), (2,10,23)(3,24,11)(4,17,12)(6,19,14)(7,15,20)(8,16,21), (1,22,9)(2,10,23)(3,24,11)(4,12,17)(5,18,13)(6,14,19)(7,20,15)(8,16,21), (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)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24) );
 
G=PermutationGroup([[(1,9,22),(3,11,24),(4,17,12),(5,18,13),(7,20,15),(8,16,21)], [(2,10,23),(3,24,11),(4,17,12),(6,19,14),(7,15,20),(8,16,21)], [(1,22,9),(2,10,23),(3,24,11),(4,12,17),(5,18,13),(6,14,19),(7,20,15),(8,16,21)], [(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),(10,12),(11,15),(14,16),(17,23),(19,21),(20,24)]])
 
G:=TransitiveGroup(24,1331);
 

►On 27 points - transitive group 27T136
Generators in S27
(1 21 25)(2 19 15)(3 10 6)(4 18 12)(5 17 7)(8 16 14)(9 11 13)(20 22 23)(24 27 26)

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

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

(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)

(4 6)(5 9)(8 10)(13 15)(14 18)(17 19)(20 24)(21 27)(23 25)
 
G:=sub<Sym(27)| (1,21,25)(2,19,15)(3,10,6)(4,18,12)(5,17,7)(8,16,14)(9,11,13)(20,22,23)(24,27,26), (1,20,24)(2,9,5)(3,18,14)(4,16,6)(7,15,13)(8,10,12)(11,17,19)(21,22,27)(23,26,25), (1,3,2)(4,13,23)(5,24,14)(6,15,25)(7,26,16)(8,17,27)(9,20,18)(10,19,21)(11,22,12), (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), (4,6)(5,9)(8,10)(13,15)(14,18)(17,19)(20,24)(21,27)(23,25)>;
 
G:=Group( (1,21,25)(2,19,15)(3,10,6)(4,18,12)(5,17,7)(8,16,14)(9,11,13)(20,22,23)(24,27,26), (1,20,24)(2,9,5)(3,18,14)(4,16,6)(7,15,13)(8,10,12)(11,17,19)(21,22,27)(23,26,25), (1,3,2)(4,13,23)(5,24,14)(6,15,25)(7,26,16)(8,17,27)(9,20,18)(10,19,21)(11,22,12), (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), (4,6)(5,9)(8,10)(13,15)(14,18)(17,19)(20,24)(21,27)(23,25) );
 
G=PermutationGroup([[(1,21,25),(2,19,15),(3,10,6),(4,18,12),(5,17,7),(8,16,14),(9,11,13),(20,22,23),(24,27,26)], [(1,20,24),(2,9,5),(3,18,14),(4,16,6),(7,15,13),(8,10,12),(11,17,19),(21,22,27),(23,26,25)], [(1,3,2),(4,13,23),(5,24,14),(6,15,25),(7,26,16),(8,17,27),(9,20,18),(10,19,21),(11,22,12)], [(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)], [(4,6),(5,9),(8,10),(13,15),(14,18),(17,19),(20,24),(21,27),(23,25)]])
 
G:=TransitiveGroup(27,136);
 

Matrix representation of C33⋊SD16 ►in GL8(𝔽73)

8800007272
064000000
0088007272
000640000
00008811
000006400
00000010
00000001
,
10000019
01000000
00646500650
00080000
000088065
000006400
00000080
000000064
,
64000006565
064000000
00640006565
000640000
00008000
00000800
00000080
00000008
,
10001000
00000100
10000000
00000001
720100000
00772007272
77200007272
01000000
,
10001000
01000000
00101000
00772007272
000072000
00000010
00000100
000077211

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,8,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,72,0,72,0,1,0,1,0,72,0,72,0,1,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,65,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,64,0,0,1,0,65,0,0,0,8,0,9,0,0,0,65,0,0,64],[64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,65,0,65,0,0,0,8,0,65,0,65,0,0,0,0,8],[1,0,1,0,72,0,7,0,0,0,0,0,0,0,72,1,0,0,0,0,1,7,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,72,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,72,0,0,0,0,1,0,1,0,72,0,0,7,0,0,0,0,0,0,1,72,0,0,0,72,0,1,0,1,0,0,0,72,0,0,0,1] >;
 

C33⋊SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes {\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:SD16");
 
// GroupNames label
 
G:=SmallGroup(432,738);
 
// by ID
 
G=gap.SmallGroup(432,738);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,84,85,135,58,2244,1971,998,165,677,2028,1363,530,14118]);
 
// 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,e*a*e=d*b*d^-1=a*b^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

Export

Subgroup lattice of C33⋊SD16 in TeX
Character table of C33⋊SD16 in TeX

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