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G = C3×C13⋊C3order 117 = 32·13

Direct product of C3 and C13⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C13 — C3×C13⋊C3
C1C13C13⋊C3 — C3×C13⋊C3
C13 — C3×C13⋊C3
C1C3

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

13C3
13C3
13C3
13C32

Character table of C3×C13⋊C3

 class 13A3B3C3D3E3F3G3H13A13B13C13D39A39B39C39D39E39F39G39H
 size 111131313131313333333333333
ρ1111111111111111111111    trivial
ρ21ζ32ζ3ζ3ζ3ζ321ζ3211111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ31ζ3ζ321ζ31ζ32ζ32ζ31111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ41ζ32ζ3ζ321ζ3ζ321ζ31111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ51ζ3ζ32ζ31ζ32ζ31ζ321111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ61ζ3ζ32ζ32ζ32ζ31ζ311111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ71ζ32ζ31ζ321ζ3ζ3ζ321111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ8111ζ3ζ32ζ32ζ32ζ3ζ3111111111111    linear of order 3
ρ9111ζ32ζ3ζ3ζ3ζ32ζ32111111111111    linear of order 3
ρ10333000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ13913313ζ136135132ζ136135132ζ13121310134    complex lifted from C13⋊C3
ρ11333000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13913313ζ136135132ζ13121310134ζ1311138137ζ1311138137ζ13913313    complex lifted from C13⋊C3
ρ12333000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ13913313ζ136135132ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13121310134ζ1311138137    complex lifted from C13⋊C3
ρ13333000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13913313ζ136135132    complex lifted from C13⋊C3
ρ143-3-3-3/2-3+3-3/2000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13    complex faithful
ρ153-3-3-3/2-3+3-3/2000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132    complex faithful
ρ163-3-3-3/2-3+3-3/2000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134    complex faithful
ρ173-3+3-3/2-3-3-3/2000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132    complex faithful
ρ183-3+3-3/2-3-3-3/2000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13    complex faithful
ρ193-3+3-3/2-3-3-3/2000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137    complex faithful
ρ203-3+3-3/2-3-3-3/2000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134    complex faithful
ρ213-3-3-3/2-3+3-3/2000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137    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

2300
0230
0023
,
69541
100
010
,
100
246854
35510
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

Export

Subgroup lattice of C3×C13⋊C3 in TeX
Character table of C3×C13⋊C3 in TeX

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