direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C26, C23⋊C39, C22⋊C78, (C2×C26)⋊6C6, (C22×C26)⋊1C3, SmallGroup(312,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C26 |
Generators and relations for A4×C26
G = < a,b,c,d | a26=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C26
►On 78 points
Generators in S78
(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)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(1 53 33)(2 54 34)(3 55 35)(4 56 36)(5 57 37)(6 58 38)(7 59 39)(8 60 40)(9 61 41)(10 62 42)(11 63 43)(12 64 44)(13 65 45)(14 66 46)(15 67 47)(16 68 48)(17 69 49)(18 70 50)(19 71 51)(20 72 52)(21 73 27)(22 74 28)(23 75 29)(24 76 30)(25 77 31)(26 78 32)
G:=sub<Sym(78)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (1,53,33)(2,54,34)(3,55,35)(4,56,36)(5,57,37)(6,58,38)(7,59,39)(8,60,40)(9,61,41)(10,62,42)(11,63,43)(12,64,44)(13,65,45)(14,66,46)(15,67,47)(16,68,48)(17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,27)(22,74,28)(23,75,29)(24,76,30)(25,77,31)(26,78,32)>;
G:=Group( (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (1,53,33)(2,54,34)(3,55,35)(4,56,36)(5,57,37)(6,58,38)(7,59,39)(8,60,40)(9,61,41)(10,62,42)(11,63,43)(12,64,44)(13,65,45)(14,66,46)(15,67,47)(16,68,48)(17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,27)(22,74,28)(23,75,29)(24,76,30)(25,77,31)(26,78,32) );
G=PermutationGroup([[(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),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78)], [(1,53,33),(2,54,34),(3,55,35),(4,56,36),(5,57,37),(6,58,38),(7,59,39),(8,60,40),(9,61,41),(10,62,42),(11,63,43),(12,64,44),(13,65,45),(14,66,46),(15,67,47),(16,68,48),(17,69,49),(18,70,50),(19,71,51),(20,72,52),(21,73,27),(22,74,28),(23,75,29),(24,76,30),(25,77,31),(26,78,32)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39X | 78A | ··· | 78X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 | A4 | C2×A4 | A4×C13 | A4×C26 |
| kernel | A4×C26 | A4×C13 | C22×C26 | C2×C26 | C2×A4 | A4 | C23 | C22 | C26 | C13 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 | 1 | 1 | 12 | 12 |
Matrix representation of A4×C26 ►in GL3(𝔽79) generated by
| 69 | 0 | 0 |
| 0 | 69 | 0 |
| 0 | 0 | 69 |
| 78 | 0 | 0 |
| 0 | 78 | 0 |
| 56 | 0 | 1 |
| 78 | 0 | 0 |
| 24 | 1 | 0 |
| 0 | 0 | 78 |
| 55 | 77 | 0 |
| 0 | 24 | 1 |
| 0 | 56 | 0 |
G:=sub<GL(3,GF(79))| [69,0,0,0,69,0,0,0,69],[78,0,56,0,78,0,0,0,1],[78,24,0,0,1,0,0,0,78],[55,0,0,77,24,56,0,1,0] >;
A4×C26 in GAP, Magma, Sage, TeX
A_4\times C_{26} % in TeX
G:=Group("A4xC26"); // GroupNames label
G:=SmallGroup(312,56);
// by ID
G=gap.SmallGroup(312,56);
# by ID
G:=PCGroup([5,-2,-3,-13,-2,2,1568,2934]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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