C13:2C8 - GroupNames

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

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

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

C₁₃⋊₂C₈ is a maximal subgroup of
C₁₃⋊C₁₆ C₈×D₁₃ C₈⋊D₁₃ C₅₂.₄C₄ D₄⋊D₁₃ D₄.D₁₃ Q₈⋊D₁₃ C₁₃⋊Q₁₆ C₁₃⋊₂C₂₄ C₃₉⋊₃C₈
C₁₃⋊₂C₈ is a maximal quotient of
C₁₃⋊₂C₁₆ C₃₉⋊₃C₈

32 conjugacy classes

class	1	2	4A	4B	8A	8B	8C	8D	13A	···	13F	26A	···	26F	52A	···	52L
order	1	2	4	4	8	8	8	8	13	···	13	26	···	26	52	···	52
size	1	1	1	1	13	13	13	13	2	···	2	2	···	2	2	···	2

32 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+			+	-
image	C₁	C₂	C₄	C₈	D₁₃	Dic₁₃	C₁₃⋊₂C₈
kernel	C₁₃⋊₂C₈	C₅₂	C₂₆	C₁₃	C₄	C₂	C₁
# reps	1	1	2	4	6	6	12

Matrix representation of C₁₃⋊₂C₈ ►in GL₃(𝔽₃₁₃) generated by

1	0	0
0	312	1
0	234	78

188	0	0
0	303	264
0	34	10

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

C₁₃⋊₂C₈ 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 C₁₃⋊₂C₈ in TeX

G = C13⋊2C8 order 104 = 23·13

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

G = C₁₃⋊₂C₈ order 104 = 2³·13

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