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G = C51⋊C8order 408 = 23·3·17

1st semidirect product of C51 and C8 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C511C8, D17.Dic3, C17⋊(C3⋊C8), C3⋊(C17⋊C8), C17⋊C4.S3, (C3×D17).1C4, (C3×C17⋊C4).1C2, SmallGroup(408,34)

Series: Derived Chief Lower central Upper central

C1C51 — C51⋊C8
C1C17C51C3×D17C3×C17⋊C4 — C51⋊C8
C51 — C51⋊C8
C1

Generators and relations for C51⋊C8
 G = < a,b | a51=b8=1, bab-1=a26 >

17C2
17C4
17C6
51C8
17C12
17C3⋊C8
3C17⋊C8

Character table of C51⋊C8

 class 1234A4B68A8B8C8D12A12B17A17B51A51B51C51D
 size 1172171734515151513434888888
ρ1111111111111111111    trivial
ρ2111111-1-1-1-111111111    linear of order 2
ρ3111-1-11i-i-ii-1-1111111    linear of order 4
ρ4111-1-11-iii-i-1-1111111    linear of order 4
ρ51-11i-i-1ζ83ζ8ζ85ζ87i-i111111    linear of order 8
ρ61-11-ii-1ζ8ζ83ζ87ζ85-ii111111    linear of order 8
ρ71-11i-i-1ζ87ζ85ζ8ζ83i-i111111    linear of order 8
ρ81-11-ii-1ζ85ζ87ζ83ζ8-ii111111    linear of order 8
ρ922-122-10000-1-122-1-1-1-1    orthogonal lifted from S3
ρ1022-1-2-2-100001122-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ112-2-1-2i2i10000i-i22-1-1-1-1    complex lifted from C3⋊C8
ρ122-2-12i-2i10000-ii22-1-1-1-1    complex lifted from C3⋊C8
ρ13808000000000-1-17/2-1+17/2-1-17/2-1+17/2-1-17/2-1+17/2    orthogonal lifted from C17⋊C8
ρ14808000000000-1+17/2-1-17/2-1+17/2-1-17/2-1+17/2-1-17/2    orthogonal lifted from C17⋊C8
ρ1580-4000000000-1-17/2-1+17/23ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ17317141712175173ζ32ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ171715179178172ζ3ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ1731711171017717632ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ171716171317417    complex faithful
ρ1680-4000000000-1+17/2-1-17/232ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ1717161713174173ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ17317141712175173ζ32ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ171715179178172ζ3ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ17317111710177176    complex faithful
ρ1780-4000000000-1+17/2-1-17/2ζ32ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ171715179178172ζ3ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ1731711171017717632ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ1717161713174173ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ17317141712175173    complex faithful
ρ1880-4000000000-1-17/2-1+17/2ζ3ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ1731711171017717632ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ1717161713174173ζ17143ζ17123ζ17113ζ17103ζ1773ζ1763ζ1753ζ17317141712175173ζ32ζ171632ζ171532ζ171332ζ17932ζ17832ζ17432ζ17232ζ171715179178172    complex faithful

Smallest permutation representation of C51⋊C8
On 51 points
Generators in S51
(1 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 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51)
(2 3 5 9 17 33 14 27)(4 7 13 25 49 46 40 28)(6 11 21 41 30 8 15 29)(10 19 37 22 43 34 16 31)(12 23 45 38 24 47 42 32)(18 35)(20 39 26 51 50 48 44 36)

G:=sub<Sym(51)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,3,5,9,17,33,14,27)(4,7,13,25,49,46,40,28)(6,11,21,41,30,8,15,29)(10,19,37,22,43,34,16,31)(12,23,45,38,24,47,42,32)(18,35)(20,39,26,51,50,48,44,36)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,3,5,9,17,33,14,27)(4,7,13,25,49,46,40,28)(6,11,21,41,30,8,15,29)(10,19,37,22,43,34,16,31)(12,23,45,38,24,47,42,32)(18,35)(20,39,26,51,50,48,44,36) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)], [(2,3,5,9,17,33,14,27),(4,7,13,25,49,46,40,28),(6,11,21,41,30,8,15,29),(10,19,37,22,43,34,16,31),(12,23,45,38,24,47,42,32),(18,35),(20,39,26,51,50,48,44,36)])

Matrix representation of C51⋊C8 in GL8(𝔽2)

10000010
00100001
01010011
10111011
00100011
00011001
00110101
10001001
,
10010001
01110100
00111011
01111100
01111110
01111011
00110111
00101010

G:=sub<GL(8,GF(2))| [1,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,1,1,1,0,0,0,0,1,1,1,1,1,1,1],[1,0,0,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0,1,1,1,1,1,0,1,0,0,1,1,0] >;

C51⋊C8 in GAP, Magma, Sage, TeX

C_{51}\rtimes C_8
% in TeX

G:=Group("C51:C8");
// GroupNames label

G:=SmallGroup(408,34);
// by ID

G=gap.SmallGroup(408,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,30,26,323,1204,3909,2414]);
// Polycyclic

G:=Group<a,b|a^51=b^8=1,b*a*b^-1=a^26>;
// generators/relations

Export

Subgroup lattice of C51⋊C8 in TeX
Character table of C51⋊C8 in TeX

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