Copied to
clipboard

G = C22×D13order 104 = 23·13

Direct product of C22 and D13

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

Aliases: C22×D13, C13⋊C23, C26⋊C22, (C2×C26)⋊3C2, SmallGroup(104,13)

Series: Derived Chief Lower central Upper central

C1C13 — C22×D13
C1C13D13D26 — C22×D13
C13 — C22×D13
C1C22

Generators and relations for C22×D13
 G = < a,b,c,d | a2=b2=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

13C2
13C2
13C2
13C2
13C22
13C22
13C22
13C22
13C22
13C22
13C23

Smallest permutation representation of C22×D13
On 52 points
Generators in S52
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 40)(12 41)(13 42)(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)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 14)(9 15)(10 16)(11 17)(12 18)(13 19)(27 50)(28 51)(29 52)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)
(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 42)(2 41)(3 40)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(26 39)

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

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

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

C22×D13 is a maximal subgroup of   D26⋊C4  D13.D4  D13⋊A4
C22×D13 is a maximal quotient of   D525C2  D42D13  D52⋊C2

32 conjugacy classes

class 1 2A2B2C2D2E2F2G13A···13F26A···26R
order1222222213···1326···26
size1111131313132···22···2

32 irreducible representations

dim11122
type+++++
imageC1C2C2D13D26
kernelC22×D13D26C2×C26C22C2
# reps161618

Matrix representation of C22×D13 in GL3(𝔽53) generated by

100
0520
0052
,
5200
010
001
,
100
001
0528
,
5200
0052
0520
G:=sub<GL(3,GF(53))| [1,0,0,0,52,0,0,0,52],[52,0,0,0,1,0,0,0,1],[1,0,0,0,0,52,0,1,8],[52,0,0,0,0,52,0,52,0] >;

C22×D13 in GAP, Magma, Sage, TeX

C_2^2\times D_{13}
% in TeX

G:=Group("C2^2xD13");
// GroupNames label

G:=SmallGroup(104,13);
// by ID

G=gap.SmallGroup(104,13);
# by ID

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

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

Export

Subgroup lattice of C22×D13 in TeX

׿
×
𝔽