metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26.1D4, (C2×C52)⋊1C4, C13⋊1(C23⋊C4), (C2×D52).2C2, D13.D4⋊1C2, (C22×D13)⋊2C4, C26.1(C22⋊C4), C2.4(D13.D4), (C22×D13).14C22, (C2×C4)⋊(C13⋊C4), (C2×C26).2(C2×C4), C22.2(C2×C13⋊C4), SmallGroup(416,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.D4
G = < a,b,c,d | a26=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=a17b, dbd-1=a4b, dcd-1=a12bc-1 >
2C2
26C2
26C2
52C2
2C4
13C22
13C22
26C22
52C4
52C4
52C22
52C22
2D13
2D13
2C26
4D13
13C23
13C23
26C2×C4
26D4
26D4
26C2×C4
2C52
2D26
4D26
4D26
13C2×D4
13C22⋊C4
13C22⋊C4
2D52
2D52
13C23⋊C4
Smallest permutation representation of D26.D4
►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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 36)(28 35)(29 34)(30 33)(31 32)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(53 61)(54 60)(55 59)(56 58)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)
(1 64 32 99)(2 59 31 104)(3 54 30 83)(4 75 29 88)(5 70 28 93)(6 65 27 98)(7 60 52 103)(8 55 51 82)(9 76 50 87)(10 71 49 92)(11 66 48 97)(12 61 47 102)(13 56 46 81)(14 77 45 86)(15 72 44 91)(16 67 43 96)(17 62 42 101)(18 57 41 80)(19 78 40 85)(20 73 39 90)(21 68 38 95)(22 63 37 100)(23 58 36 79)(24 53 35 84)(25 74 34 89)(26 69 33 94)
(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 44 37 46)(28 39 36 51)(29 34 35 30)(31 50 33 40)(32 45)(38 41 52 49)(42 47 48 43)(53 102 62 83)(54 97 61 88)(55 92 60 93)(56 87 59 98)(57 82 58 103)(63 104 78 81)(64 99 77 86)(65 94 76 91)(66 89 75 96)(67 84 74 101)(68 79 73 80)(69 100 72 85)(70 95 71 90)
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,36)(28,35)(29,34)(30,33)(31,32)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(53,61)(54,60)(55,59)(56,58)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93), (1,64,32,99)(2,59,31,104)(3,54,30,83)(4,75,29,88)(5,70,28,93)(6,65,27,98)(7,60,52,103)(8,55,51,82)(9,76,50,87)(10,71,49,92)(11,66,48,97)(12,61,47,102)(13,56,46,81)(14,77,45,86)(15,72,44,91)(16,67,43,96)(17,62,42,101)(18,57,41,80)(19,78,40,85)(20,73,39,90)(21,68,38,95)(22,63,37,100)(23,58,36,79)(24,53,35,84)(25,74,34,89)(26,69,33,94), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,44,37,46)(28,39,36,51)(29,34,35,30)(31,50,33,40)(32,45)(38,41,52,49)(42,47,48,43)(53,102,62,83)(54,97,61,88)(55,92,60,93)(56,87,59,98)(57,82,58,103)(63,104,78,81)(64,99,77,86)(65,94,76,91)(66,89,75,96)(67,84,74,101)(68,79,73,80)(69,100,72,85)(70,95,71,90)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,36)(28,35)(29,34)(30,33)(31,32)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(53,61)(54,60)(55,59)(56,58)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93), (1,64,32,99)(2,59,31,104)(3,54,30,83)(4,75,29,88)(5,70,28,93)(6,65,27,98)(7,60,52,103)(8,55,51,82)(9,76,50,87)(10,71,49,92)(11,66,48,97)(12,61,47,102)(13,56,46,81)(14,77,45,86)(15,72,44,91)(16,67,43,96)(17,62,42,101)(18,57,41,80)(19,78,40,85)(20,73,39,90)(21,68,38,95)(22,63,37,100)(23,58,36,79)(24,53,35,84)(25,74,34,89)(26,69,33,94), (2,22,26,6)(3,17,25,11)(4,12,24,16)(5,7,23,21)(8,18,20,10)(9,13,19,15)(27,44,37,46)(28,39,36,51)(29,34,35,30)(31,50,33,40)(32,45)(38,41,52,49)(42,47,48,43)(53,102,62,83)(54,97,61,88)(55,92,60,93)(56,87,59,98)(57,82,58,103)(63,104,78,81)(64,99,77,86)(65,94,76,91)(66,89,75,96)(67,84,74,101)(68,79,73,80)(69,100,72,85)(70,95,71,90) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(27,36),(28,35),(29,34),(30,33),(31,32),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(53,61),(54,60),(55,59),(56,58),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93)], [(1,64,32,99),(2,59,31,104),(3,54,30,83),(4,75,29,88),(5,70,28,93),(6,65,27,98),(7,60,52,103),(8,55,51,82),(9,76,50,87),(10,71,49,92),(11,66,48,97),(12,61,47,102),(13,56,46,81),(14,77,45,86),(15,72,44,91),(16,67,43,96),(17,62,42,101),(18,57,41,80),(19,78,40,85),(20,73,39,90),(21,68,38,95),(22,63,37,100),(23,58,36,79),(24,53,35,84),(25,74,34,89),(26,69,33,94)], [(2,22,26,6),(3,17,25,11),(4,12,24,16),(5,7,23,21),(8,18,20,10),(9,13,19,15),(27,44,37,46),(28,39,36,51),(29,34,35,30),(31,50,33,40),(32,45),(38,41,52,49),(42,47,48,43),(53,102,62,83),(54,97,61,88),(55,92,60,93),(56,87,59,98),(57,82,58,103),(63,104,78,81),(64,99,77,86),(65,94,76,91),(66,89,75,96),(67,84,74,101),(68,79,73,80),(69,100,72,85),(70,95,71,90)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 26 | 26 | 52 | 4 | 52 | 52 | 52 | 52 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | D4 | C23⋊C4 | C13⋊C4 | C2×C13⋊C4 | D13.D4 | D26.D4 |
| kernel | D26.D4 | D13.D4 | C2×D52 | C2×C52 | C22×D13 | D26 | C13 | C2×C4 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 3 | 3 | 6 | 12 |
Matrix representation of D26.D4 ►in GL4(𝔽53) generated by
| 12 | 39 | 11 | 46 |
| 7 | 8 | 8 | 7 |
| 46 | 11 | 39 | 12 |
| 41 | 15 | 49 | 8 |
| 12 | 39 | 11 | 46 |
| 8 | 49 | 15 | 41 |
| 46 | 5 | 38 | 45 |
| 1 | 7 | 41 | 7 |
| 16 | 29 | 47 | 5 |
| 23 | 18 | 43 | 28 |
| 47 | 17 | 14 | 28 |
| 41 | 17 | 19 | 5 |
| 1 | 0 | 0 | 0 |
| 7 | 48 | 15 | 8 |
| 12 | 38 | 4 | 45 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(53))| [12,7,46,41,39,8,11,15,11,8,39,49,46,7,12,8],[12,8,46,1,39,49,5,7,11,15,38,41,46,41,45,7],[16,23,47,41,29,18,17,17,47,43,14,19,5,28,28,5],[1,7,12,0,0,48,38,0,0,15,4,1,0,8,45,0] >;
D26.D4 in GAP, Magma, Sage, TeX
D_{26}.D_4 % in TeX
G:=Group("D26.D4"); // GroupNames label
G:=SmallGroup(416,74);
// by ID
G=gap.SmallGroup(416,74);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,188,579,9221,3473]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^17*b,d*b*d^-1=a^4*b,d*c*d^-1=a^12*b*c^-1>;
// generators/relations
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