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G = C3×F13  order 468 = 22·32·13

Direct product of C3 and F13

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

Aliases: C3×F13, C39⋊2C12, C13⋊C3⋊C12, C13⋊(C3×C12), C13⋊C6.C6, C13⋊C4⋊C32, D13.(C3×C6), (C3×D13).3C6, (C3×C13⋊C4)⋊C3, (C3×C13⋊C3)⋊2C4, (C3×C13⋊C6).2C2, SmallGroup(468,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C3×F13
Chief series
C1 — C13 — D13 — C3×D13 — C3×C13⋊C6 — C3×F13
Lower central
C13 — C3×F13
Upper central
C1 — C3

Generators and relations for C3×F13
 G = < a,b,c | a3=b13=c12=1, ab=ba, ac=ca, cbc-1=b6 >

C1
13C2
C3
13C3
13C3
13C3
C13
13C4
13C6
13C6
13C6
13C6
13C32
D13
C13⋊C3
C39
C13⋊C3
C13⋊C3
13C12
13C12
13C12
13C12
13C3×C6
C13⋊C4
C13⋊C6
C13⋊C6
C13⋊C6
C3×D13
C3×C13⋊C3
13C3×C12
F13
C3×C13⋊C4
F13
F13
C3×C13⋊C6
C3×F13

Smallest permutation representation of C3×F13
►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 12 5 6 4 8 13 3 10 9 11 7)(15 25 18 19 17 21 26 16 23 22 24 20)(28 38 31 32 30 34 39 29 36 35 37 33)
 
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), (2,12,5,6,4,8,13,3,10,9,11,7)(15,25,18,19,17,21,26,16,23,22,24,20)(28,38,31,32,30,34,39,29,36,35,37,33)>;
 
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), (2,12,5,6,4,8,13,3,10,9,11,7)(15,25,18,19,17,21,26,16,23,22,24,20)(28,38,31,32,30,34,39,29,36,35,37,33) );
 
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)], [(2,12,5,6,4,8,13,3,10,9,11,7),(15,25,18,19,17,21,26,16,23,22,24,20),(28,38,31,32,30,34,39,29,36,35,37,33)]])
 

39 conjugacy classes

class 1  2 3A3B3C···3H4A4B6A···6H12A···12P 13 39A39B
order12333···3446···612···12133939
size1131113···13131313···1313···13121212

39 irreducible representations

dim1111111111212
type+++
imageC1C2C3C3C4C6C6C12C12F13C3×F13
kernelC3×F13C3×C13⋊C6F13C3×C13⋊C4C3×C13⋊C3C13⋊C6C3×D13C13⋊C3C39C3C1
# reps116226212412

Matrix representation of C3×F13 ►in GL12(𝔽157)

14400000000000
01440000000000
00144000000000
00014400000000
00001440000000
00000144000000
00000014400000
00000001440000
00000000144000
00000000014400
00000000001440
00000000000144
,
00000000000156
10000000000156
01000000000156
00100000000156
00010000000156
00001000000156
00000100000156
00000010000156
00000001000156
00000000100156
00000000010156
00000000001156
,
000001000000
000000000001
000010000000
000000000010
000100000000
000000000100
001000000000
000000001000
010000000000
000000010000
100000000000
000000100000

G:=sub<GL(12,GF(157))| [144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,0,0,0,144],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,156,156,156,156,156,156,156,156,156,156,156,156],[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0] >;
 

C3×F13 in GAP, Magma, Sage, TeX 

C_3\times F_{13}
 
% in TeX
 
G:=Group("C3xF13");
 
// GroupNames label
 
G:=SmallGroup(468,29);
 
// by ID
 
G=gap.SmallGroup(468,29);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-2,-13,90,7204,1359,619]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^13=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C3×F13 in TeX

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