Copied to
clipboard

G = C3×C17⋊C4order 204 = 22·3·17

Direct product of C3 and C17⋊C4

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×C17⋊C4, C17⋊C12, C512C4, D17.C6, (C3×D17).2C2, SmallGroup(204,5)

Series: Derived Chief Lower central Upper central

C1C17 — C3×C17⋊C4
C1C17D17C3×D17 — C3×C17⋊C4
C17 — C3×C17⋊C4
C1C3

Generators and relations for C3×C17⋊C4
 G = < a,b,c | a3=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >

17C2
17C4
17C6
17C12

Character table of C3×C17⋊C4

 class 123A3B4A4B6A6B12A12B12C12D17A17B17C17D51A51B51C51D51E51F51G51H
 size 117111717171717171717444444444444
ρ1111111111111111111111111    trivial
ρ21111-1-111-1-1-1-1111111111111    linear of order 2
ρ311ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ31111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ411ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ65ζ61111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ511ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ321111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ611ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ6ζ651111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ71-111-ii-1-1-ii-ii111111111111    linear of order 4
ρ81-111i-i-1-1i-ii-i111111111111    linear of order 4
ρ91-1ζ3ζ32-iiζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ31111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 12
ρ101-1ζ32ζ3-iiζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ321111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 12
ρ111-1ζ3ζ32i-iζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ31111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 12
ρ121-1ζ32ζ3i-iζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ321111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 12
ρ13404400000000ζ1716171317417ζ17141712175173ζ1715179178172ζ17111710177176ζ1716171317417ζ1715179178172ζ1716171317417ζ17141712175173ζ17111710177176ζ17141712175173ζ17111710177176ζ1715179178172    orthogonal lifted from C17⋊C4
ρ14404400000000ζ17111710177176ζ1716171317417ζ17141712175173ζ1715179178172ζ17111710177176ζ17141712175173ζ17111710177176ζ1716171317417ζ1715179178172ζ1716171317417ζ1715179178172ζ17141712175173    orthogonal lifted from C17⋊C4
ρ15404400000000ζ17141712175173ζ1715179178172ζ17111710177176ζ1716171317417ζ17141712175173ζ17111710177176ζ17141712175173ζ1715179178172ζ1716171317417ζ1715179178172ζ1716171317417ζ17111710177176    orthogonal lifted from C17⋊C4
ρ16404400000000ζ1715179178172ζ17111710177176ζ1716171317417ζ17141712175173ζ1715179178172ζ1716171317417ζ1715179178172ζ17111710177176ζ17141712175173ζ17111710177176ζ17141712175173ζ1716171317417    orthogonal lifted from C17⋊C4
ρ1740-2-2-3-2+2-300000000ζ1715179178172ζ17111710177176ζ1716171317417ζ17141712175173ζ32ζ171532ζ17932ζ17832ζ172ζ3ζ17163ζ17133ζ1743ζ17ζ3ζ17153ζ1793ζ1783ζ172ζ3ζ17113ζ17103ζ1773ζ176ζ3ζ17143ζ17123ζ1753ζ173ζ32ζ171132ζ171032ζ17732ζ176ζ32ζ171432ζ171232ζ17532ζ173ζ32ζ171632ζ171332ζ17432ζ17    complex faithful
ρ1840-2+2-3-2-2-300000000ζ1715179178172ζ17111710177176ζ1716171317417ζ17141712175173ζ3ζ17153ζ1793ζ1783ζ172ζ32ζ171632ζ171332ζ17432ζ17ζ32ζ171532ζ17932ζ17832ζ172ζ32ζ171132ζ171032ζ17732ζ176ζ32ζ171432ζ171232ζ17532ζ173ζ3ζ17113ζ17103ζ1773ζ176ζ3ζ17143ζ17123ζ1753ζ173ζ3ζ17163ζ17133ζ1743ζ17    complex faithful
ρ1940-2-2-3-2+2-300000000ζ17141712175173ζ1715179178172ζ17111710177176ζ1716171317417ζ32ζ171432ζ171232ζ17532ζ173ζ3ζ17113ζ17103ζ1773ζ176ζ3ζ17143ζ17123ζ1753ζ173ζ3ζ17153ζ1793ζ1783ζ172ζ3ζ17163ζ17133ζ1743ζ17ζ32ζ171532ζ17932ζ17832ζ172ζ32ζ171632ζ171332ζ17432ζ17ζ32ζ171132ζ171032ζ17732ζ176    complex faithful
ρ2040-2-2-3-2+2-300000000ζ1716171317417ζ17141712175173ζ1715179178172ζ17111710177176ζ32ζ171632ζ171332ζ17432ζ17ζ3ζ17153ζ1793ζ1783ζ172ζ3ζ17163ζ17133ζ1743ζ17ζ3ζ17143ζ17123ζ1753ζ173ζ3ζ17113ζ17103ζ1773ζ176ζ32ζ171432ζ171232ζ17532ζ173ζ32ζ171132ζ171032ζ17732ζ176ζ32ζ171532ζ17932ζ17832ζ172    complex faithful
ρ2140-2+2-3-2-2-300000000ζ17141712175173ζ1715179178172ζ17111710177176ζ1716171317417ζ3ζ17143ζ17123ζ1753ζ173ζ32ζ171132ζ171032ζ17732ζ176ζ32ζ171432ζ171232ζ17532ζ173ζ32ζ171532ζ17932ζ17832ζ172ζ32ζ171632ζ171332ζ17432ζ17ζ3ζ17153ζ1793ζ1783ζ172ζ3ζ17163ζ17133ζ1743ζ17ζ3ζ17113ζ17103ζ1773ζ176    complex faithful
ρ2240-2-2-3-2+2-300000000ζ17111710177176ζ1716171317417ζ17141712175173ζ1715179178172ζ32ζ171132ζ171032ζ17732ζ176ζ3ζ17143ζ17123ζ1753ζ173ζ3ζ17113ζ17103ζ1773ζ176ζ3ζ17163ζ17133ζ1743ζ17ζ3ζ17153ζ1793ζ1783ζ172ζ32ζ171632ζ171332ζ17432ζ17ζ32ζ171532ζ17932ζ17832ζ172ζ32ζ171432ζ171232ζ17532ζ173    complex faithful
ρ2340-2+2-3-2-2-300000000ζ1716171317417ζ17141712175173ζ1715179178172ζ17111710177176ζ3ζ17163ζ17133ζ1743ζ17ζ32ζ171532ζ17932ζ17832ζ172ζ32ζ171632ζ171332ζ17432ζ17ζ32ζ171432ζ171232ζ17532ζ173ζ32ζ171132ζ171032ζ17732ζ176ζ3ζ17143ζ17123ζ1753ζ173ζ3ζ17113ζ17103ζ1773ζ176ζ3ζ17153ζ1793ζ1783ζ172    complex faithful
ρ2440-2+2-3-2-2-300000000ζ17111710177176ζ1716171317417ζ17141712175173ζ1715179178172ζ3ζ17113ζ17103ζ1773ζ176ζ32ζ171432ζ171232ζ17532ζ173ζ32ζ171132ζ171032ζ17732ζ176ζ32ζ171632ζ171332ζ17432ζ17ζ32ζ171532ζ17932ζ17832ζ172ζ3ζ17163ζ17133ζ1743ζ17ζ3ζ17153ζ1793ζ1783ζ172ζ3ζ17143ζ17123ζ1753ζ173    complex faithful

Smallest permutation representation of C3×C17⋊C4
On 51 points
Generators in S51
(1 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)
(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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)

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

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

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

C3×C17⋊C4 is a maximal subgroup of   C51⋊C8

Matrix representation of C3×C17⋊C4 in GL4(𝔽409) generated by

53000
05300
00530
00053
,
000408
100392
01048
001392
,
114917
037714288
0378103394
01832337
G:=sub<GL(4,GF(409))| [53,0,0,0,0,53,0,0,0,0,53,0,0,0,0,53],[0,1,0,0,0,0,1,0,0,0,0,1,408,392,48,392],[1,0,0,0,1,377,378,18,49,14,103,32,17,288,394,337] >;

C3×C17⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{17}\rtimes C_4
% in TeX

G:=Group("C3xC17:C4");
// GroupNames label

G:=SmallGroup(204,5);
// by ID

G=gap.SmallGroup(204,5);
# by ID

G:=PCGroup([4,-2,-3,-2,-17,24,2499,523]);
// Polycyclic

G:=Group<a,b,c|a^3=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C3×C17⋊C4 in TeX
Character table of C3×C17⋊C4 in TeX

׿
×
𝔽