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G = C13⋊C18order 234 = 2·32·13

The semidirect product of C13 and C18 acting via C18/C3=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C13⋊C18, D13⋊C9, C39.C6, C13⋊C9⋊C2, C3.(C13⋊C6), (C3×D13).C3, SmallGroup(234,1)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C18
C1C13C39C13⋊C9 — C13⋊C18
C13 — C13⋊C18
C1C3

Generators and relations for C13⋊C18
 G = < a,b | a13=b18=1, bab-1=a10 >

13C2
13C6
13C9
13C18

Character table of C13⋊C18

 class 123A3B6A6B9A9B9C9D9E9F13A13B18A18B18C18D18E18F39A39B39C39D
 size 113111313131313131313661313131313136666
ρ1111111111111111111111111    trivial
ρ21-111-1-111111111-1-1-1-1-1-11111    linear of order 2
ρ3111111ζ32ζ32ζ32ζ3ζ3ζ311ζ32ζ3ζ3ζ32ζ32ζ31111    linear of order 3
ρ41-111-1-1ζ3ζ3ζ3ζ32ζ32ζ3211ζ65ζ6ζ6ζ65ζ65ζ61111    linear of order 6
ρ51-111-1-1ζ32ζ32ζ32ζ3ζ3ζ311ζ6ζ65ζ65ζ6ζ6ζ651111    linear of order 6
ρ6111111ζ3ζ3ζ3ζ32ζ32ζ3211ζ3ζ32ζ32ζ3ζ3ζ321111    linear of order 3
ρ71-1ζ3ζ32ζ65ζ6ζ94ζ9ζ97ζ98ζ95ζ921197989594992ζ32ζ32ζ3ζ3    linear of order 18
ρ811ζ32ζ3ζ32ζ3ζ98ζ92ζ95ζ97ζ9ζ9411ζ95ζ97ζ9ζ98ζ92ζ94ζ3ζ3ζ32ζ32    linear of order 9
ρ91-1ζ3ζ32ζ65ζ6ζ9ζ97ζ94ζ92ζ98ζ951194929899795ζ32ζ32ζ3ζ3    linear of order 18
ρ101-1ζ32ζ3ζ6ζ65ζ98ζ92ζ95ζ97ζ9ζ941195979989294ζ3ζ3ζ32ζ32    linear of order 18
ρ1111ζ3ζ32ζ3ζ32ζ94ζ9ζ97ζ98ζ95ζ9211ζ97ζ98ζ95ζ94ζ9ζ92ζ32ζ32ζ3ζ3    linear of order 9
ρ1211ζ3ζ32ζ3ζ32ζ9ζ97ζ94ζ92ζ98ζ9511ζ94ζ92ζ98ζ9ζ97ζ95ζ32ζ32ζ3ζ3    linear of order 9
ρ131-1ζ32ζ3ζ6ζ65ζ95ζ98ζ92ζ9ζ94ζ971192994959897ζ3ζ3ζ32ζ32    linear of order 18
ρ1411ζ3ζ32ζ3ζ32ζ97ζ94ζ9ζ95ζ92ζ9811ζ9ζ95ζ92ζ97ζ94ζ98ζ32ζ32ζ3ζ3    linear of order 9
ρ1511ζ32ζ3ζ32ζ3ζ95ζ98ζ92ζ9ζ94ζ9711ζ92ζ9ζ94ζ95ζ98ζ97ζ3ζ3ζ32ζ32    linear of order 9
ρ161-1ζ3ζ32ζ65ζ6ζ97ζ94ζ9ζ95ζ92ζ981199592979498ζ32ζ32ζ3ζ3    linear of order 18
ρ1711ζ32ζ3ζ32ζ3ζ92ζ95ζ98ζ94ζ97ζ911ζ98ζ94ζ97ζ92ζ95ζ9ζ3ζ3ζ32ζ32    linear of order 9
ρ181-1ζ32ζ3ζ6ζ65ζ92ζ95ζ98ζ94ζ97ζ91198949792959ζ3ζ3ζ32ζ32    linear of order 18
ρ19606600000000-1-13/2-1+13/2000000-1+13/2-1-13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ20606600000000-1+13/2-1-13/2000000-1-13/2-1+13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ2160-3+3-3-3-3-300000000-1+13/2-1-13/2000000ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132    complex faithful, Schur index 3
ρ2260-3-3-3-3+3-300000000-1+13/2-1-13/2000000ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132    complex faithful, Schur index 3
ρ2360-3+3-3-3-3-300000000-1-13/2-1+13/2000000ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13    complex faithful, Schur index 3
ρ2460-3-3-3-3+3-300000000-1-13/2-1+13/2000000ζ3ζ13123ζ13103ζ1393ζ1343ζ1333ζ13ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132ζ32ζ131132ζ13832ζ13732ζ13632ζ13532ζ132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13    complex faithful, Schur index 3

Smallest permutation representation of C13⋊C18
On 117 points
Generators in S117
(1 40 89 22 62 101 78 69 110 53 13 98 31)(2 14 70 63 41 99 111 102 90 32 54 79 23)(3 55 103 42 15 80 91 82 71 24 33 112 46)(4 34 83 16 56 113 72 81 104 47 25 92 43)(5 26 64 57 35 93 105 114 84 44 48 73 17)(6 49 115 36 27 74 85 94 65 18 45 106 58)(7 28 95 10 50 107 66 75 116 59 19 86 37)(8 20 76 51 29 87 117 108 96 38 60 67 11)(9 61 109 30 21 68 97 88 77 12 39 100 52)
(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)

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

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

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

C13⋊C18 is a maximal subgroup of   C13⋊C36
C13⋊C18 is a maximal quotient of   C132C36

Matrix representation of C13⋊C18 in GL6(𝔽937)

93610000
93601000
93600100
93600010
93600001
274663273664274662
,
675868532596501708
661742366571195276
561619591729834516
624518692666674516
296223885936366276
62751588564435708

G:=sub<GL(6,GF(937))| [936,936,936,936,936,274,1,0,0,0,0,663,0,1,0,0,0,273,0,0,1,0,0,664,0,0,0,1,0,274,0,0,0,0,1,662],[675,661,561,624,296,627,868,742,619,518,223,515,532,366,591,692,885,885,596,571,729,666,936,644,501,195,834,674,366,35,708,276,516,516,276,708] >;

C13⋊C18 in GAP, Magma, Sage, TeX

C_{13}\rtimes C_{18}
% in TeX

G:=Group("C13:C18");
// GroupNames label

G:=SmallGroup(234,1);
// by ID

G=gap.SmallGroup(234,1);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,29,3459,439]);
// Polycyclic

G:=Group<a,b|a^13=b^18=1,b*a*b^-1=a^10>;
// generators/relations

Export

Subgroup lattice of C13⋊C18 in TeX
Character table of C13⋊C18 in TeX

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