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G = C49⋊C6order 294 = 2·3·72

The semidirect product of C49 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C49⋊C6, D49⋊C3, C7.F7, C49⋊C3⋊C2, SmallGroup(294,1)

Series: Derived Chief Lower central Upper central

C1C49 — C49⋊C6
C1C7C49C49⋊C3 — C49⋊C6
C49 — C49⋊C6
C1

Generators and relations for C49⋊C6
 G = < a,b | a49=b6=1, bab-1=a31 >

49C2
49C3
49C6
7D7
7C7⋊C3
7F7

Character table of C49⋊C6

 class 123A3B6A6B749A49B49C49D49E49F49G
 size 1494949494966666666
ρ111111111111111    trivial
ρ21-111-1-111111111    linear of order 2
ρ311ζ32ζ3ζ32ζ311111111    linear of order 3
ρ411ζ3ζ32ζ3ζ3211111111    linear of order 3
ρ51-1ζ3ζ32ζ65ζ611111111    linear of order 6
ρ61-1ζ32ζ3ζ6ζ6511111111    linear of order 6
ρ76000006-1-1-1-1-1-1-1    orthogonal lifted from F7
ρ8600000-1ζ4948493149304919491849ζ49474938493649134911492ζ49434939493349164910496ζ493749324929492049174912ζ49454927492649234922494ζ49404934492549244915499ζ494649444941498495493    orthogonal faithful
ρ9600000-1ζ49454927492649234922494ζ494649444941498495493ζ49404934492549244915499ζ4948493149304919491849ζ49434939493349164910496ζ49474938493649134911492ζ493749324929492049174912    orthogonal faithful
ρ10600000-1ζ49434939493349164910496ζ493749324929492049174912ζ49474938493649134911492ζ49454927492649234922494ζ49404934492549244915499ζ494649444941498495493ζ4948493149304919491849    orthogonal faithful
ρ11600000-1ζ493749324929492049174912ζ49404934492549244915499ζ49454927492649234922494ζ494649444941498495493ζ4948493149304919491849ζ49434939493349164910496ζ49474938493649134911492    orthogonal faithful
ρ12600000-1ζ494649444941498495493ζ49434939493349164910496ζ4948493149304919491849ζ49474938493649134911492ζ493749324929492049174912ζ49454927492649234922494ζ49404934492549244915499    orthogonal faithful
ρ13600000-1ζ49404934492549244915499ζ4948493149304919491849ζ494649444941498495493ζ49434939493349164910496ζ49474938493649134911492ζ493749324929492049174912ζ49454927492649234922494    orthogonal faithful
ρ14600000-1ζ49474938493649134911492ζ49454927492649234922494ζ493749324929492049174912ζ49404934492549244915499ζ494649444941498495493ζ4948493149304919491849ζ49434939493349164910496    orthogonal faithful

Smallest permutation representation of C49⋊C6
On 49 points
Generators in S49
(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)
(2 20 19 49 31 32)(3 39 37 48 12 14)(4 9 6 47 42 45)(5 28 24 46 23 27)(7 17 11 44 34 40)(8 36 29 43 15 22)(10 25 16 41 26 35)(13 33 21 38 18 30)

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

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

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

Matrix representation of C49⋊C6 in GL6(𝔽883)

650876836306797814
697196222375866
17867367939392
491508577344570530
3538448614769740
8433138048217657
,
39183685492821413
329804391782344476
44557710143022492
4756245315147554
13253986846047438
47086142355579408

G:=sub<GL(6,GF(883))| [650,69,17,491,353,843,876,719,86,508,844,313,836,62,736,577,861,804,306,22,79,344,47,821,797,375,39,570,697,7,814,866,392,530,40,657],[391,329,445,475,132,470,836,804,577,62,539,861,85,391,101,453,868,423,492,782,430,15,460,555,821,344,22,147,47,79,413,476,492,554,438,408] >;

C49⋊C6 in GAP, Magma, Sage, TeX

C_{49}\rtimes C_6
% in TeX

G:=Group("C49:C6");
// GroupNames label

G:=SmallGroup(294,1);
// by ID

G=gap.SmallGroup(294,1);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,938,1410,514,4035,1351]);
// Polycyclic

G:=Group<a,b|a^49=b^6=1,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C49⋊C6 in TeX
Character table of C49⋊C6 in TeX

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