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G = S3×C13⋊C3order 234 = 2·32·13

Direct product of S3 and C13⋊C3

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

Aliases: S3×C13⋊C3, C393C6, (S3×C13)⋊C3, C132(C3×S3), C3⋊(C2×C13⋊C3), (C3×C13⋊C3)⋊3C2, SmallGroup(234,8)

Series: Derived Chief Lower central Upper central

C1C39 — S3×C13⋊C3
C1C13C39C3×C13⋊C3 — S3×C13⋊C3
C39 — S3×C13⋊C3
C1

Generators and relations for S3×C13⋊C3
 G = < a,b,c,d | a3=b2=c13=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

3C2
13C3
26C3
39C6
13C32
3C26
2C13⋊C3
13C3×S3
3C2×C13⋊C3

Character table of S3×C13⋊C3

 class 123A3B3C3D3E6A6B13A13B13C13D26A26B26C26D39A39B39C39D
 size 132131326263939333399996666
ρ1111111111111111111111    trivial
ρ21-111111-1-11111-1-1-1-11111    linear of order 2
ρ31-11ζ3ζ32ζ32ζ3ζ6ζ651111-1-1-1-11111    linear of order 6
ρ41-11ζ32ζ3ζ3ζ32ζ65ζ61111-1-1-1-11111    linear of order 6
ρ5111ζ3ζ32ζ32ζ3ζ32ζ3111111111111    linear of order 3
ρ6111ζ32ζ3ζ3ζ32ζ3ζ32111111111111    linear of order 3
ρ720-122-1-10022220000-1-1-1-1    orthogonal lifted from S3
ρ820-1-1--3-1+-3ζ65ζ60022220000-1-1-1-1    complex lifted from C3×S3
ρ920-1-1+-3-1--3ζ6ζ650022220000-1-1-1-1    complex lifted from C3×S3
ρ103-33000000ζ13913313ζ136135132ζ13121310134ζ131113813713913313131213101341311138137136135132ζ13913313ζ1311138137ζ13121310134ζ136135132    complex lifted from C2×C13⋊C3
ρ11333000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ136135132ζ1311138137ζ13913313ζ13121310134ζ136135132ζ13913313ζ1311138137ζ13121310134    complex lifted from C13⋊C3
ρ12333000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13913313ζ136135132ζ1311138137ζ13121310134ζ136135132ζ13913313ζ1311138137    complex lifted from C13⋊C3
ρ13333000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ13913313ζ1311138137ζ13121310134ζ136135132ζ13913313    complex lifted from C13⋊C3
ρ143-33000000ζ136135132ζ13121310134ζ1311138137ζ1391331313613513213111381371391331313121310134ζ136135132ζ13913313ζ1311138137ζ13121310134    complex lifted from C2×C13⋊C3
ρ15333000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ136135132ζ13913313ζ1311138137ζ13121310134ζ136135132    complex lifted from C13⋊C3
ρ163-33000000ζ13121310134ζ1311138137ζ13913313ζ13613513213121310134139133131361351321311138137ζ13121310134ζ136135132ζ13913313ζ1311138137    complex lifted from C2×C13⋊C3
ρ173-33000000ζ1311138137ζ13913313ζ136135132ζ1312131013413111381371361351321312131013413913313ζ1311138137ζ13121310134ζ136135132ζ13913313    complex lifted from C2×C13⋊C3
ρ1860-3000000139+2ζ133+2ζ13136+2ζ135+2ζ1321312+2ζ1310+2ζ1341311+2ζ138+2ζ137000013913313131113813713121310134136135132    complex faithful
ρ1960-30000001311+2ζ138+2ζ137139+2ζ133+2ζ13136+2ζ135+2ζ1321312+2ζ1310+2ζ134000013111381371312131013413613513213913313    complex faithful
ρ2060-30000001312+2ζ1310+2ζ1341311+2ζ138+2ζ137139+2ζ133+2ζ13136+2ζ135+2ζ132000013121310134136135132139133131311138137    complex faithful
ρ2160-3000000136+2ζ135+2ζ1321312+2ζ1310+2ζ1341311+2ζ138+2ζ137139+2ζ133+2ζ13000013613513213913313131113813713121310134    complex faithful

Smallest permutation representation of S3×C13⋊C3
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 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)| (1,14,27)(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), (2,4,10)(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( (1,14,27)(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), (2,4,10)(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([[(1,14,27),(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)], [(2,4,10),(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)]])

Matrix representation of S3×C13⋊C3 in GL5(𝔽79)

7878000
10000
00100
00010
00001
,
10000
7878000
00100
00010
00001
,
10000
01000
0091347
001052
00014
,
550000
055000
00162815
0067655
00723457

G:=sub<GL(5,GF(79))| [78,1,0,0,0,78,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,78,0,0,0,0,78,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,1,0,0,0,13,0,1,0,0,47,52,4],[55,0,0,0,0,0,55,0,0,0,0,0,16,67,72,0,0,28,6,34,0,0,15,55,57] >;

S3×C13⋊C3 in GAP, Magma, Sage, TeX

S_3\times C_{13}\rtimes C_3
% in TeX

G:=Group("S3xC13:C3");
// GroupNames label

G:=SmallGroup(234,8);
// by ID

G=gap.SmallGroup(234,8);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,146,439]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^13=d^3=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^9>;
// generators/relations

Export

Subgroup lattice of S3×C13⋊C3 in TeX
Character table of S3×C13⋊C3 in TeX

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