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G = C13⋊C9order 117 = 32·13

The semidirect product of C13 and C9 acting via C9/C3=C3

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

Aliases: C13⋊C9, C39.C3, C3.(C13⋊C3), SmallGroup(117,1)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C9
C1C13C39 — C13⋊C9
C13 — C13⋊C9
C1C3

Generators and relations for C13⋊C9
 G = < a,b | a13=b9=1, bab-1=a9 >

13C9

Character table of C13⋊C9

 class 13A3B9A9B9C9D9E9F13A13B13C13D39A39B39C39D39E39F39G39H
 size 111131313131313333333333333
ρ1111111111111111111111    trivial
ρ2111ζ3ζ32ζ32ζ32ζ3ζ3111111111111    linear of order 3
ρ3111ζ32ζ3ζ3ζ3ζ32ζ32111111111111    linear of order 3
ρ41ζ32ζ3ζ92ζ97ζ9ζ94ζ95ζ981111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ51ζ3ζ32ζ94ζ95ζ92ζ98ζ9ζ971111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ61ζ3ζ32ζ9ζ98ζ95ζ92ζ97ζ941111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ71ζ32ζ3ζ98ζ9ζ94ζ97ζ92ζ951111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ81ζ3ζ32ζ97ζ92ζ98ζ95ζ94ζ91111ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ91ζ32ζ3ζ95ζ94ζ97ζ9ζ98ζ921111ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ10333000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ13913313ζ136135132ζ136135132ζ13121310134    complex lifted from C13⋊C3
ρ11333000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13913313ζ136135132ζ13121310134ζ1311138137ζ1311138137ζ13913313    complex lifted from C13⋊C3
ρ12333000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ13913313ζ136135132ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13121310134ζ1311138137    complex lifted from C13⋊C3
ρ13333000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13913313ζ136135132    complex lifted from C13⋊C3
ρ143-3-3-3/2-3+3-3/2000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13    complex faithful, Schur index 3
ρ153-3-3-3/2-3+3-3/2000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132    complex faithful, Schur index 3
ρ163-3-3-3/2-3+3-3/2000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134    complex faithful, Schur index 3
ρ173-3+3-3/2-3-3-3/2000000ζ13121310134ζ1311138137ζ13913313ζ136135132ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132    complex faithful, Schur index 3
ρ183-3+3-3/2-3-3-3/2000000ζ136135132ζ13121310134ζ1311138137ζ13913313ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13    complex faithful, Schur index 3
ρ193-3+3-3/2-3-3-3/2000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ134ζ3ζ13123ζ13103ζ134ζ3ζ13113ζ1383ζ137    complex faithful, Schur index 3
ρ203-3+3-3/2-3-3-3/2000000ζ1311138137ζ13913313ζ136135132ζ13121310134ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134    complex faithful, Schur index 3
ρ213-3-3-3/2-3+3-3/2000000ζ13913313ζ136135132ζ13121310134ζ1311138137ζ32ζ13932ζ13332ζ13ζ32ζ13632ζ13532ζ132ζ3ζ13113ζ1383ζ137ζ3ζ1393ζ1333ζ13ζ3ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ134ζ32ζ131232ζ131032ζ134ζ32ζ131132ζ13832ζ137    complex faithful, Schur index 3

Smallest permutation representation of C13⋊C9
Regular action on 117 points
Generators in S117
(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 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)
(1 116 78 34 98 64 19 83 48)(2 106 74 35 101 60 20 86 44)(3 109 70 36 104 56 21 89 40)(4 112 66 37 94 65 22 79 49)(5 115 75 38 97 61 23 82 45)(6 105 71 39 100 57 24 85 41)(7 108 67 27 103 53 25 88 50)(8 111 76 28 93 62 26 91 46)(9 114 72 29 96 58 14 81 42)(10 117 68 30 99 54 15 84 51)(11 107 77 31 102 63 16 87 47)(12 110 73 32 92 59 17 90 43)(13 113 69 33 95 55 18 80 52)

G:=sub<Sym(117)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,116,78,34,98,64,19,83,48)(2,106,74,35,101,60,20,86,44)(3,109,70,36,104,56,21,89,40)(4,112,66,37,94,65,22,79,49)(5,115,75,38,97,61,23,82,45)(6,105,71,39,100,57,24,85,41)(7,108,67,27,103,53,25,88,50)(8,111,76,28,93,62,26,91,46)(9,114,72,29,96,58,14,81,42)(10,117,68,30,99,54,15,84,51)(11,107,77,31,102,63,16,87,47)(12,110,73,32,92,59,17,90,43)(13,113,69,33,95,55,18,80,52)>;

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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,116,78,34,98,64,19,83,48)(2,106,74,35,101,60,20,86,44)(3,109,70,36,104,56,21,89,40)(4,112,66,37,94,65,22,79,49)(5,115,75,38,97,61,23,82,45)(6,105,71,39,100,57,24,85,41)(7,108,67,27,103,53,25,88,50)(8,111,76,28,93,62,26,91,46)(9,114,72,29,96,58,14,81,42)(10,117,68,30,99,54,15,84,51)(11,107,77,31,102,63,16,87,47)(12,110,73,32,92,59,17,90,43)(13,113,69,33,95,55,18,80,52) );

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),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,116,78,34,98,64,19,83,48),(2,106,74,35,101,60,20,86,44),(3,109,70,36,104,56,21,89,40),(4,112,66,37,94,65,22,79,49),(5,115,75,38,97,61,23,82,45),(6,105,71,39,100,57,24,85,41),(7,108,67,27,103,53,25,88,50),(8,111,76,28,93,62,26,91,46),(9,114,72,29,96,58,14,81,42),(10,117,68,30,99,54,15,84,51),(11,107,77,31,102,63,16,87,47),(12,110,73,32,92,59,17,90,43),(13,113,69,33,95,55,18,80,52)])

C13⋊C9 is a maximal subgroup of   C13⋊C18  C9×C13⋊C3  C117⋊C3  C1173C3  C39.C32  C39.A4
C13⋊C9 is a maximal quotient of   C13⋊C27  C39.A4

Matrix representation of C13⋊C9 in GL3(𝔽937) generated by

010
001
1730454
,
643281633
85427250
532848804
G:=sub<GL(3,GF(937))| [0,0,1,1,0,730,0,1,454],[643,85,532,281,427,848,633,250,804] >;

C13⋊C9 in GAP, Magma, Sage, TeX

C_{13}\rtimes C_9
% in TeX

G:=Group("C13:C9");
// GroupNames label

G:=SmallGroup(117,1);
// by ID

G=gap.SmallGroup(117,1);
# by ID

G:=PCGroup([3,-3,-3,-13,9,245]);
// Polycyclic

G:=Group<a,b|a^13=b^9=1,b*a*b^-1=a^9>;
// generators/relations

Export

Subgroup lattice of C13⋊C9 in TeX
Character table of C13⋊C9 in TeX

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