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G = C13⋊C36order 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

C1C13 — C13⋊C36
C1C13C39C3×D13C13⋊C18 — C13⋊C36
C13 — C13⋊C36
C1C3

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

13C2
13C4
13C6
13C9
13C12
13C18
13C36

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