Copied to
clipboard

G = C53⋊C4order 212 = 22·53

The semidirect product of C53 and C4 acting faithfully

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

Aliases: C53⋊C4, D53.C2, SmallGroup(212,3)

Series: Derived Chief Lower central Upper central

C1C53 — C53⋊C4
C1C53D53 — C53⋊C4
C53 — C53⋊C4
C1

Generators and relations for C53⋊C4
 G = < a,b | a53=b4=1, bab-1=a23 >

53C2
53C4

Character table of C53⋊C4

 class 124A4B53A53B53C53D53E53F53G53H53I53J53K53L53M
 size 15353534444444444444
ρ111111111111111111    trivial
ρ211-1-11111111111111    linear of order 2
ρ31-1i-i1111111111111    linear of order 4
ρ41-1-ii1111111111111    linear of order 4
ρ54000ζ5336533353205317ζ534553285325538ζ53525330532353ζ5340533453195313ζ535053375316533ζ53515346537532ζ5342534153125311ζ5331532953245322ζ534753325321536ζ534953395314534ζ53485344539535ζ5343533553185310ζ5338532753265315    orthogonal faithful
ρ64000ζ5331532953245322ζ53525330532353ζ5336533353205317ζ53485344539535ζ53515346537532ζ5340533453195313ζ534553285325538ζ535053375316533ζ534953395314534ζ5338532753265315ζ534753325321536ζ5342534153125311ζ5343533553185310    orthogonal faithful
ρ74000ζ53515346537532ζ5331532953245322ζ535053375316533ζ534953395314534ζ53485344539535ζ534753325321536ζ5336533353205317ζ5340533453195313ζ5343533553185310ζ5342534153125311ζ5338532753265315ζ53525330532353ζ534553285325538    orthogonal faithful
ρ84000ζ5338532753265315ζ534753325321536ζ534953395314534ζ53525330532353ζ5342534153125311ζ534553285325538ζ53485344539535ζ5343533553185310ζ5331532953245322ζ535053375316533ζ5336533353205317ζ5340533453195313ζ53515346537532    orthogonal faithful
ρ94000ζ53485344539535ζ53515346537532ζ5340533453195313ζ5343533553185310ζ534953395314534ζ5338532753265315ζ535053375316533ζ534753325321536ζ534553285325538ζ53525330532353ζ5342534153125311ζ5331532953245322ζ5336533353205317    orthogonal faithful
ρ104000ζ5342534153125311ζ5338532753265315ζ5343533553185310ζ5331532953245322ζ53525330532353ζ5336533353205317ζ534953395314534ζ534553285325538ζ53515346537532ζ5340533453195313ζ535053375316533ζ534753325321536ζ53485344539535    orthogonal faithful
ρ114000ζ5340533453195313ζ535053375316533ζ53515346537532ζ5338532753265315ζ534753325321536ζ534953395314534ζ5331532953245322ζ53485344539535ζ5342534153125311ζ534553285325538ζ5343533553185310ζ5336533353205317ζ53525330532353    orthogonal faithful
ρ124000ζ534953395314534ζ53485344539535ζ534753325321536ζ534553285325538ζ5343533553185310ζ5342534153125311ζ5340533453195313ζ5338532753265315ζ5336533353205317ζ5331532953245322ζ53525330532353ζ53515346537532ζ535053375316533    orthogonal faithful
ρ134000ζ53525330532353ζ5342534153125311ζ534553285325538ζ53515346537532ζ5331532953245322ζ535053375316533ζ5343533553185310ζ5336533353205317ζ53485344539535ζ534753325321536ζ5340533453195313ζ5338532753265315ζ534953395314534    orthogonal faithful
ρ144000ζ534753325321536ζ5340533453195313ζ53485344539535ζ5342534153125311ζ5338532753265315ζ5343533553185310ζ53515346537532ζ534953395314534ζ53525330532353ζ5336533353205317ζ534553285325538ζ535053375316533ζ5331532953245322    orthogonal faithful
ρ154000ζ534553285325538ζ5343533553185310ζ5342534153125311ζ535053375316533ζ5336533353205317ζ5331532953245322ζ5338532753265315ζ53525330532353ζ5340533453195313ζ53485344539535ζ53515346537532ζ534953395314534ζ534753325321536    orthogonal faithful
ρ164000ζ535053375316533ζ5336533353205317ζ5331532953245322ζ534753325321536ζ5340533453195313ζ53485344539535ζ53525330532353ζ53515346537532ζ5338532753265315ζ5343533553185310ζ534953395314534ζ534553285325538ζ5342534153125311    orthogonal faithful
ρ174000ζ5343533553185310ζ534953395314534ζ5338532753265315ζ5336533353205317ζ534553285325538ζ53525330532353ζ534753325321536ζ5342534153125311ζ535053375316533ζ53515346537532ζ5331532953245322ζ53485344539535ζ5340533453195313    orthogonal faithful

Smallest permutation representation of C53⋊C4
On 53 points: primitive
Generators in S53
(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 52 53)
(2 31 53 24)(3 8 52 47)(4 38 51 17)(5 15 50 40)(6 45 49 10)(7 22 48 33)(9 29 46 26)(11 36 44 19)(12 13 43 42)(14 20 41 35)(16 27 39 28)(18 34 37 21)(23 25 32 30)

G:=sub<Sym(53)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53), (2,31,53,24)(3,8,52,47)(4,38,51,17)(5,15,50,40)(6,45,49,10)(7,22,48,33)(9,29,46,26)(11,36,44,19)(12,13,43,42)(14,20,41,35)(16,27,39,28)(18,34,37,21)(23,25,32,30)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53), (2,31,53,24)(3,8,52,47)(4,38,51,17)(5,15,50,40)(6,45,49,10)(7,22,48,33)(9,29,46,26)(11,36,44,19)(12,13,43,42)(14,20,41,35)(16,27,39,28)(18,34,37,21)(23,25,32,30) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)], [(2,31,53,24),(3,8,52,47),(4,38,51,17),(5,15,50,40),(6,45,49,10),(7,22,48,33),(9,29,46,26),(11,36,44,19),(12,13,43,42),(14,20,41,35),(16,27,39,28),(18,34,37,21),(23,25,32,30)])

C53⋊C4 is a maximal quotient of   C53⋊C8

Matrix representation of C53⋊C4 in GL4(𝔽1061) generated by

0100
0010
0001
1060435297435
,
1000
56051194184
733735831466
232156545779
G:=sub<GL(4,GF(1061))| [0,0,0,1060,1,0,0,435,0,1,0,297,0,0,1,435],[1,560,733,232,0,511,735,156,0,94,831,545,0,184,466,779] >;

C53⋊C4 in GAP, Magma, Sage, TeX

C_{53}\rtimes C_4
% in TeX

G:=Group("C53:C4");
// GroupNames label

G:=SmallGroup(212,3);
// by ID

G=gap.SmallGroup(212,3);
# by ID

G:=PCGroup([3,-2,-2,-53,6,1082,941]);
// Polycyclic

G:=Group<a,b|a^53=b^4=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C53⋊C4 in TeX
Character table of C53⋊C4 in TeX

׿
×
𝔽