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G = C132C8order 104 = 23·13

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

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

Aliases: C132C8, C26.2C4, C52.2C2, C4.2D13, C2.Dic13, SmallGroup(104,1)

Series: Derived Chief Lower central Upper central

C1C13 — C132C8
C1C13C26C52 — C132C8
C13 — C132C8
C1C4

Generators and relations for C132C8
 G = < a,b | a13=b8=1, bab-1=a-1 >

13C8

Smallest permutation representation of C132C8
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 92 42 66 17 79 36 53)(2 104 43 78 18 91 37 65)(3 103 44 77 19 90 38 64)(4 102 45 76 20 89 39 63)(5 101 46 75 21 88 27 62)(6 100 47 74 22 87 28 61)(7 99 48 73 23 86 29 60)(8 98 49 72 24 85 30 59)(9 97 50 71 25 84 31 58)(10 96 51 70 26 83 32 57)(11 95 52 69 14 82 33 56)(12 94 40 68 15 81 34 55)(13 93 41 67 16 80 35 54)

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

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

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

C132C8 is a maximal subgroup of
C13⋊C16  C8×D13  C8⋊D13  C52.4C4  D4⋊D13  D4.D13  Q8⋊D13  C13⋊Q16  C132C24  C393C8
C132C8 is a maximal quotient of
C132C16  C393C8

32 conjugacy classes

class 1  2 4A4B8A8B8C8D13A···13F26A···26F52A···52L
order1244888813···1326···2652···52
size1111131313132···22···22···2

32 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D13Dic13C132C8
kernelC132C8C52C26C13C4C2C1
# reps11246612

Matrix representation of C132C8 in GL3(𝔽313) generated by

100
03121
023478
,
18800
0303264
03410
G:=sub<GL(3,GF(313))| [1,0,0,0,312,234,0,1,78],[188,0,0,0,303,34,0,264,10] >;

C132C8 in GAP, Magma, Sage, TeX

C_{13}\rtimes_2C_8
% in TeX

G:=Group("C13:2C8");
// GroupNames label

G:=SmallGroup(104,1);
// by ID

G=gap.SmallGroup(104,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,8,21,1539]);
// Polycyclic

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

Export

Subgroup lattice of C132C8 in TeX

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