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## G = C22×D13order 104 = 23·13

### Direct product of C22 and D13

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×D13
 Chief series C1 — C13 — D13 — D26 — C22×D13
 Lower central C13 — C22×D13
 Upper central C1 — C22

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 >

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

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

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

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

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 2A 2B 2C 2D 2E 2F 2G 13A ··· 13F 26A ··· 26R order 1 2 2 2 2 2 2 2 13 ··· 13 26 ··· 26 size 1 1 1 1 13 13 13 13 2 ··· 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D13 D26 kernel C22×D13 D26 C2×C26 C22 C2 # reps 1 6 1 6 18

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

 1 0 0 0 52 0 0 0 52
,
 52 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 52 8
,
 52 0 0 0 0 52 0 52 0
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

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