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G = C33⋊SD16order 432 = 24·33

2nd semidirect product of C33 and SD16 acting faithfully

non-abelian, soluble, monomial

Aliases: C332SD16, C32AΓL1(𝔽9), S3≀C2.S3, C3⋊F91C2, C32⋊C4.1D6, C32⋊(D4.S3), C33⋊Q82C2, (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

C1C32C3×C32⋊C4 — C33⋊SD16
C1C3C33C3×C3⋊S3C3×C32⋊C4C3⋊F9 — C33⋊SD16
C33C3×C3⋊S3C3×C32⋊C4 — C33⋊SD16
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 >

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

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

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

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

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

G:=TransitiveGroup(24,1331);

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

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

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

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

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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