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G = C338SD16order 432 = 24·33

4th semidirect product of C33 and SD16 acting via SD16/C2=D4

non-abelian, soluble, monomial

Aliases: C338SD16, C6.10S3≀C2, C322C82S3, (C3×C6).11D12, C335Q81C2, C3⋊Dic3.15D6, (C32×C6).16D4, C339D4.1C2, C324(C24⋊C2), C2.4(C322D12), C31(C322SD16), (C3×C322C8)⋊2C2, (C3×C3⋊Dic3).2C22, SmallGroup(432,589)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊Dic3 — C338SD16
C1C3C33C32×C6C3×C3⋊Dic3C339D4 — C338SD16
C33C32×C6C3×C3⋊Dic3 — C338SD16
C1C2

Generators and relations for C338SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=b, bc=cb, dbd-1=a-1, ebe=a, cd=dc, ece=c-1, ede=d3 >

Subgroups: 640 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3 [×4], C6, C6 [×5], C8, D4, Q8, C32, C32 [×4], Dic3 [×5], C12 [×2], D6 [×3], C2×C6, SD16, C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×4], C24, Dic6 [×2], D12, C3⋊D4, C33, C3×Dic3 [×5], C3⋊Dic3, C3⋊Dic3, S3×C6 [×3], C2×C3⋊S3, C24⋊C2, C3×C3⋊S3, C32×C6, C322C8, D6⋊S3, C3⋊D12, C322Q8 [×2], C3×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C322SD16, C3×C322C8, C339D4, C335Q8, C338SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, D12, C24⋊C2, S3≀C2, C322SD16, C322D12, C338SD16

Character table of C338SD16

 class 12A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A12B12C12D24A24B24C24D
 size 113624488183624488363618181818363618181818
ρ1111111111111111111111111111    trivial
ρ211-1111111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ311-1111111-111111-1-11111-1-11111    linear of order 2
ρ4111111111-11111111-1-111-1-1-1-1-1-1    linear of order 2
ρ5220-122-1-120-122-1-100-2-2-1-1001111    orthogonal lifted from D6
ρ622022222-20222220000-2-2000000    orthogonal lifted from D4
ρ7220-122-1-120-122-1-10022-1-100-1-1-1-1    orthogonal lifted from S3
ρ8220-122-1-1-20-122-1-100001100-3-333    orthogonal lifted from D12
ρ9220-122-1-1-20-122-1-10000110033-3-3    orthogonal lifted from D12
ρ102-202222200-2-2-2-2-200-2--20000-2--2--2-2    complex lifted from SD16
ρ112-202222200-2-2-2-2-200--2-20000--2-2-2--2    complex lifted from SD16
ρ122-20-122-1-1001-2-21100--2-23-300ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32    complex lifted from C24⋊C2
ρ132-20-122-1-1001-2-21100-2--23-300ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ142-20-122-1-1001-2-21100-2--2-3300ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3    complex lifted from C24⋊C2
ρ152-20-122-1-1001-2-21100--2-2-3300ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ1644-24-21-21004-21-21110000000000    orthogonal lifted from S3≀C2
ρ174424-21-21004-21-21-1-10000000000    orthogonal lifted from S3≀C2
ρ1844041-21-20-241-21-2000000110000    orthogonal lifted from S3≀C2
ρ1944041-21-20241-21-2000000-1-10000    orthogonal lifted from S3≀C2
ρ204-4041-21-200-4-12-120000003-30000    symplectic lifted from C322SD16, Schur index 2
ρ214-4041-21-200-4-12-12000000-330000    symplectic lifted from C322SD16, Schur index 2
ρ224-404-21-2100-42-12-1-3--30000000000    complex lifted from C322SD16
ρ234-404-21-2100-42-12-1--3-30000000000    complex lifted from C322SD16
ρ248-80-4-422-10044-2-21000000000000    orthogonal faithful
ρ25880-4-422-100-4-422-1000000000000    orthogonal lifted from C322D12
ρ26880-42-4-1200-42-4-12000000000000    orthogonal lifted from C322D12
ρ278-80-42-4-12004-241-2000000000000    symplectic faithful, Schur index 2

Permutation representations of C338SD16
On 24 points - transitive group 24T1307
Generators in S24
(1 13 21)(2 22 14)(3 23 15)(4 16 24)(5 9 17)(6 18 10)(7 19 11)(8 12 20)
(1 21 13)(2 22 14)(3 15 23)(4 16 24)(5 17 9)(6 18 10)(7 11 19)(8 12 20)
(1 21 13)(2 22 14)(3 23 15)(4 24 16)(5 17 9)(6 18 10)(7 19 11)(8 20 12)
(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 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)

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

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

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

G:=TransitiveGroup(24,1307);

Matrix representation of C338SD16 in GL6(𝔽73)

100000
010000
0000072
0000720
0001720
0010072
,
100000
010000
0000072
0007210
0007200
0010072
,
0720000
1720000
001000
000100
000010
000001
,
36620000
11250000
000010
001000
000001
000100
,
0720000
7200000
001000
000010
000100
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72,72,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,11,0,0,0,0,62,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C338SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("C3^3:8SD16");
// GroupNames label

G:=SmallGroup(432,589);
// by ID

G=gap.SmallGroup(432,589);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,36,254,58,1684,1691,298,677,348,1027,14118]);
// 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=e*a*e=b,b*c=c*b,d*b*d^-1=a^-1,e*b*e=a,c*d=d*c,e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

Export

Character table of C338SD16 in TeX

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