metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26.3Q8, D26.10D4, C26.5C42, (C2×C52)⋊2C4, D13.(C4⋊C4), C26.7(C4⋊C4), D26.7(C2×C4), (C2×Dic13)⋊6C4, C13⋊(C2.C42), D13.(C22⋊C4), C2.3(C52⋊C4), C26.4(C22⋊C4), C2.2(D13.D4), (C22×D13).35C22, (C2×C13⋊C4)⋊C4, C2.5(C4×C13⋊C4), (C2×C4)⋊2(C13⋊C4), (C2×C4×D13).9C2, (C2×C26).9(C2×C4), (C22×C13⋊C4).1C2, C22.13(C2×C13⋊C4), SmallGroup(416,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.Q8
G = < a,b,c,d | a26=b2=c4=1, d2=a12bc2, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=a13c-1 >
Subgroups: 604 in 76 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C22×C4, D13, C26, C2.C42, Dic13, C52, C13⋊C4, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×C52, C2×C13⋊C4, C2×C13⋊C4, C22×D13, C2×C4×D13, C22×C13⋊C4, D26.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C13⋊C4, C2×C13⋊C4, C4×C13⋊C4, C52⋊C4, D13.D4, D26.Q8
►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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)
(1 49 66 89)(2 50 67 90)(3 51 68 91)(4 52 69 92)(5 27 70 93)(6 28 71 94)(7 29 72 95)(8 30 73 96)(9 31 74 97)(10 32 75 98)(11 33 76 99)(12 34 77 100)(13 35 78 101)(14 36 53 102)(15 37 54 103)(16 38 55 104)(17 39 56 79)(18 40 57 80)(19 41 58 81)(20 42 59 82)(21 43 60 83)(22 44 61 84)(23 45 62 85)(24 46 63 86)(25 47 64 87)(26 48 65 88)
(2 22 26 6)(3 17 25 11)(4 12 24 16)(5 7 23 21)(8 18 20 10)(9 13 19 15)(27 82 45 96)(28 103 44 101)(29 98 43 80)(30 93 42 85)(31 88 41 90)(32 83 40 95)(33 104 39 100)(34 99 38 79)(35 94 37 84)(36 89)(46 91 52 87)(47 86 51 92)(48 81 50 97)(49 102)(54 74 78 58)(55 69 77 63)(56 64 76 68)(57 59 75 73)(60 70 72 62)(61 65 71 67)
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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98), (1,49,66,89)(2,50,67,90)(3,51,68,91)(4,52,69,92)(5,27,70,93)(6,28,71,94)(7,29,72,95)(8,30,73,96)(9,31,74,97)(10,32,75,98)(11,33,76,99)(12,34,77,100)(13,35,78,101)(14,36,53,102)(15,37,54,103)(16,38,55,104)(17,39,56,79)(18,40,57,80)(19,41,58,81)(20,42,59,82)(21,43,60,83)(22,44,61,84)(23,45,62,85)(24,46,63,86)(25,47,64,87)(26,48,65,88), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,82,45,96)(28,103,44,101)(29,98,43,80)(30,93,42,85)(31,88,41,90)(32,83,40,95)(33,104,39,100)(34,99,38,79)(35,94,37,84)(36,89)(46,91,52,87)(47,86,51,92)(48,81,50,97)(49,102)(54,74,78,58)(55,69,77,63)(56,64,76,68)(57,59,75,73)(60,70,72,62)(61,65,71,67)>;
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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98), (1,49,66,89)(2,50,67,90)(3,51,68,91)(4,52,69,92)(5,27,70,93)(6,28,71,94)(7,29,72,95)(8,30,73,96)(9,31,74,97)(10,32,75,98)(11,33,76,99)(12,34,77,100)(13,35,78,101)(14,36,53,102)(15,37,54,103)(16,38,55,104)(17,39,56,79)(18,40,57,80)(19,41,58,81)(20,42,59,82)(21,43,60,83)(22,44,61,84)(23,45,62,85)(24,46,63,86)(25,47,64,87)(26,48,65,88), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,82,45,96)(28,103,44,101)(29,98,43,80)(30,93,42,85)(31,88,41,90)(32,83,40,95)(33,104,39,100)(34,99,38,79)(35,94,37,84)(36,89)(46,91,52,87)(47,86,51,92)(48,81,50,97)(49,102)(54,74,78,58)(55,69,77,63)(56,64,76,68)(57,59,75,73)(60,70,72,62)(61,65,71,67) );
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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98)], [(1,49,66,89),(2,50,67,90),(3,51,68,91),(4,52,69,92),(5,27,70,93),(6,28,71,94),(7,29,72,95),(8,30,73,96),(9,31,74,97),(10,32,75,98),(11,33,76,99),(12,34,77,100),(13,35,78,101),(14,36,53,102),(15,37,54,103),(16,38,55,104),(17,39,56,79),(18,40,57,80),(19,41,58,81),(20,42,59,82),(21,43,60,83),(22,44,61,84),(23,45,62,85),(24,46,63,86),(25,47,64,87),(26,48,65,88)], [(2,22,26,6),(3,17,25,11),(4,12,24,16),(5,7,23,21),(8,18,20,10),(9,13,19,15),(27,82,45,96),(28,103,44,101),(29,98,43,80),(30,93,42,85),(31,88,41,90),(32,83,40,95),(33,104,39,100),(34,99,38,79),(35,94,37,84),(36,89),(46,91,52,87),(47,86,51,92),(48,81,50,97),(49,102),(54,74,78,58),(55,69,77,63),(56,64,76,68),(57,59,75,73),(60,70,72,62),(61,65,71,67)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4L | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | 2 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | C13⋊C4 | C2×C13⋊C4 | C4×C13⋊C4 | C52⋊C4 | D13.D4 |
| kernel | D26.Q8 | C2×C4×D13 | C22×C13⋊C4 | C2×Dic13 | C2×C52 | C2×C13⋊C4 | D26 | D26 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 3 | 1 | 3 | 3 | 6 | 6 | 6 |
Matrix representation of D26.Q8 ►in GL8(𝔽53)
| 52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 28 | 23 | 14 |
| 0 | 0 | 0 | 0 | 39 | 38 | 38 | 39 |
| 0 | 0 | 0 | 0 | 14 | 23 | 28 | 22 |
| 0 | 0 | 0 | 0 | 31 | 24 | 16 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 28 | 23 | 14 |
| 0 | 0 | 0 | 0 | 38 | 16 | 24 | 31 |
| 0 | 0 | 0 | 0 | 14 | 36 | 29 | 15 |
| 0 | 0 | 0 | 0 | 52 | 39 | 31 | 39 |
| 32 | 43 | 0 | 0 | 0 | 0 | 0 | 0 |
| 44 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 36 | 29 | 15 |
| 0 | 0 | 0 | 0 | 31 | 24 | 16 | 38 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(53))| [52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,39,14,31,0,0,0,0,28,38,23,24,0,0,0,0,23,38,28,16,0,0,0,0,14,39,22,38],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,38,14,52,0,0,0,0,28,16,36,39,0,0,0,0,23,24,29,31,0,0,0,0,14,31,15,39],[32,44,0,0,0,0,0,0,43,21,0,0,0,0,0,0,0,0,10,20,0,0,0,0,0,0,40,43,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52],[1,17,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,23,15,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,14,31,0,0,0,0,0,0,36,24,0,0,0,0,0,0,29,16,1,0,0,0,0,0,15,38,0] >;
D26.Q8 in GAP, Magma, Sage, TeX
D_{26}.Q_8 % in TeX
G:=Group("D26.Q8"); // GroupNames label
G:=SmallGroup(416,81);
// by ID
G=gap.SmallGroup(416,81);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,9221,3473]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^12*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^13*c^-1>;
// generators/relations