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G = C337SD16order 432 = 24·33

3rd semidirect product of C33 and SD16 acting via SD16/C2=D4

non-abelian, soluble, monomial

Aliases: C337SD16, C6.17S3≀C2, D6⋊S3.S3, C334C83C2, C335Q82C2, C3⋊Dic3.12D6, (C32×C6).11D4, C324(D4.S3), C2.6(C33⋊D4), C32(C322SD16), (C3×D6⋊S3).1C2, (C3×C6).17(C3⋊D4), (C3×C3⋊Dic3).9C22, SmallGroup(432,584)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊Dic3 — C337SD16
C1C3C33C32×C6C3×C3⋊Dic3C335Q8 — C337SD16
C33C32×C6C3×C3⋊Dic3 — C337SD16
C1C2

Generators and relations for C337SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 508 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3, C6, C6 [×8], C8, D4, Q8, C32, C32 [×4], Dic3 [×5], C12 [×2], D6, C2×C6 [×3], SD16, C3×S3 [×4], C3×C6, C3×C6 [×5], C3⋊C8, Dic6 [×2], C3⋊D4, C3×D4, C33, C3×Dic3 [×5], C3⋊Dic3, C3⋊Dic3, S3×C6 [×3], C62, D4.S3, S3×C32, C32×C6, C322C8, D6⋊S3, C322Q8 [×2], C3×C3⋊D4, C3×C3⋊Dic3, C3×C3⋊Dic3, S3×C3×C6, C322SD16, C334C8, C3×D6⋊S3, C335Q8, C337SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, C3⋊D4, D4.S3, S3≀C2, C322SD16, C33⋊D4, C337SD16

Character table of C337SD16

 class 12A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B12A12B12C
 size 1112244448183624444812121212121212125454363636
ρ1111111111111111111111111111111    trivial
ρ211-11111111-1111111-1-1-1-1-1-1-1-111-11-1    linear of order 2
ρ31111111111-111111111111111-1-1-11-1    linear of order 2
ρ411-111111111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ5220222222-2022222200000000000-20    orthogonal lifted from D4
ρ6222-1-12-12-120-1-122-1-1-12-1-12-1-1-1000-10    orthogonal lifted from S3
ρ722-2-1-12-12-120-1-122-1-11-211-2111000-10    orthogonal lifted from D6
ρ8220-1-12-12-1-20-1-122-1-1--30-3-30-3--3--300010    complex lifted from C3⋊D4
ρ9220-1-12-12-1-20-1-122-1-1-30--3--30--3-3-300010    complex lifted from C3⋊D4
ρ102-2022222200-2-2-2-2-2-200000000--2-2000    complex lifted from SD16
ρ112-2022222200-2-2-2-2-2-200000000-2--2000    complex lifted from SD16
ρ1244241-211-20041-211-2-1-1-12-1-1-1200000    orthogonal lifted from S3≀C2
ρ134404-21-2-210-24-21-2-210000000000101    orthogonal lifted from S3≀C2
ρ144404-21-2-21024-21-2-210000000000-10-1    orthogonal lifted from S3≀C2
ρ1544-241-211-20041-211-2111-2111-200000    orthogonal lifted from S3≀C2
ρ164-40-2-24-24-20022-4-4220000000000000    symplectic lifted from D4.S3, Schur index 2
ρ174-404-21-2-2100-42-122-10000000000-303    symplectic lifted from C322SD16, Schur index 2
ρ184-404-21-2-2100-42-122-1000000000030-3    symplectic lifted from C322SD16, Schur index 2
ρ194-4041-211-200-4-12-1-12--3--3--30-3-3-3000000    complex lifted from C322SD16
ρ2044-2-2-1-3-3/2-2-1+3-3/21100-2-1+3-3/2-21-1-3-3/21ζ31ζ321+-31ζ32ζ31--300000    complex lifted from C33⋊D4
ρ214-40-2-1-3-3/2-2-1+3-3/2110021-3-3/22-11+3-3/2-13+-3/2--3-3+-3/20-33--3/2-3--3/2000000    complex faithful
ρ22442-2-1-3-3/2-2-1+3-3/21100-2-1+3-3/2-21-1-3-3/21ζ65-1ζ6-1--3-1ζ6ζ65-1+-300000    complex lifted from C33⋊D4
ρ234-40-2-1-3-3/2-2-1+3-3/2110021-3-3/22-11+3-3/2-1-3--3/2-33--3/20--3-3+-3/23+-3/2000000    complex faithful
ρ244-40-2-1+3-3/2-2-1-3-3/2110021+3-3/22-11-3-3/2-1-3+-3/2--33+-3/20-3-3--3/23--3/2000000    complex faithful
ρ2544-2-2-1+3-3/2-2-1-3-3/21100-2-1-3-3/2-21-1+3-3/21ζ321ζ31--31ζ3ζ321+-300000    complex lifted from C33⋊D4
ρ264-40-2-1+3-3/2-2-1-3-3/2110021+3-3/22-11-3-3/2-13--3/2-3-3--3/20--33+-3/2-3+-3/2000000    complex faithful
ρ274-4041-211-200-4-12-1-12-3-3-30--3--3--3000000    complex lifted from C322SD16
ρ28442-2-1+3-3/2-2-1-3-3/21100-2-1-3-3/2-21-1+3-3/21ζ6-1ζ65-1+-3-1ζ65ζ6-1--300000    complex lifted from C33⋊D4
ρ29880-4222-4-100-422-42-10000000000000    orthogonal lifted from C33⋊D4
ρ308-80-4222-4-1004-2-24-210000000000000    symplectic faithful, Schur index 2

Permutation representations of C337SD16
On 24 points - transitive group 24T1291
Generators in S24
(2 23 12)(4 14 17)(6 19 16)(8 10 21)
(1 11 22)(3 24 13)(5 15 18)(7 20 9)
(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)| (2,23,12)(4,14,17)(6,19,16)(8,10,21), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (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( (2,23,12)(4,14,17)(6,19,16)(8,10,21), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (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([(2,23,12),(4,14,17),(6,19,16),(8,10,21)], [(1,11,22),(3,24,13),(5,15,18),(7,20,9)], [(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(24,1291);

Matrix representation of C337SD16 in GL4(𝔽7) generated by

1040
5614
4406
0001
,
5353
3523
0010
0004
,
3632
6342
0020
0004
,
0630
0245
3425
2263
,
1005
5602
4416
0006
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[0,0,3,2,6,2,4,2,3,4,2,6,0,5,5,3],[1,5,4,0,0,6,4,0,0,0,1,0,5,2,6,6] >;

C337SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("C3^3:7SD16");
// GroupNames label

G:=SmallGroup(432,584);
// by ID

G=gap.SmallGroup(432,584);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,254,135,58,1684,571,298,677,1027,14118]);
// 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^-1,e*a*e=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

Export

Character table of C337SD16 in TeX

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