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G = C3×C13⋊C6order 234 = 2·32·13

Direct product of C3 and C13⋊C6

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

Aliases: C3×C13⋊C6, C392C6, D13⋊C32, C13⋊C3⋊C6, C13⋊(C3×C6), (C3×D13)⋊C3, (C3×C13⋊C3)⋊2C2, SmallGroup(234,7)

Series: Derived Chief Lower central Upper central

C1C13 — C3×C13⋊C6
C1C13C39C3×C13⋊C3 — C3×C13⋊C6
C13 — C3×C13⋊C6
C1C3

Generators and relations for C3×C13⋊C6
 G = < a,b,c | a3=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >

13C2
13C3
13C3
13C3
13C6
13C6
13C6
13C6
13C32
13C3×C6

Character table of C3×C13⋊C6

 class 123A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H13A13B39A39B39C39D
 size 113111313131313131313131313131313666666
ρ1111111111111111111111111    trivial
ρ21-111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ31-1ζ3ζ32ζ31ζ321ζ32ζ3ζ6ζ6ζ65ζ6ζ65-1ζ65-111ζ3ζ32ζ3ζ32    linear of order 6
ρ41-1ζ32ζ3ζ3ζ3ζ32ζ3211-1ζ65ζ6ζ6-1ζ65ζ65ζ611ζ32ζ3ζ32ζ3    linear of order 6
ρ51-111ζ3ζ32ζ32ζ3ζ3ζ32ζ65-1-1ζ6ζ6ζ6ζ65ζ65111111    linear of order 6
ρ611ζ32ζ3ζ321ζ31ζ3ζ32ζ3ζ3ζ32ζ3ζ321ζ32111ζ32ζ3ζ32ζ3    linear of order 3
ρ71111ζ3ζ32ζ32ζ3ζ3ζ32ζ311ζ32ζ32ζ32ζ3ζ3111111    linear of order 3
ρ811ζ32ζ31ζ321ζ3ζ32ζ3ζ32ζ3ζ321ζ3ζ321ζ311ζ32ζ3ζ32ζ3    linear of order 3
ρ91-1ζ3ζ321ζ31ζ32ζ3ζ32ζ65ζ6ζ65-1ζ6ζ65-1ζ611ζ3ζ32ζ3ζ32    linear of order 6
ρ101-1ζ3ζ32ζ32ζ32ζ3ζ311-1ζ6ζ65ζ65-1ζ6ζ6ζ6511ζ3ζ32ζ3ζ32    linear of order 6
ρ111-111ζ32ζ3ζ3ζ32ζ32ζ3ζ6-1-1ζ65ζ65ζ65ζ6ζ6111111    linear of order 6
ρ121-1ζ32ζ31ζ321ζ3ζ32ζ3ζ6ζ65ζ6-1ζ65ζ6-1ζ6511ζ32ζ3ζ32ζ3    linear of order 6
ρ1311ζ3ζ32ζ32ζ32ζ3ζ3111ζ32ζ3ζ31ζ32ζ32ζ311ζ3ζ32ζ3ζ32    linear of order 3
ρ141-1ζ32ζ3ζ321ζ31ζ3ζ32ζ65ζ65ζ6ζ65ζ6-1ζ6-111ζ32ζ3ζ32ζ3    linear of order 6
ρ1511ζ32ζ3ζ3ζ3ζ32ζ32111ζ3ζ32ζ321ζ3ζ3ζ3211ζ32ζ3ζ32ζ3    linear of order 3
ρ1611ζ3ζ321ζ31ζ32ζ3ζ32ζ3ζ32ζ31ζ32ζ31ζ3211ζ3ζ32ζ3ζ32    linear of order 3
ρ1711ζ3ζ32ζ31ζ321ζ32ζ3ζ32ζ32ζ3ζ32ζ31ζ3111ζ3ζ32ζ3ζ32    linear of order 3
ρ181111ζ32ζ3ζ3ζ32ζ32ζ3ζ3211ζ3ζ3ζ3ζ32ζ32111111    linear of order 3
ρ19606600000000000000-1+13/2-1-13/2-1-13/2-1+13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ20606600000000000000-1-13/2-1+13/2-1+13/2-1-13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ2160-3-3-3-3+3-300000000000000-1+13/2-1-13/2ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132    complex faithful
ρ2260-3+3-3-3-3-300000000000000-1-13/2-1+13/2ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13    complex faithful
ρ2360-3+3-3-3-3-300000000000000-1+13/2-1-13/2ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132    complex faithful
ρ2460-3-3-3-3+3-300000000000000-1-13/2-1+13/2ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13    complex faithful

Smallest permutation representation of C3×C13⋊C6
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)
(1 27 14)(2 31 17 13 36 24)(3 35 20 12 32 21)(4 39 23 11 28 18)(5 30 26 10 37 15)(6 34 16 9 33 25)(7 38 19 8 29 22)

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

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

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

C3×C13⋊C6 is a maximal subgroup of   C3⋊F13

Matrix representation of C3×C13⋊C6 in GL6(𝔽79)

5500000
0550000
0055000
0005500
0000550
0000055
,
0000078
1000063
0100077
0010064
0001077
0000163
,
1368011660
6148042660
68440115555
48485555110
268066420
3724066110

G:=sub<GL(6,GF(79))| [55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,78,63,77,64,77,63],[13,61,68,48,2,37,68,48,44,48,68,24,0,0,0,55,0,0,11,42,11,55,66,66,66,66,55,11,42,11,0,0,55,0,0,0] >;

C3×C13⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_{13}\rtimes C_6
% in TeX

G:=Group("C3xC13:C6");
// GroupNames label

G:=SmallGroup(234,7);
// by ID

G=gap.SmallGroup(234,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,3459,439]);
// Polycyclic

G:=Group<a,b,c|a^3=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

Export

Subgroup lattice of C3×C13⋊C6 in TeX
Character table of C3×C13⋊C6 in TeX

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