direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C26, C23⋊C26, C52⋊4C22, C26.11C23, C4⋊(C2×C26), (C2×C4)⋊2C26, (C2×C52)⋊6C2, C22⋊(C2×C26), (C22×C26)⋊1C2, (C2×C26)⋊2C22, C2.1(C22×C26), SmallGroup(208,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C26
G = < a,b,c | a26=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C13, C2×D4, C26, C26, C26, C52, C2×C26, C2×C26, C2×C26, C2×C52, D4×C13, C22×C26, D4×C26
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C2×C26, D4×C13, C22×C26, D4×C26
►On 104 points
Generators in S104
(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)(79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 56 44 93)(2 57 45 94)(3 58 46 95)(4 59 47 96)(5 60 48 97)(6 61 49 98)(7 62 50 99)(8 63 51 100)(9 64 52 101)(10 65 27 102)(11 66 28 103)(12 67 29 104)(13 68 30 79)(14 69 31 80)(15 70 32 81)(16 71 33 82)(17 72 34 83)(18 73 35 84)(19 74 36 85)(20 75 37 86)(21 76 38 87)(22 77 39 88)(23 78 40 89)(24 53 41 90)(25 54 42 91)(26 55 43 92)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)
G:=sub<Sym(104)| (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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,56,44,93)(2,57,45,94)(3,58,46,95)(4,59,47,96)(5,60,48,97)(6,61,49,98)(7,62,50,99)(8,63,51,100)(9,64,52,101)(10,65,27,102)(11,66,28,103)(12,67,29,104)(13,68,30,79)(14,69,31,80)(15,70,32,81)(16,71,33,82)(17,72,34,83)(18,73,35,84)(19,74,36,85)(20,75,37,86)(21,76,38,87)(22,77,39,88)(23,78,40,89)(24,53,41,90)(25,54,42,91)(26,55,43,92), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)>;
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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,56,44,93)(2,57,45,94)(3,58,46,95)(4,59,47,96)(5,60,48,97)(6,61,49,98)(7,62,50,99)(8,63,51,100)(9,64,52,101)(10,65,27,102)(11,66,28,103)(12,67,29,104)(13,68,30,79)(14,69,31,80)(15,70,32,81)(16,71,33,82)(17,72,34,83)(18,73,35,84)(19,74,36,85)(20,75,37,86)(21,76,38,87)(22,77,39,88)(23,78,40,89)(24,53,41,90)(25,54,42,91)(26,55,43,92), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88) );
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),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,56,44,93),(2,57,45,94),(3,58,46,95),(4,59,47,96),(5,60,48,97),(6,61,49,98),(7,62,50,99),(8,63,51,100),(9,64,52,101),(10,65,27,102),(11,66,28,103),(12,67,29,104),(13,68,30,79),(14,69,31,80),(15,70,32,81),(16,71,33,82),(17,72,34,83),(18,73,35,84),(19,74,36,85),(20,75,37,86),(21,76,38,87),(22,77,39,88),(23,78,40,89),(24,53,41,90),(25,54,42,91),(26,55,43,92)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88)]])
D4×C26 is a maximal subgroup of
D4⋊Dic13 C52.D4 C23⋊Dic13 D52⋊6C22 C23.18D26 C52.17D4 C23⋊D26 C52⋊2D4 Dic13⋊D4 C52⋊D4 D4⋊6D26
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26CF | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | D4 | D4×C13 |
| kernel | D4×C26 | C2×C52 | D4×C13 | C22×C26 | C2×D4 | C2×C4 | D4 | C23 | C26 | C2 |
| # reps | 1 | 1 | 4 | 2 | 12 | 12 | 48 | 24 | 2 | 24 |
Matrix representation of D4×C26 ►in GL3(𝔽53) generated by
| 6 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 52 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 52 | 0 |
| 52 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(53))| [6,0,0,0,9,0,0,0,9],[52,0,0,0,0,52,0,1,0],[52,0,0,0,0,1,0,1,0] >;
D4×C26 in GAP, Magma, Sage, TeX
D_4\times C_{26} % in TeX
G:=Group("D4xC26"); // GroupNames label
G:=SmallGroup(208,46);
// by ID
G=gap.SmallGroup(208,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-13,-2,1061]);
// Polycyclic
G:=Group<a,b,c|a^26=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations