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G = C4×D13order 104 = 23·13

Direct product of C4 and D13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C13 — C4×D13
C1C13C26D26 — C4×D13
C13 — C4×D13
C1C4

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

13C2
13C2
13C22
13C4
13C2×C4

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 2A2B2C4A4B4C4D13A···13F26A···26F52A···52L
order1222444413···1326···2652···52
size1113131113132···22···22···2

32 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D13D26C4×D13
kernelC4×D13Dic13C52D26D13C4C2C1
# reps111146612

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

3000
010
001
,
100
001
0528
,
5200
001
010
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

Export

Subgroup lattice of C4×D13 in TeX

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