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G = C51⋊C4order 204 = 22·3·17

1st semidirect product of C51 and C4 acting faithfully

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

Aliases: C511C4, C17⋊Dic3, D17.S3, C3⋊(C17⋊C4), (C3×D17).1C2, SmallGroup(204,6)

Series: Derived Chief Lower central Upper central

C1C51 — C51⋊C4
C1C17C51C3×D17 — C51⋊C4
C51 — C51⋊C4
C1

Generators and relations for C51⋊C4
 G = < a,b | a51=b4=1, bab-1=a47 >

17C2
51C4
17C6
17Dic3
3C17⋊C4

Character table of C51⋊C4

 class 1234A4B617A17B17C17D51A51B51C51D51E51F51G51H
 size 1172515134444444444444
ρ1111111111111111111    trivial
ρ2111-1-11111111111111    linear of order 2
ρ31-11-ii-1111111111111    linear of order 4
ρ41-11i-i-1111111111111    linear of order 4
ρ522-100-12222-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-2-10012222-1-1-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ7404000ζ1716171317417ζ1715179178172ζ17141712175173ζ17111710177176ζ17141712175173ζ17111710177176ζ17111710177176ζ1715179178172ζ1716171317417ζ1715179178172ζ17141712175173ζ1716171317417    orthogonal lifted from C17⋊C4
ρ8404000ζ17111710177176ζ17141712175173ζ1716171317417ζ1715179178172ζ1716171317417ζ1715179178172ζ1715179178172ζ17141712175173ζ17111710177176ζ17141712175173ζ1716171317417ζ17111710177176    orthogonal lifted from C17⋊C4
ρ9404000ζ17141712175173ζ17111710177176ζ1715179178172ζ1716171317417ζ1715179178172ζ1716171317417ζ1716171317417ζ17111710177176ζ17141712175173ζ17111710177176ζ1715179178172ζ17141712175173    orthogonal lifted from C17⋊C4
ρ10404000ζ1715179178172ζ1716171317417ζ17111710177176ζ17141712175173ζ17111710177176ζ17141712175173ζ17141712175173ζ1716171317417ζ1715179178172ζ1716171317417ζ17111710177176ζ1715179178172    orthogonal lifted from C17⋊C4
ρ1140-2000ζ17111710177176ζ17141712175173ζ1716171317417ζ17151791781723ζ17163ζ17133ζ1743ζ1717161732ζ171532ζ17932ζ17832ζ1721715172ζ32ζ171532ζ17932ζ17832ζ172179178ζ3ζ17143ζ17123ζ1753ζ173171217532ζ171132ζ171032ζ17732ζ17617111763ζ17143ζ17123ζ1753ζ1731714173ζ3ζ17163ζ17133ζ1743ζ171713174ζ32ζ171132ζ171032ζ17732ζ1761710177    complex faithful
ρ1240-2000ζ1716171317417ζ1715179178172ζ17141712175173ζ17111710177176ζ3ζ17143ζ17123ζ1753ζ1731712175ζ32ζ171132ζ171032ζ17732ζ176171017732ζ171132ζ171032ζ17732ζ176171117632ζ171532ζ17932ζ17832ζ17217151723ζ17163ζ17133ζ1743ζ17171617ζ32ζ171532ζ17932ζ17832ζ1721791783ζ17143ζ17123ζ1753ζ1731714173ζ3ζ17163ζ17133ζ1743ζ171713174    complex faithful
ρ1340-2000ζ17111710177176ζ17141712175173ζ1716171317417ζ1715179178172ζ3ζ17163ζ17133ζ1743ζ171713174ζ32ζ171532ζ17932ζ17832ζ17217917832ζ171532ζ17932ζ17832ζ17217151723ζ17143ζ17123ζ1753ζ1731714173ζ32ζ171132ζ171032ζ17732ζ1761710177ζ3ζ17143ζ17123ζ1753ζ17317121753ζ17163ζ17133ζ1743ζ1717161732ζ171132ζ171032ζ17732ζ1761711176    complex faithful
ρ1440-2000ζ1715179178172ζ1716171317417ζ17111710177176ζ17141712175173ζ32ζ171132ζ171032ζ17732ζ17617101773ζ17143ζ17123ζ1753ζ1731714173ζ3ζ17143ζ17123ζ1753ζ1731712175ζ3ζ17163ζ17133ζ1743ζ17171317432ζ171532ζ17932ζ17832ζ17217151723ζ17163ζ17133ζ1743ζ1717161732ζ171132ζ171032ζ17732ζ1761711176ζ32ζ171532ζ17932ζ17832ζ172179178    complex faithful
ρ1540-2000ζ17141712175173ζ17111710177176ζ1715179178172ζ1716171317417ζ32ζ171532ζ17932ζ17832ζ1721791783ζ17163ζ17133ζ1743ζ17171617ζ3ζ17163ζ17133ζ1743ζ17171317432ζ171132ζ171032ζ17732ζ17617111763ζ17143ζ17123ζ1753ζ1731714173ζ32ζ171132ζ171032ζ17732ζ176171017732ζ171532ζ17932ζ17832ζ1721715172ζ3ζ17143ζ17123ζ1753ζ1731712175    complex faithful
ρ1640-2000ζ1716171317417ζ1715179178172ζ17141712175173ζ171117101771763ζ17143ζ17123ζ1753ζ173171417332ζ171132ζ171032ζ17732ζ1761711176ζ32ζ171132ζ171032ζ17732ζ1761710177ζ32ζ171532ζ17932ζ17832ζ172179178ζ3ζ17163ζ17133ζ1743ζ17171317432ζ171532ζ17932ζ17832ζ1721715172ζ3ζ17143ζ17123ζ1753ζ17317121753ζ17163ζ17133ζ1743ζ17171617    complex faithful
ρ1740-2000ζ1715179178172ζ1716171317417ζ17111710177176ζ1714171217517332ζ171132ζ171032ζ17732ζ1761711176ζ3ζ17143ζ17123ζ1753ζ17317121753ζ17143ζ17123ζ1753ζ17317141733ζ17163ζ17133ζ1743ζ17171617ζ32ζ171532ζ17932ζ17832ζ172179178ζ3ζ17163ζ17133ζ1743ζ171713174ζ32ζ171132ζ171032ζ17732ζ176171017732ζ171532ζ17932ζ17832ζ1721715172    complex faithful
ρ1840-2000ζ17141712175173ζ17111710177176ζ1715179178172ζ171617131741732ζ171532ζ17932ζ17832ζ1721715172ζ3ζ17163ζ17133ζ1743ζ1717131743ζ17163ζ17133ζ1743ζ17171617ζ32ζ171132ζ171032ζ17732ζ1761710177ζ3ζ17143ζ17123ζ1753ζ173171217532ζ171132ζ171032ζ17732ζ1761711176ζ32ζ171532ζ17932ζ17832ζ1721791783ζ17143ζ17123ζ1753ζ1731714173    complex faithful

Smallest permutation representation of C51⋊C4
On 51 points
Generators in S51
(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 39 17 48)(3 26 33 44)(4 13 49 40)(5 51 14 36)(6 38 30 32)(7 25 46 28)(8 12 11 24)(9 50 27 20)(10 37 43 16)(15 23 21 47)(18 35)(19 22 34 31)(29 45 41 42)

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

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

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

C51⋊C4 is a maximal subgroup of   S3×C17⋊C4
C51⋊C4 is a maximal quotient of   C513C8

Matrix representation of C51⋊C4 in GL4(𝔽409) generated by

88293232354
5546286190
21938633208
201125234348
,
1000
1303304320
16195170106
319328213344
G:=sub<GL(4,GF(409))| [88,55,219,201,293,46,386,125,232,286,33,234,354,190,208,348],[1,1,16,319,0,303,195,328,0,304,170,213,0,320,106,344] >;

C51⋊C4 in GAP, Magma, Sage, TeX

C_{51}\rtimes C_4
% in TeX

G:=Group("C51:C4");
// GroupNames label

G:=SmallGroup(204,6);
// by ID

G=gap.SmallGroup(204,6);
# by ID

G:=PCGroup([4,-2,-2,-3,-17,8,98,771,1543]);
// Polycyclic

G:=Group<a,b|a^51=b^4=1,b*a*b^-1=a^47>;
// generators/relations

Export

Subgroup lattice of C51⋊C4 in TeX
Character table of C51⋊C4 in TeX

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