Copied to
clipboard

G = C3×C13⋊C4order 156 = 22·3·13

Direct product of C3 and C13⋊C4

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

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

Series: Derived Chief Lower central Upper central

C1C13 — C3×C13⋊C4
C1C13D13C3×D13 — C3×C13⋊C4
C13 — C3×C13⋊C4
C1C3

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

13C2
13C4
13C6
13C12

Character table of C3×C13⋊C4

 class 123A3B4A4B6A6B12A12B12C12D13A13B13C39A39B39C39D39E39F
 size 113111313131313131313444444444
ρ1111111111111111111111    trivial
ρ21111-1-111-1-1-1-1111111111    linear of order 2
ρ311ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ3111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ411ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ65ζ6111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 6
ρ511ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ6ζ65111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 6
ρ611ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ32111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ71-111-ii-1-1-ii-ii111111111    linear of order 4
ρ81-111i-i-1-1i-ii-i111111111    linear of order 4
ρ91-1ζ32ζ3-iiζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 12
ρ101-1ζ3ζ32-iiζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 12
ρ111-1ζ32ζ3i-iζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 12
ρ121-1ζ3ζ32i-iζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 12
ρ13404400000000ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513    orthogonal lifted from C13⋊C4
ρ14404400000000ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132    orthogonal lifted from C13⋊C4
ρ15404400000000ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134    orthogonal lifted from C13⋊C4
ρ1640-2-2-3-2+2-300000000ζ13111310133132ζ131213813513ζ139137136134ζ3ζ13113ζ13103ζ1333ζ132ζ3ζ1393ζ1373ζ1363ζ134ζ32ζ131132ζ131032ζ13332ζ132ζ32ζ13932ζ13732ζ13632ζ134ζ32ζ131232ζ13832ζ13532ζ13ζ3ζ13123ζ1383ζ1353ζ13    complex faithful
ρ1740-2-2-3-2+2-300000000ζ139137136134ζ13111310133132ζ131213813513ζ3ζ1393ζ1373ζ1363ζ134ζ3ζ13123ζ1383ζ1353ζ13ζ32ζ13932ζ13732ζ13632ζ134ζ32ζ131232ζ13832ζ13532ζ13ζ32ζ131132ζ131032ζ13332ζ132ζ3ζ13113ζ13103ζ1333ζ132    complex faithful
ρ1840-2+2-3-2-2-300000000ζ131213813513ζ139137136134ζ13111310133132ζ32ζ131232ζ13832ζ13532ζ13ζ32ζ131132ζ131032ζ13332ζ132ζ3ζ13123ζ1383ζ1353ζ13ζ3ζ13113ζ13103ζ1333ζ132ζ3ζ1393ζ1373ζ1363ζ134ζ32ζ13932ζ13732ζ13632ζ134    complex faithful
ρ1940-2+2-3-2-2-300000000ζ139137136134ζ13111310133132ζ131213813513ζ32ζ13932ζ13732ζ13632ζ134ζ32ζ131232ζ13832ζ13532ζ13ζ3ζ1393ζ1373ζ1363ζ134ζ3ζ13123ζ1383ζ1353ζ13ζ3ζ13113ζ13103ζ1333ζ132ζ32ζ131132ζ131032ζ13332ζ132    complex faithful
ρ2040-2+2-3-2-2-300000000ζ13111310133132ζ131213813513ζ139137136134ζ32ζ131132ζ131032ζ13332ζ132ζ32ζ13932ζ13732ζ13632ζ134ζ3ζ13113ζ13103ζ1333ζ132ζ3ζ1393ζ1373ζ1363ζ134ζ3ζ13123ζ1383ζ1353ζ13ζ32ζ131232ζ13832ζ13532ζ13    complex faithful
ρ2140-2-2-3-2+2-300000000ζ131213813513ζ139137136134ζ13111310133132ζ3ζ13123ζ1383ζ1353ζ13ζ3ζ13113ζ13103ζ1333ζ132ζ32ζ131232ζ13832ζ13532ζ13ζ32ζ131132ζ131032ζ13332ζ132ζ32ζ13932ζ13732ζ13632ζ134ζ3ζ1393ζ1373ζ1363ζ134    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

144000
014400
001440
000144
,
103122103156
1000
0100
0010
,
1000
548910955
1224867102
0010
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

Export

Subgroup lattice of C3×C13⋊C4 in TeX
Character table of C3×C13⋊C4 in TeX

׿
×
𝔽