direct product, metabelian, soluble, monomial, A-group
Aliases: A4×D13, C13⋊3(C2×A4), (C2×C26)⋊1C6, C22⋊(C3×D13), (A4×C13)⋊2C2, (C22×D13)⋊1C3, SmallGroup(312,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C26 — A4×D13 |
Generators and relations for A4×D13
G = < a,b,c,d,e | a2=b2=c3=d13=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of A4×D13
►On 52 points
Generators in S52
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 14)(11 15)(12 16)(13 17)(27 51)(28 52)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)
(1 36)(2 37)(3 38)(4 39)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 40)(25 41)(26 42)
(14 32 43)(15 33 44)(16 34 45)(17 35 46)(18 36 47)(19 37 48)(20 38 49)(21 39 50)(22 27 51)(23 28 52)(24 29 40)(25 30 41)(26 31 42)
(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 21)(15 20)(16 19)(17 18)(22 26)(23 25)(27 31)(28 30)(32 39)(33 38)(34 37)(35 36)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)
G:=sub<Sym(52)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,14)(11,15)(12,16)(13,17)(27,51)(28,52)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50), (1,36)(2,37)(3,38)(4,39)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,40)(25,41)(26,42), (14,32,43)(15,33,44)(16,34,45)(17,35,46)(18,36,47)(19,37,48)(20,38,49)(21,39,50)(22,27,51)(23,28,52)(24,29,40)(25,30,41)(26,31,42), (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)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)>;
G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,14)(11,15)(12,16)(13,17)(27,51)(28,52)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50), (1,36)(2,37)(3,38)(4,39)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,40)(25,41)(26,42), (14,32,43)(15,33,44)(16,34,45)(17,35,46)(18,36,47)(19,37,48)(20,38,49)(21,39,50)(22,27,51)(23,28,52)(24,29,40)(25,30,41)(26,31,42), (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)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47) );
G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,14),(11,15),(12,16),(13,17),(27,51),(28,52),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50)], [(1,36),(2,37),(3,38),(4,39),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,40),(25,41),(26,42)], [(14,32,43),(15,33,44),(16,34,45),(17,35,46),(18,36,47),(19,37,48),(20,38,49),(21,39,50),(22,27,51),(23,28,52),(24,29,40),(25,30,41),(26,31,42)], [(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),(14,21),(15,20),(16,19),(17,18),(22,26),(23,25),(27,31),(28,30),(32,39),(33,38),(34,37),(35,36),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 |
| size | 1 | 3 | 13 | 39 | 4 | 4 | 52 | 52 | 2 | ··· | 2 | 6 | ··· | 6 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D13 | C3×D13 | A4 | C2×A4 | A4×D13 |
| kernel | A4×D13 | A4×C13 | C22×D13 | C2×C26 | A4 | C22 | D13 | C13 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 12 | 1 | 1 | 6 |
Matrix representation of A4×D13 ►in GL5(𝔽79)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 78 | 78 | 78 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 78 | 78 | 78 |
| 0 | 0 | 1 | 0 | 0 |
| 23 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 78 | 78 | 78 |
| 0 | 0 | 0 | 1 | 0 |
| 61 | 4 | 0 | 0 | 0 |
| 78 | 66 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 19 | 13 | 0 | 0 | 0 |
| 27 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(79))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,78,0,0,1,0,78,0,0,0,0,78],[1,0,0,0,0,0,1,0,0,0,0,0,0,78,1,0,0,0,78,0,0,0,1,78,0],[23,0,0,0,0,0,23,0,0,0,0,0,1,78,0,0,0,0,78,1,0,0,0,78,0],[61,78,0,0,0,4,66,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[19,27,0,0,0,13,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
A4×D13 in GAP, Magma, Sage, TeX
A_4\times D_{13} % in TeX
G:=Group("A4xD13"); // GroupNames label
G:=SmallGroup(312,50);
// by ID
G=gap.SmallGroup(312,50);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-13,142,68,7204]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^13=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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