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G = C49⋊C3order 147 = 3·72

The semidirect product of C49 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C49⋊C3, C7.(C7⋊C3), SmallGroup(147,1)

Series: Derived Chief Lower central Upper central

C1C49 — C49⋊C3
C1C7C49 — C49⋊C3
C49 — C49⋊C3
C1

Generators and relations for C49⋊C3
 G = < a,b | a49=b3=1, bab-1=a18 >

49C3
7C7⋊C3

Character table of C49⋊C3

 class 13A3B7A7B49A49B49C49D49E49F49G49H49I49J49K49L49M49N
 size 149493333333333333333
ρ11111111111111111111    trivial
ρ21ζ32ζ31111111111111111    linear of order 3
ρ31ζ3ζ321111111111111111    linear of order 3
ρ430033-1+-7/2-1--7/2-1--7/2-1--7/2-1--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2-1--7/2    complex lifted from C7⋊C3
ρ530033-1--7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1+-7/2-1+-7/2    complex lifted from C7⋊C3
ρ6300-1+-7/2-1--7/2ζ493749324929ζ4941495493ζ494049344924ζ494549274926ζ492049174912ζ494349394916ζ494749384913ζ49254915499ζ4930491849ζ49364911492ζ49234922494ζ49464944498ζ49334910496ζ494849314919    complex faithful
ρ7300-1--7/2-1+-7/2ζ49334910496ζ49234922494ζ493749324929ζ49364911492ζ494349394916ζ4941495493ζ4930491849ζ492049174912ζ494049344924ζ494849314919ζ494749384913ζ494549274926ζ49464944498ζ49254915499    complex faithful
ρ8300-1--7/2-1+-7/2ζ494549274926ζ4930491849ζ49464944498ζ49254915499ζ49234922494ζ494749384913ζ493749324929ζ4941495493ζ49334910496ζ492049174912ζ494049344924ζ494849314919ζ49364911492ζ494349394916    complex faithful
ρ9300-1--7/2-1+-7/2ζ494049344924ζ494349394916ζ4930491849ζ49464944498ζ49254915499ζ492049174912ζ49234922494ζ494849314919ζ494749384913ζ494549274926ζ4941495493ζ49334910496ζ493749324929ζ49364911492    complex faithful
ρ10300-1--7/2-1+-7/2ζ494849314919ζ493749324929ζ49364911492ζ494349394916ζ4930491849ζ494049344924ζ49464944498ζ494749384913ζ494549274926ζ4941495493ζ49334910496ζ492049174912ζ49254915499ζ49234922494    complex faithful
ρ11300-1--7/2-1+-7/2ζ494749384913ζ49254915499ζ49234922494ζ493749324929ζ49364911492ζ494849314919ζ494349394916ζ494549274926ζ4941495493ζ49334910496ζ492049174912ζ494049344924ζ4930491849ζ49464944498    complex faithful
ρ12300-1+-7/2-1--7/2ζ49234922494ζ494849314919ζ4941495493ζ494049344924ζ494549274926ζ49364911492ζ492049174912ζ49464944498ζ494349394916ζ493749324929ζ49254915499ζ4930491849ζ494749384913ζ49334910496    complex faithful
ρ13300-1+-7/2-1--7/2ζ49464944498ζ494749384913ζ49334910496ζ494849314919ζ4941495493ζ49234922494ζ494049344924ζ494349394916ζ493749324929ζ49254915499ζ4930491849ζ49364911492ζ494549274926ζ492049174912    complex faithful
ρ14300-1--7/2-1+-7/2ζ492049174912ζ49464944498ζ49254915499ζ49234922494ζ493749324929ζ49334910496ζ49364911492ζ494049344924ζ494849314919ζ494749384913ζ494549274926ζ4941495493ζ494349394916ζ4930491849    complex faithful
ρ15300-1+-7/2-1--7/2ζ49254915499ζ49334910496ζ494849314919ζ4941495493ζ494049344924ζ493749324929ζ494549274926ζ4930491849ζ49364911492ζ49234922494ζ49464944498ζ494349394916ζ492049174912ζ494749384913    complex faithful
ρ16300-1+-7/2-1--7/2ζ4930491849ζ492049174912ζ494749384913ζ49334910496ζ494849314919ζ49254915499ζ4941495493ζ49364911492ζ49234922494ζ49464944498ζ494349394916ζ493749324929ζ494049344924ζ494549274926    complex faithful
ρ17300-1--7/2-1+-7/2ζ4941495493ζ49364911492ζ494349394916ζ4930491849ζ49464944498ζ494549274926ζ49254915499ζ49334910496ζ492049174912ζ494049344924ζ494849314919ζ494749384913ζ49234922494ζ493749324929    complex faithful
ρ18300-1+-7/2-1--7/2ζ49364911492ζ494049344924ζ494549274926ζ492049174912ζ494749384913ζ4930491849ζ49334910496ζ49234922494ζ49464944498ζ494349394916ζ493749324929ζ49254915499ζ494849314919ζ4941495493    complex faithful
ρ19300-1+-7/2-1--7/2ζ494349394916ζ494549274926ζ492049174912ζ494749384913ζ49334910496ζ49464944498ζ494849314919ζ493749324929ζ49254915499ζ4930491849ζ49364911492ζ49234922494ζ4941495493ζ494049344924    complex faithful

Smallest permutation representation of C49⋊C3
On 49 points
Generators in S49
(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)
(2 31 19)(3 12 37)(4 42 6)(5 23 24)(7 34 11)(8 15 29)(9 45 47)(10 26 16)(13 18 21)(14 48 39)(17 40 44)(20 32 49)(22 43 36)(25 35 41)(27 46 28)(30 38 33)

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

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

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

C49⋊C3 is a maximal subgroup of   C49⋊C6
C49⋊C3 is a maximal quotient of   C49⋊C9

Matrix representation of C49⋊C3 in GL3(𝔽883) generated by

584610587
587275597
597366340
,
325364
857135141
364204745
G:=sub<GL(3,GF(883))| [584,587,597,610,275,366,587,597,340],[3,857,364,25,135,204,364,141,745] >;

C49⋊C3 in GAP, Magma, Sage, TeX

C_{49}\rtimes C_3
% in TeX

G:=Group("C49:C3");
// GroupNames label

G:=SmallGroup(147,1);
// by ID

G=gap.SmallGroup(147,1);
# by ID

G:=PCGroup([3,-3,-7,-7,541,46,380]);
// Polycyclic

G:=Group<a,b|a^49=b^3=1,b*a*b^-1=a^18>;
// generators/relations

Export

Subgroup lattice of C49⋊C3 in TeX
Character table of C49⋊C3 in TeX

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