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## G = C4×D13order 104 = 23·13

### Direct product of C4 and D13

Aliases: C4×D13, C522C2, C4Dic13, C2.1D26, D26.2C2, Dic132C2, C26.2C22, C132(C2×C4), SmallGroup(104,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C4×D13
 Chief series C1 — C13 — C26 — D26 — C4×D13
 Lower central C13 — C4×D13
 Upper central C1 — C4

Generators and relations for C4×D13
G = < a,b,c | a4=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C4×D13
On 52 points
Generators in S52
(1 52 20 31)(2 40 21 32)(3 41 22 33)(4 42 23 34)(5 43 24 35)(6 44 25 36)(7 45 26 37)(8 46 14 38)(9 47 15 39)(10 48 16 27)(11 49 17 28)(12 50 18 29)(13 51 19 30)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(27 34)(28 33)(29 32)(30 31)(35 39)(36 38)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)

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

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

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

C4×D13 is a maximal subgroup of
C8⋊D13  D13⋊C8  C52.C4  C52⋊C4  D525C2  D42D13  D52⋊C2  D78.C2
C4×D13 is a maximal quotient of
C8⋊D13  C26.D4  D26⋊C4  D78.C2

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D13 D26 C4×D13 kernel C4×D13 Dic13 C52 D26 D13 C4 C2 C1 # reps 1 1 1 1 4 6 6 12

Matrix representation of C4×D13 in GL3(𝔽53) generated by

 30 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 52 8
,
 52 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(53))| [30,0,0,0,1,0,0,0,1],[1,0,0,0,0,52,0,1,8],[52,0,0,0,0,1,0,1,0] >;

C4×D13 in GAP, Magma, Sage, TeX

C_4\times D_{13}
% in TeX

G:=Group("C4xD13");
// GroupNames label

G:=SmallGroup(104,5);
// by ID

G=gap.SmallGroup(104,5);
# by ID

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

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

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