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## G = C3×C13⋊C4order 156 = 22·3·13

### Direct product of C3 and C13⋊C4

Aliases: C3×C13⋊C4, C392C4, C133C12, D13.2C6, (C3×D13).2C2, SmallGroup(156,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C3×C13⋊C4
 Chief series C1 — C13 — D13 — C3×D13 — C3×C13⋊C4
 Lower central C13 — C3×C13⋊C4
 Upper central C1 — C3

Generators and relations for C3×C13⋊C4
G = < a,b,c | a3=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C3×C13⋊C4

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 13A 13B 13C 39A 39B 39C 39D 39E 39F size 1 13 1 1 13 13 13 13 13 13 13 13 4 4 4 4 4 4 4 4 4 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 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 6 ρ5 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 -1 1 1 -i i -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 1 1 i -i -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 1 -1 ζ32 ζ3 -i i ζ6 ζ65 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i ζ65 ζ6 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 12 ρ11 1 -1 ζ32 ζ3 i -i ζ6 ζ65 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i ζ65 ζ6 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 12 ρ13 4 0 4 4 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ14 4 0 4 4 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ15 4 0 4 4 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ16 4 0 -2-2√-3 -2+2√-3 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 complex faithful ρ17 4 0 -2-2√-3 -2+2√-3 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 complex faithful ρ18 4 0 -2+2√-3 -2-2√-3 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 complex faithful ρ19 4 0 -2+2√-3 -2-2√-3 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 complex faithful ρ20 4 0 -2+2√-3 -2-2√-3 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 complex faithful ρ21 4 0 -2-2√-3 -2+2√-3 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ3ζ1312+ζ3ζ138+ζ3ζ135+ζ3ζ13 ζ3ζ1311+ζ3ζ1310+ζ3ζ133+ζ3ζ132 ζ32ζ1312+ζ32ζ138+ζ32ζ135+ζ32ζ13 ζ32ζ1311+ζ32ζ1310+ζ32ζ133+ζ32ζ132 ζ32ζ139+ζ32ζ137+ζ32ζ136+ζ32ζ134 ζ3ζ139+ζ3ζ137+ζ3ζ136+ζ3ζ134 complex faithful

Smallest permutation representation of C3×C13⋊C4
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 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)

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

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

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

C3×C13⋊C4 is a maximal subgroup of   C13⋊C36

Matrix representation of C3×C13⋊C4 in GL4(𝔽157) generated by

 144 0 0 0 0 144 0 0 0 0 144 0 0 0 0 144
,
 103 122 103 156 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 54 89 109 55 122 48 67 102 0 0 1 0
G:=sub<GL(4,GF(157))| [144,0,0,0,0,144,0,0,0,0,144,0,0,0,0,144],[103,1,0,0,122,0,1,0,103,0,0,1,156,0,0,0],[1,54,122,0,0,89,48,0,0,109,67,1,0,55,102,0] >;

C3×C13⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{13}\rtimes C_4
% in TeX

G:=Group("C3xC13:C4");
// GroupNames label

G:=SmallGroup(156,9);
// by ID

G=gap.SmallGroup(156,9);
# by ID

G:=PCGroup([4,-2,-3,-2,-13,24,1539,395]);
// Polycyclic

G:=Group<a,b,c|a^3=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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