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G = C3×C19⋊C3order 171 = 32·19

Direct product of C3 and C19⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C3×C19⋊C3, C57⋊C3, C19⋊C32, SmallGroup(171,4)

Series: Derived Chief Lower central Upper central

C1C19 — C3×C19⋊C3
C1C19C19⋊C3 — C3×C19⋊C3
C19 — C3×C19⋊C3
C1C3

Generators and relations for C3×C19⋊C3
 G = < a,b,c | a3=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

19C3
19C3
19C3
19C32

Character table of C3×C19⋊C3

 class 13A3B3C3D3E3F3G3H19A19B19C19D19E19F57A57B57C57D57E57F57G57H57I57J57K57L
 size 111191919191919333333333333333333
ρ1111111111111111111111111111    trivial
ρ21ζ3ζ32ζ32ζ3ζ321ζ31111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 3
ρ31ζ32ζ3ζ3ζ32ζ31ζ321111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 3
ρ41ζ32ζ3ζ32ζ31ζ31ζ32111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 3
ρ51ζ3ζ3211ζ3ζ3ζ32ζ32111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 3
ρ61ζ3ζ32ζ3ζ321ζ321ζ3111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 3
ρ7111ζ32ζ3ζ3ζ32ζ32ζ3111111111111111111    linear of order 3
ρ81ζ32ζ311ζ32ζ32ζ3ζ3111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 3
ρ9111ζ3ζ32ζ32ζ3ζ3ζ32111111111111111111    linear of order 3
ρ10333000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ1914193192ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ19171916195ζ19181912198ζ191519131910ζ199196194ζ191119719ζ191119719    complex lifted from C19⋊C3
ρ11333000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ19181912198ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ191119719ζ191519131910ζ1914193192ζ19171916195ζ199196194ζ199196194    complex lifted from C19⋊C3
ρ12333000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ191119719ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ19181912198ζ199196194ζ19171916195ζ1914193192ζ191519131910ζ191519131910    complex lifted from C19⋊C3
ρ13333000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ191519131910ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ199196194ζ1914193192ζ19181912198ζ191119719ζ19171916195ζ19171916195    complex lifted from C19⋊C3
ρ14333000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ199196194ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ191519131910ζ19171916195ζ191119719ζ19181912198ζ1914193192ζ1914193192    complex lifted from C19⋊C3
ρ15333000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ19171916195ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ1914193192ζ191119719ζ199196194ζ191519131910ζ19181912198ζ19181912198    complex lifted from C19⋊C3
ρ163-3+3-3/2-3-3-3/2000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ3ζ19113ζ1973ζ19ζ32ζ191132ζ19732ζ19ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ3ζ19183ζ19123ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ3ζ19143ζ1933ζ192ζ32ζ191532ζ191332ζ1910ζ3ζ19153ζ19133ζ1910    complex faithful
ρ173-3-3-3/2-3+3-3/2000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ32ζ19932ζ19632ζ194ζ3ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ32ζ191832ζ191232ζ198ζ3ζ19143ζ1933ζ192ζ32ζ191432ζ19332ζ192    complex faithful
ρ183-3+3-3/2-3-3-3/2000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ3ζ19153ζ19133ζ1910ζ32ζ191532ζ191332ζ1910ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ3ζ19113ζ1973ζ19ζ32ζ191732ζ191632ζ195ζ3ζ19173ζ19163ζ195    complex faithful
ρ193-3+3-3/2-3-3-3/2000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ3ζ19173ζ19163ζ195ζ32ζ191732ζ191632ζ195ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ3ζ19153ζ19133ζ1910ζ32ζ191832ζ191232ζ198ζ3ζ19183ζ19123ζ198    complex faithful
ρ203-3+3-3/2-3-3-3/2000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ3ζ19143ζ1933ζ192ζ32ζ191432ζ19332ζ192ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ3ζ1993ζ1963ζ194ζ32ζ191132ζ19732ζ19ζ3ζ19113ζ1973ζ19    complex faithful
ρ213-3-3-3/2-3+3-3/2000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ32ζ191832ζ191232ζ198ζ3ζ19183ζ19123ζ198ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ32ζ191132ζ19732ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ32ζ191732ζ191632ζ195ζ3ζ1993ζ1963ζ194ζ32ζ19932ζ19632ζ194    complex faithful
ρ223-3-3-3/2-3+3-3/2000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ32ζ191532ζ191332ζ1910ζ3ζ19153ζ19133ζ1910ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ32ζ191132ζ19732ζ19ζ3ζ19173ζ19163ζ195ζ32ζ191732ζ191632ζ195    complex faithful
ρ233-3-3-3/2-3+3-3/2000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ32ζ191432ζ19332ζ192ζ3ζ19143ζ1933ζ192ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ32ζ19932ζ19632ζ194ζ3ζ19113ζ1973ζ19ζ32ζ191132ζ19732ζ19    complex faithful
ρ243-3-3-3/2-3+3-3/2000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ32ζ191732ζ191632ζ195ζ3ζ19173ζ19163ζ195ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ32ζ191532ζ191332ζ1910ζ3ζ19183ζ19123ζ198ζ32ζ191832ζ191232ζ198    complex faithful
ρ253-3+3-3/2-3-3-3/2000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ3ζ19183ζ19123ζ198ζ32ζ191832ζ191232ζ198ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ3ζ19113ζ1973ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ3ζ19173ζ19163ζ195ζ32ζ19932ζ19632ζ194ζ3ζ1993ζ1963ζ194    complex faithful
ρ263-3+3-3/2-3-3-3/2000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ3ζ1993ζ1963ζ194ζ32ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ3ζ19183ζ19123ζ198ζ32ζ191432ζ19332ζ192ζ3ζ19143ζ1933ζ192    complex faithful
ρ273-3-3-3/2-3+3-3/2000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ32ζ191132ζ19732ζ19ζ3ζ19113ζ1973ζ19ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ32ζ191832ζ191232ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ32ζ191432ζ19332ζ192ζ3ζ19153ζ19133ζ1910ζ32ζ191532ζ191332ζ1910    complex faithful

Smallest permutation representation of C3×C19⋊C3
On 57 points
Generators in S57
(1 39 20)(2 40 21)(3 41 22)(4 42 23)(5 43 24)(6 44 25)(7 45 26)(8 46 27)(9 47 28)(10 48 29)(11 49 30)(12 50 31)(13 51 32)(14 52 33)(15 53 34)(16 54 35)(17 55 36)(18 56 37)(19 57 38)
(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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)

G:=sub<Sym(57)| (1,39,20)(2,40,21)(3,41,22)(4,42,23)(5,43,24)(6,44,25)(7,45,26)(8,46,27)(9,47,28)(10,48,29)(11,49,30)(12,50,31)(13,51,32)(14,52,33)(15,53,34)(16,54,35)(17,55,36)(18,56,37)(19,57,38), (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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)>;

G:=Group( (1,39,20)(2,40,21)(3,41,22)(4,42,23)(5,43,24)(6,44,25)(7,45,26)(8,46,27)(9,47,28)(10,48,29)(11,49,30)(12,50,31)(13,51,32)(14,52,33)(15,53,34)(16,54,35)(17,55,36)(18,56,37)(19,57,38), (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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54) );

G=PermutationGroup([[(1,39,20),(2,40,21),(3,41,22),(4,42,23),(5,43,24),(6,44,25),(7,45,26),(8,46,27),(9,47,28),(10,48,29),(11,49,30),(12,50,31),(13,51,32),(14,52,33),(15,53,34),(16,54,35),(17,55,36),(18,56,37),(19,57,38)], [(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),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54)]])

C3×C19⋊C3 is a maximal subgroup of   D57⋊C3

Matrix representation of C3×C19⋊C3 in GL3(𝔽7) generated by

200
020
002
,
420
631
050
,
105
006
016
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,2],[4,6,0,2,3,5,0,1,0],[1,0,0,0,0,1,5,6,6] >;

C3×C19⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_{19}\rtimes C_3
% in TeX

G:=Group("C3xC19:C3");
// GroupNames label

G:=SmallGroup(171,4);
// by ID

G=gap.SmallGroup(171,4);
# by ID

G:=PCGroup([3,-3,-3,-19,569]);
// Polycyclic

G:=Group<a,b,c|a^3=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

Export

Subgroup lattice of C3×C19⋊C3 in TeX
Character table of C3×C19⋊C3 in TeX

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