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G = C13⋊C8order 104 = 23·13

The semidirect product of C13 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C13⋊C8, C26.C4, Dic13.2C2, C2.(C13⋊C4), SmallGroup(104,3)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C8
C1C13C26Dic13 — C13⋊C8
C13 — C13⋊C8
C1C2

Generators and relations for C13⋊C8
 G = < a,b | a13=b8=1, bab-1=a5 >

13C4
13C8

Character table of C13⋊C8

 class 124A4B8A8B8C8D13A13B13C26A26B26C
 size 11131313131313444444
ρ111111111111111    trivial
ρ21111-1-1-1-1111111    linear of order 2
ρ311-1-1-ii-ii111111    linear of order 4
ρ411-1-1i-ii-i111111    linear of order 4
ρ51-1-iiζ85ζ87ζ8ζ83111-1-1-1    linear of order 8
ρ61-1-iiζ8ζ83ζ85ζ87111-1-1-1    linear of order 8
ρ71-1i-iζ83ζ8ζ87ζ85111-1-1-1    linear of order 8
ρ81-1i-iζ87ζ85ζ83ζ8111-1-1-1    linear of order 8
ρ944000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132    orthogonal lifted from C13⋊C4
ρ1044000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513ζ139137136134    orthogonal lifted from C13⋊C4
ρ1144000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13111310133132ζ131213813513    orthogonal lifted from C13⋊C4
ρ124-4000000ζ131213813513ζ13111310133132ζ13913713613413111310133132131213813513139137136134    symplectic faithful, Schur index 2
ρ134-4000000ζ13111310133132ζ139137136134ζ13121381351313913713613413111310133132131213813513    symplectic faithful, Schur index 2
ρ144-4000000ζ139137136134ζ131213813513ζ1311131013313213121381351313913713613413111310133132    symplectic faithful, Schur index 2

Smallest permutation representation of C13⋊C8
Regular action on 104 points
Generators in S104
(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)
(1 96 40 78 26 91 27 65)(2 104 52 70 14 86 39 57)(3 99 51 75 15 81 38 62)(4 94 50 67 16 89 37 54)(5 102 49 72 17 84 36 59)(6 97 48 77 18 79 35 64)(7 92 47 69 19 87 34 56)(8 100 46 74 20 82 33 61)(9 95 45 66 21 90 32 53)(10 103 44 71 22 85 31 58)(11 98 43 76 23 80 30 63)(12 93 42 68 24 88 29 55)(13 101 41 73 25 83 28 60)

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

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

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

C13⋊C8 is a maximal subgroup of   D13⋊C8  C52.C4  C13⋊M4(2)  C13⋊C24  C39⋊C8
C13⋊C8 is a maximal quotient of   C13⋊C16  C39⋊C8

Matrix representation of C13⋊C8 in GL4(𝔽5) generated by

2140
3323
4402
1323
,
0012
0044
1023
0123
G:=sub<GL(4,GF(5))| [2,3,4,1,1,3,4,3,4,2,0,2,0,3,2,3],[0,0,1,0,0,0,0,1,1,4,2,2,2,4,3,3] >;

C13⋊C8 in GAP, Magma, Sage, TeX

C_{13}\rtimes C_8
% in TeX

G:=Group("C13:C8");
// GroupNames label

G:=SmallGroup(104,3);
// by ID

G=gap.SmallGroup(104,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,8,21,1027,775]);
// Polycyclic

G:=Group<a,b|a^13=b^8=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C13⋊C8 in TeX
Character table of C13⋊C8 in TeX

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