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G = S3×C13⋊C6  order 468 = 22·32·13

Direct product of S3 and C13⋊C6

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

Aliases: S3×C13⋊C6, D39⋊C6, C39⋊(C2×C6), (S3×C13)⋊C6, D39⋊C3⋊C2, C13⋊1(S3×C6), D13⋊(C3×S3), (S3×D13)⋊C3, (C3×D13)⋊C6, C13⋊C3⋊1D6, (C3×C13⋊C6)⋊C2, (S3×C13⋊C3)⋊C2, C3⋊1(C2×C13⋊C6), (C3×C13⋊C3)⋊C22, SmallGroup(468,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C39 — S3×C13⋊C6
Chief series
C1 — C13 — C39 — C3×C13⋊C3 — C3×C13⋊C6 — S3×C13⋊C6
Lower central
C39 — S3×C13⋊C6
Upper central
C1

Generators and relations for S3×C13⋊C6
 G = < a,b,c,d | a3=b2=c13=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >

C1
3C2
13C2
39C2
C3
13C3
26C3
C13
39C22
S3
13C6
13C6
13S3
26C6
39C6
39C6
13C32
D13
3C26
3D13
C39
C13⋊C3
2C13⋊C3
13D6
39C2×C6
13C3×S3
13C3×S3
13C3×C6
3D26
S3×C13
C13⋊C6
C3×D13
D39
2C13⋊C6
3C2×C13⋊C3
3C13⋊C6
C3×C13⋊C3
13S3×C6
S3×D13
3C2×C13⋊C6
C3×C13⋊C6
D39⋊C3
S3×C13⋊C3
S3×C13⋊C6

Character table of S3×C13⋊C6

 class 12A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I13A13B26A26B39A39B
 size 1313392131326261313262626393939396618181212
ρ1111111111111111111111111    trivial
ρ211-1-111111-1-1-1-1-1-11-11111111    linear of order 2
ρ31-11-11111111111-1-1-1-111-1-111    linear of order 2
ρ41-1-1111111-1-1-1-1-11-11-111-1-111    linear of order 2
ρ511-1-11ζ3ζ32ζ32ζ3ζ6ζ65ζ6ζ65-1ζ65ζ32ζ6ζ3111111    linear of order 6
ρ61-1-111ζ3ζ32ζ32ζ3ζ6ζ65ζ6ζ65-1ζ3ζ6ζ32ζ6511-1-111    linear of order 6
ρ711111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ31ζ3ζ32ζ32ζ3111111    linear of order 3
ρ81-11-11ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ31ζ65ζ6ζ6ζ6511-1-111    linear of order 6
ρ911111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ321ζ32ζ3ζ3ζ32111111    linear of order 3
ρ101-1-111ζ32ζ3ζ3ζ32ζ65ζ6ζ65ζ6-1ζ32ζ65ζ3ζ611-1-111    linear of order 6
ρ1111-1-11ζ32ζ3ζ3ζ32ζ65ζ6ζ65ζ6-1ζ6ζ3ζ65ζ32111111    linear of order 6
ρ121-11-11ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ321ζ6ζ65ζ65ζ611-1-111    linear of order 6
ρ132020-122-1-122-1-1-100002200-1-1    orthogonal lifted from S3
ρ1420-20-122-1-1-2-211100002200-1-1    orthogonal lifted from D6
ρ152020-1-1-√-3-1+√-3ζ65ζ6-1+√-3-1-√-3ζ65ζ6-100002200-1-1    complex lifted from C3×S3
ρ1620-20-1-1+√-3-1-√-3ζ6ζ651+√-31-√-3ζ32ζ3100002200-1-1    complex lifted from S3×C6
ρ172020-1-1+√-3-1-√-3ζ6ζ65-1-√-3-1+√-3ζ6ζ65-100002200-1-1    complex lifted from C3×S3
ρ1820-20-1-1-√-3-1+√-3ζ65ζ61-√-31+√-3ζ3ζ32100002200-1-1    complex lifted from S3×C6
ρ196-60060000000000000-1-√13/2-1+√13/21+√13/21-√13/2-1+√13/2-1-√13/2    orthogonal lifted from C2×C13⋊C6
ρ206-60060000000000000-1+√13/2-1-√13/21-√13/21+√13/2-1-√13/2-1+√13/2    orthogonal lifted from C2×C13⋊C6
ρ21660060000000000000-1+√13/2-1-√13/2-1+√13/2-1-√13/2-1-√13/2-1+√13/2    orthogonal lifted from C13⋊C6
ρ22660060000000000000-1-√13/2-1+√13/2-1-√13/2-1+√13/2-1+√13/2-1-√13/2    orthogonal lifted from C13⋊C6
ρ2312000-60000000000000-1+√13-1-√13001+√13/21-√13/2    orthogonal faithful
ρ2412000-60000000000000-1-√13-1+√13001-√13/21+√13/2    orthogonal faithful

Smallest permutation representation of S3×C13⋊C6
►On 39 points
Generators in S39
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)

(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)

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

Matrix representation of S3×C13⋊C6 ►in GL8(𝔽79)

078000000
178000000
00100000
00010000
00001000
00000100
00000010
00000001
,
078000000
780000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00637764776378
00100000
00010000
00001000
00000100
00000010
,
240000000
024000000
00100000
00614677636062
0016181611717
00000010
00010000
0017171161816

G:=sub<GL(8,GF(79))| [0,1,0,0,0,0,0,0,78,78,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,63,1,0,0,0,0,0,0,77,0,1,0,0,0,0,0,64,0,0,1,0,0,0,0,77,0,0,0,1,0,0,0,63,0,0,0,0,1,0,0,78,0,0,0,0,0],[24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,1,61,16,0,0,17,0,0,0,46,18,0,1,17,0,0,0,77,16,0,0,1,0,0,0,63,1,0,0,16,0,0,0,60,17,1,0,18,0,0,0,62,17,0,0,16] >;
 

S3×C13⋊C6 in GAP, Magma, Sage, TeX 

S_3\times C_{13}\rtimes C_6
 
% in TeX
 
G:=Group("S3xC13:C6");
 
// GroupNames label
 
G:=SmallGroup(468,31);
 
// by ID
 
G=gap.SmallGroup(468,31);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-13,248,10804,2039]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^13=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^10>;
 
// generators/relations
 

Export

Subgroup lattice of S3×C13⋊C6 in TeX
Character table of S3×C13⋊C6 in TeX

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