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G = C13⋊C36  order 468 = 22·32·13

The semidirect product of C13 and C36 acting via C36/C3=C12

metacyclic, supersoluble, monomial, Z-group

Aliases: C13⋊C36, C3.F13, C39.C12, D13.C18, C13⋊C9⋊C4, C13⋊C4⋊C9, C13⋊C18.C2, (C3×D13).2C6, (C3×C13⋊C4).C3, SmallGroup(468,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C13⋊C36
Chief series
C1 — C13 — C39 — C3×D13 — C13⋊C18 — C13⋊C36
Lower central
C13 — C13⋊C36
Upper central
C1 — C3

Generators and relations for C13⋊C36
 G = < a,b | a13=b36=1, bab-1=a6 >

C1
13C2
C3
C13
13C4
13C6
13C9
D13
C39
13C12
13C18
C13⋊C4
C3×D13
C13⋊C9
13C36
C3×C13⋊C4
C13⋊C18
C13⋊C36

Smallest permutation representation of C13⋊C36
►On 117 points
Generators in S117
(1 94 58 67 35 85 44 26 103 17 49 76 112)(2 45 113 86 77 36 50 68 18 59 104 95 27)(3 51 28 37 96 78 105 87 60 114 19 10 69)(4 106 70 79 11 97 20 38 115 29 61 52 88)(5 21 89 98 53 12 62 80 30 71 116 107 39)(6 63 40 13 108 54 117 99 72 90 31 22 81)(7 82 46 55 23 109 32 14 91 41 73 64 100)(8 33 101 110 65 24 74 56 42 47 92 83 15)(9 75 16 25 84 66 93 111 48 102 43 34 57)

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

39 conjugacy classes

class 1  2 3A3B4A4B6A6B9A···9F12A12B12C12D 13 18A···18F36A···36L39A39B
order123344669···9121212121318···1836···363939
size113111313131313···13131313131213···1313···131212

39 irreducible representations

dim1111111111212
type+++
imageC1C2C3C4C6C9C12C18C36F13C13⋊C36
kernelC13⋊C36C13⋊C18C3×C13⋊C4C13⋊C9C3×D13C13⋊C4C39D13C13C3C1
# reps112226461212

Matrix representation of C13⋊C36 ►in GL12(𝔽937)

010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
936936936936936936936936936936936936
,
846031131184663784631131108460
40240291091910914024020728
91402402072804024029109191
062653562662653562600535326535
311311846637846311311084600846
626535626626535626005353265350
053532653500626535626626535626
846008460311311846637846311311
535326535006265356266265356260
91910914024020728040240291
72804024029109191091402402
084603113118466378463113110846

G:=sub<GL(12,GF(937))| [0,0,0,0,0,0,0,0,0,0,0,936,1,0,0,0,0,0,0,0,0,0,0,936,0,1,0,0,0,0,0,0,0,0,0,936,0,0,1,0,0,0,0,0,0,0,0,936,0,0,0,1,0,0,0,0,0,0,0,936,0,0,0,0,1,0,0,0,0,0,0,936,0,0,0,0,0,1,0,0,0,0,0,936,0,0,0,0,0,0,1,0,0,0,0,936,0,0,0,0,0,0,0,1,0,0,0,936,0,0,0,0,0,0,0,0,1,0,0,936,0,0,0,0,0,0,0,0,0,1,0,936,0,0,0,0,0,0,0,0,0,0,1,936],[846,402,91,0,311,626,0,846,535,91,728,0,0,402,402,626,311,535,535,0,326,91,0,846,311,91,402,535,846,626,326,0,535,0,402,0,311,0,0,626,637,626,535,846,0,91,402,311,846,91,728,626,846,535,0,0,0,402,91,311,637,91,0,535,311,626,0,311,626,402,0,846,846,0,402,626,311,0,626,311,535,0,91,637,311,91,402,0,0,0,535,846,626,728,91,846,311,402,91,0,846,535,626,637,626,0,0,311,0,402,0,535,0,326,626,846,535,402,91,311,846,0,91,326,0,535,535,311,626,402,402,0,0,728,91,535,846,0,626,311,0,91,402,846] >;
 

C13⋊C36 in GAP, Magma, Sage, TeX 

C_{13}\rtimes C_{36}
 
% in TeX
 
G:=Group("C13:C36");
 
// GroupNames label
 
G:=SmallGroup(468,7);
 
// by ID
 
G=gap.SmallGroup(468,7);
 
# by ID
 
G:=PCGroup([5,-2,-3,-2,-3,-13,30,66,7204,1359,1814]);
 
// Polycyclic
 
G:=Group<a,b|a^13=b^36=1,b*a*b^-1=a^6>;
 
// generators/relations
 

Export

Subgroup lattice of C13⋊C36 in TeX

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