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G = C33⋊C13  order 351 = 33·13

The semidirect product of C33 and C13 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C33⋊C13, SmallGroup(351,12)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C33⋊C13
Chief series
C1 — C33 — C33⋊C13
Lower central
C33 — C33⋊C13
Upper central
C1

Generators and relations for C33⋊C13
 G = < a,b,c,d | a3=b3=c3=d13=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=bc-1, dcd-1=a >

C1
13C3
27C13
13C32
C33
C33⋊C13

Character table of C33⋊C13

 class 13A3B13A13B13C13D13E13F13G13H13I13J13K13L
 size 11313272727272727272727272727
ρ1111111111111111    trivial
ρ2111ζ1312ζ132ζ133ζ134ζ135ζ136ζ137ζ138ζ139ζ1310ζ1311ζ13    linear of order 13
ρ3111ζ132ζ139ζ137ζ135ζ133ζ13ζ1312ζ1310ζ138ζ136ζ134ζ1311    linear of order 13
ρ4111ζ1311ζ134ζ136ζ138ζ1310ζ1312ζ13ζ133ζ135ζ137ζ139ζ132    linear of order 13
ρ5111ζ1310ζ136ζ139ζ1312ζ132ζ135ζ138ζ1311ζ13ζ134ζ137ζ133    linear of order 13
ρ6111ζ133ζ137ζ134ζ13ζ1311ζ138ζ135ζ132ζ1312ζ139ζ136ζ1310    linear of order 13
ρ7111ζ137ζ1312ζ135ζ1311ζ134ζ1310ζ133ζ139ζ132ζ138ζ13ζ136    linear of order 13
ρ8111ζ135ζ133ζ1311ζ136ζ13ζ139ζ134ζ1312ζ137ζ132ζ1310ζ138    linear of order 13
ρ9111ζ136ζ13ζ138ζ132ζ139ζ133ζ1310ζ134ζ1311ζ135ζ1312ζ137    linear of order 13
ρ10111ζ13ζ1311ζ1310ζ139ζ138ζ137ζ136ζ135ζ134ζ133ζ132ζ1312    linear of order 13
ρ11111ζ139ζ138ζ1312ζ133ζ137ζ1311ζ132ζ136ζ1310ζ13ζ135ζ134    linear of order 13
ρ12111ζ134ζ135ζ13ζ1310ζ136ζ132ζ1311ζ137ζ133ζ1312ζ138ζ139    linear of order 13
ρ13111ζ138ζ1310ζ132ζ137ζ1312ζ134ζ139ζ13ζ136ζ1311ζ133ζ135    linear of order 13
ρ1413-1+3√-3/2-1-3√-3/2000000000000    complex faithful
ρ1513-1-3√-3/2-1+3√-3/2000000000000    complex faithful

Permutation representations of C33⋊C13
►On 27 points: primitive - transitive group 27T134
Generators in S27
(1 19 8)(2 15 7)(3 20 5)(4 26 18)(6 22 12)(9 27 16)(10 13 14)(11 17 23)(21 25 24)

(1 2 26)(3 23 21)(4 8 7)(5 17 24)(6 16 13)(9 14 22)(10 12 27)(11 25 20)(15 18 19)

(1 20 9)(2 11 14)(3 16 8)(4 21 6)(5 27 19)(7 23 13)(10 15 17)(12 18 24)(22 26 25)

(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)
 
G:=sub<Sym(27)| (1,19,8)(2,15,7)(3,20,5)(4,26,18)(6,22,12)(9,27,16)(10,13,14)(11,17,23)(21,25,24), (1,2,26)(3,23,21)(4,8,7)(5,17,24)(6,16,13)(9,14,22)(10,12,27)(11,25,20)(15,18,19), (1,20,9)(2,11,14)(3,16,8)(4,21,6)(5,27,19)(7,23,13)(10,15,17)(12,18,24)(22,26,25), (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)>;
 
G:=Group( (1,19,8)(2,15,7)(3,20,5)(4,26,18)(6,22,12)(9,27,16)(10,13,14)(11,17,23)(21,25,24), (1,2,26)(3,23,21)(4,8,7)(5,17,24)(6,16,13)(9,14,22)(10,12,27)(11,25,20)(15,18,19), (1,20,9)(2,11,14)(3,16,8)(4,21,6)(5,27,19)(7,23,13)(10,15,17)(12,18,24)(22,26,25), (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) );
 
G=PermutationGroup([[(1,19,8),(2,15,7),(3,20,5),(4,26,18),(6,22,12),(9,27,16),(10,13,14),(11,17,23),(21,25,24)], [(1,2,26),(3,23,21),(4,8,7),(5,17,24),(6,16,13),(9,14,22),(10,12,27),(11,25,20),(15,18,19)], [(1,20,9),(2,11,14),(3,16,8),(4,21,6),(5,27,19),(7,23,13),(10,15,17),(12,18,24),(22,26,25)], [(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)]])
 
G:=TransitiveGroup(27,134);
 

Matrix representation of C33⋊C13 ►in GL13(𝔽79)

0000000010000
0000000001000
78550024235615624000
0000000000100
24230000237878552300
78787823024555655565500
0000000000010
0002423012424230230
0000000000001
05624237855472307878781
07801240565600232455
5610555500232401023
1000000000000
,
0001000000000
0000100000000
0000010000000
0000001000000
0000000100000
242424565656787800000
1000000000000
0100000000000
0000000000100
24230000237878552300
0000000000010
0000000010000
07801240565600232455
,
0100000000000
0010000000000
1000000000000
0000100000000
0000010000000
0001000000000
0000000100000
242424565656787800000
0000000001000
78550024235615624000
24230000237878552300
0002423012424230230
05624237855472307878781
,
1000000000000
0000000010000
0000000000001
0000010000000
78787823024555655565500
56056007812402324055
0000000100000
0002423012424230230
0000000000010
5610555500232401023
78550024235615624000
24230000237878552300
07801240565600232455

G:=sub<GL(13,GF(79))| [0,0,78,0,24,78,0,0,0,0,0,56,1,0,0,55,0,23,78,0,0,0,56,78,1,0,0,0,0,0,0,78,0,0,0,24,0,0,0,0,0,0,0,0,23,0,24,0,23,1,55,0,0,0,24,0,0,0,0,23,0,78,24,55,0,0,0,23,0,0,24,0,0,0,55,0,0,0,0,0,56,0,23,55,0,1,0,47,56,0,0,0,0,1,0,78,56,0,24,0,23,56,23,0,1,0,56,0,78,55,0,24,0,0,0,24,0,0,1,24,0,55,56,0,23,0,78,0,0,0,0,0,0,1,23,55,0,0,0,78,23,1,0,0,0,0,0,0,0,1,23,0,78,24,0,0,0,0,0,0,0,0,0,0,1,1,55,23,0],[0,0,0,0,0,24,1,0,0,24,0,0,0,0,0,0,0,0,24,0,1,0,23,0,0,78,0,0,0,0,0,24,0,0,0,0,0,0,0,1,0,0,0,0,56,0,0,0,0,0,0,1,0,1,0,0,0,56,0,0,0,0,0,0,24,0,0,1,0,0,56,0,0,0,0,0,0,0,0,0,0,1,0,78,0,0,0,23,0,0,56,0,0,0,0,1,78,0,0,0,78,0,0,56,0,0,0,0,0,0,0,0,0,78,0,1,0,0,0,0,0,0,0,0,0,0,55,0,0,0,0,0,0,0,0,0,0,0,1,23,0,0,23,0,0,0,0,0,0,0,0,0,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,55],[0,0,1,0,0,0,0,24,0,78,24,0,0,1,0,0,0,0,0,0,24,0,55,23,0,56,0,1,0,0,0,0,0,24,0,0,0,0,24,0,0,0,0,0,1,0,56,0,0,0,24,23,0,0,0,1,0,0,0,56,0,24,0,23,78,0,0,0,0,1,0,0,56,0,23,0,0,55,0,0,0,0,0,0,0,78,0,56,23,1,47,0,0,0,0,0,0,1,78,0,1,78,24,23,0,0,0,0,0,0,0,0,0,56,78,24,0,0,0,0,0,0,0,0,0,1,24,55,23,78,0,0,0,0,0,0,0,0,0,0,23,0,78,0,0,0,0,0,0,0,0,0,0,0,23,78,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,78,56,0,0,0,56,78,24,0,0,0,0,0,78,0,0,0,0,1,55,23,78,0,0,0,0,78,56,0,0,0,0,0,0,0,0,0,0,0,23,0,0,24,0,55,0,0,1,0,0,0,0,0,0,0,23,0,55,24,0,24,0,0,0,1,24,78,0,0,0,0,23,0,0,0,0,0,0,55,1,0,1,0,0,56,23,56,0,0,0,0,56,24,1,24,0,23,1,78,56,0,1,0,0,55,0,0,24,0,24,56,78,0,0,0,0,0,56,23,0,23,0,0,24,55,0,0,0,0,0,55,24,0,0,0,1,0,23,23,0,0,0,0,0,0,0,23,1,0,0,0,24,0,0,1,0,0,55,0,0,0,23,0,0,55] >;
 

C33⋊C13 in GAP, Magma, Sage, TeX 

C_3^3\rtimes C_{13}
 
% in TeX
 
G:=Group("C3^3:C13");
 
// GroupNames label
 
G:=SmallGroup(351,12);
 
// by ID
 
G=gap.SmallGroup(351,12);
 
# by ID
 
G:=PCGroup([4,-13,-3,3,3,937,1718,3539]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^13=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=a>;
 
// generators/relations
 

Export

Subgroup lattice of C33⋊C13 in TeX
Character table of C33⋊C13 in TeX

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