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G = C7⋊C9order 63 = 32·7

The semidirect product of C7 and C9 acting via C9/C3=C3

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

Aliases: C7⋊C9, C21.C3, C3.(C7⋊C3), SmallGroup(63,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C9
C1C7C21 — C7⋊C9
C7 — C7⋊C9
C1C3

Generators and relations for C7⋊C9
 G = < a,b | a7=b9=1, bab-1=a4 >

7C9

Character table of C7⋊C9

 class 13A3B7A7B9A9B9C9D9E9F21A21B21C21D
 size 111337777773333
ρ1111111111111111    trivial
ρ211111ζ3ζ32ζ32ζ3ζ3ζ321111    linear of order 3
ρ311111ζ32ζ3ζ3ζ32ζ32ζ31111    linear of order 3
ρ41ζ3ζ3211ζ92ζ9ζ94ζ95ζ98ζ97ζ32ζ32ζ3ζ3    linear of order 9
ρ51ζ3ζ3211ζ98ζ94ζ97ζ92ζ95ζ9ζ32ζ32ζ3ζ3    linear of order 9
ρ61ζ32ζ311ζ94ζ92ζ98ζ9ζ97ζ95ζ3ζ3ζ32ζ32    linear of order 9
ρ71ζ3ζ3211ζ95ζ97ζ9ζ98ζ92ζ94ζ32ζ32ζ3ζ3    linear of order 9
ρ81ζ32ζ311ζ97ζ98ζ95ζ94ζ9ζ92ζ3ζ3ζ32ζ32    linear of order 9
ρ91ζ32ζ311ζ9ζ95ζ92ζ97ζ94ζ98ζ3ζ3ζ32ζ32    linear of order 9
ρ10333-1+-7/2-1--7/2000000-1+-7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ11333-1--7/2-1+-7/2000000-1--7/2-1+-7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ123-3-3-3/2-3+3-3/2-1--7/2-1+-7/2000000ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73    complex faithful, Schur index 3
ρ133-3+3-3/2-3-3-3/2-1--7/2-1+-7/2000000ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73    complex faithful, Schur index 3
ρ143-3-3-3/2-3+3-3/2-1+-7/2-1--7/2000000ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7    complex faithful, Schur index 3
ρ153-3+3-3/2-3-3-3/2-1+-7/2-1--7/2000000ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7    complex faithful, Schur index 3

Smallest permutation representation of C7⋊C9
Regular action on 63 points
Generators in S63
(1 56 16 27 47 29 38)(2 48 57 30 17 39 19)(3 18 49 40 58 20 31)(4 59 10 21 50 32 41)(5 51 60 33 11 42 22)(6 12 52 43 61 23 34)(7 62 13 24 53 35 44)(8 54 63 36 14 45 25)(9 15 46 37 55 26 28)
(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 58 59 60 61 62 63)

G:=sub<Sym(63)| (1,56,16,27,47,29,38)(2,48,57,30,17,39,19)(3,18,49,40,58,20,31)(4,59,10,21,50,32,41)(5,51,60,33,11,42,22)(6,12,52,43,61,23,34)(7,62,13,24,53,35,44)(8,54,63,36,14,45,25)(9,15,46,37,55,26,28), (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)(55,56,57,58,59,60,61,62,63)>;

G:=Group( (1,56,16,27,47,29,38)(2,48,57,30,17,39,19)(3,18,49,40,58,20,31)(4,59,10,21,50,32,41)(5,51,60,33,11,42,22)(6,12,52,43,61,23,34)(7,62,13,24,53,35,44)(8,54,63,36,14,45,25)(9,15,46,37,55,26,28), (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)(55,56,57,58,59,60,61,62,63) );

G=PermutationGroup([(1,56,16,27,47,29,38),(2,48,57,30,17,39,19),(3,18,49,40,58,20,31),(4,59,10,21,50,32,41),(5,51,60,33,11,42,22),(6,12,52,43,61,23,34),(7,62,13,24,53,35,44),(8,54,63,36,14,45,25),(9,15,46,37,55,26,28)], [(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),(55,56,57,58,59,60,61,62,63)])

C7⋊C9 is a maximal subgroup of
C7⋊C18  C9×C7⋊C3  C63⋊C3  C633C3  C21.C32  C21.A4  C49⋊C9  C72⋊C9  C723C9
C7⋊C9 is a maximal quotient of
C7⋊C27  C21.A4  C49⋊C9  C72⋊C9  C723C9

Matrix representation of C7⋊C9 in GL3(𝔽127) generated by

010
001
12322
,
49253
1115385
5312525
G:=sub<GL(3,GF(127))| [0,0,1,1,0,23,0,1,22],[49,111,53,2,53,125,53,85,25] >;

C7⋊C9 in GAP, Magma, Sage, TeX

C_7\rtimes C_9
% in TeX

G:=Group("C7:C9");
// GroupNames label

G:=SmallGroup(63,1);
// by ID

G=gap.SmallGroup(63,1);
# by ID

G:=PCGroup([3,-3,-3,-7,9,164]);
// Polycyclic

G:=Group<a,b|a^7=b^9=1,b*a*b^-1=a^4>;
// generators/relations

Export

Subgroup lattice of C7⋊C9 in TeX
Character table of C7⋊C9 in TeX

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