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## G = C3×C13⋊C3order 117 = 32·13

### Direct product of C3 and C13⋊C3

Aliases: C3×C13⋊C3, C39⋊C3, C13⋊C32, SmallGroup(117,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C3×C13⋊C3
 Chief series C1 — C13 — C13⋊C3 — C3×C13⋊C3
 Lower central C13 — C3×C13⋊C3
 Upper central C1 — C3

Generators and relations for C3×C13⋊C3
G = < a,b,c | a3=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Character table of C3×C13⋊C3

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 13A 13B 13C 13D 39A 39B 39C 39D 39E 39F 39G 39H size 1 1 1 13 13 13 13 13 13 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ3 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ8 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ9 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ10 3 3 3 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ11 3 3 3 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ12 3 3 3 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ13 3 3 3 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ14 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 complex faithful ρ15 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 complex faithful ρ16 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 complex faithful ρ17 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 complex faithful ρ18 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 complex faithful ρ19 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 complex faithful ρ20 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 complex faithful ρ21 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ32ζ139+ζ32ζ133+ζ32ζ13 ζ32ζ136+ζ32ζ135+ζ32ζ132 ζ3ζ1311+ζ3ζ138+ζ3ζ137 ζ3ζ139+ζ3ζ133+ζ3ζ13 ζ3ζ136+ζ3ζ135+ζ3ζ132 ζ3ζ1312+ζ3ζ1310+ζ3ζ134 ζ32ζ1312+ζ32ζ1310+ζ32ζ134 ζ32ζ1311+ζ32ζ138+ζ32ζ137 complex faithful

Smallest permutation representation of C3×C13⋊C3
On 39 points
Generators in S39
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)

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

G:=Group( (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38) );

G=PermutationGroup([[(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38)]])

C3×C13⋊C3 is a maximal subgroup of   D39⋊C3  C117⋊C3  C1173C3  C13⋊He3
C3×C13⋊C3 is a maximal quotient of   C117⋊C3  C1173C3  C39.C32  C13⋊He3

Matrix representation of C3×C13⋊C3 in GL3(𝔽79) generated by

 23 0 0 0 23 0 0 0 23
,
 69 54 1 1 0 0 0 1 0
,
 1 0 0 24 68 54 3 55 10
G:=sub<GL(3,GF(79))| [23,0,0,0,23,0,0,0,23],[69,1,0,54,0,1,1,0,0],[1,24,3,0,68,55,0,54,10] >;

C3×C13⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_{13}\rtimes C_3
% in TeX

G:=Group("C3xC13:C3");
// GroupNames label

G:=SmallGroup(117,3);
// by ID

G=gap.SmallGroup(117,3);
# by ID

G:=PCGroup([3,-3,-3,-13,245]);
// Polycyclic

G:=Group<a,b,c|a^3=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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