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
| C13 — C22×D13 |
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 D52⋊5C2 D4⋊2D13 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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