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G = D4×C13order 104 = 23·13

Direct product of C13 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C13, C4⋊C26, C523C2, C22⋊C26, C26.6C22, (C2×C26)⋊1C2, C2.1(C2×C26), SmallGroup(104,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C13
C1C2C26C2×C26 — D4×C13
C1C2 — D4×C13
C1C26 — D4×C13

Generators and relations for D4×C13
 G = < a,b,c | a13=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C26
2C26

Smallest permutation representation of D4×C13
On 52 points
Generators in S52
(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)
(1 16 40 29)(2 17 41 30)(3 18 42 31)(4 19 43 32)(5 20 44 33)(6 21 45 34)(7 22 46 35)(8 23 47 36)(9 24 48 37)(10 25 49 38)(11 26 50 39)(12 14 51 27)(13 15 52 28)
(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)

G:=sub<Sym(52)| (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), (1,16,40,29)(2,17,41,30)(3,18,42,31)(4,19,43,32)(5,20,44,33)(6,21,45,34)(7,22,46,35)(8,23,47,36)(9,24,48,37)(10,25,49,38)(11,26,50,39)(12,14,51,27)(13,15,52,28), (14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)>;

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), (1,16,40,29)(2,17,41,30)(3,18,42,31)(4,19,43,32)(5,20,44,33)(6,21,45,34)(7,22,46,35)(8,23,47,36)(9,24,48,37)(10,25,49,38)(11,26,50,39)(12,14,51,27)(13,15,52,28), (14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39) );

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)], [(1,16,40,29),(2,17,41,30),(3,18,42,31),(4,19,43,32),(5,20,44,33),(6,21,45,34),(7,22,46,35),(8,23,47,36),(9,24,48,37),(10,25,49,38),(11,26,50,39),(12,14,51,27),(13,15,52,28)], [(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39)])

D4×C13 is a maximal subgroup of   D4⋊D13  D4.D13  D42D13

65 conjugacy classes

class 1 2A2B2C 4 13A···13L26A···26L26M···26AJ52A···52L
order1222413···1326···2626···2652···52
size112221···11···12···22···2

65 irreducible representations

dim11111122
type++++
imageC1C2C2C13C26C26D4D4×C13
kernelD4×C13C52C2×C26D4C4C22C13C1
# reps112121224112

Matrix representation of D4×C13 in GL2(𝔽53) generated by

150
015
,
052
10
,
10
052
G:=sub<GL(2,GF(53))| [15,0,0,15],[0,1,52,0],[1,0,0,52] >;

D4×C13 in GAP, Magma, Sage, TeX

D_4\times C_{13}
% in TeX

G:=Group("D4xC13");
// GroupNames label

G:=SmallGroup(104,10);
// by ID

G=gap.SmallGroup(104,10);
# by ID

G:=PCGroup([4,-2,-2,-13,-2,433]);
// Polycyclic

G:=Group<a,b,c|a^13=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C13 in TeX

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