metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊8D26, Q8⋊7D26, D52⋊11C22, C26.12C24, C52.26C23, D26.7C23, C13⋊22+ 1+4, Dic26⋊12C22, Dic13.7C23, (C2×C4)⋊4D26, C4○D4⋊3D13, (D4×D13)⋊5C2, (C2×D52)⋊13C2, (C2×C52)⋊5C22, D52⋊5C2⋊8C2, D52⋊C2⋊5C2, (C4×D13)⋊2C22, (D4×C13)⋊9C22, C13⋊D4⋊5C22, (C2×C26).4C23, (Q8×C13)⋊8C22, C4.33(C22×D13), C2.13(C23×D13), (C22×D13)⋊4C22, C22.3(C22×D13), (C13×C4○D4)⋊4C2, SmallGroup(416,223)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊8D26
G = < a,b,c,d | a4=b2=c26=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >
Subgroups: 1264 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C13, C2×D4, C4○D4, C4○D4, D13, C26, C26, 2+ 1+4, Dic13, C52, C52, D26, D26, C2×C26, Dic26, C4×D13, D52, C13⋊D4, C2×C52, D4×C13, Q8×C13, C22×D13, C2×D52, D52⋊5C2, D4×D13, D52⋊C2, C13×C4○D4, D4⋊8D26
Quotients: C1, C2, C22, C23, C24, D13, 2+ 1+4, D26, C22×D13, C23×D13, D4⋊8D26
►On 104 points
Generators in S104
(1 46 17 35)(2 47 18 36)(3 48 19 37)(4 49 20 38)(5 50 21 39)(6 51 22 27)(7 52 23 28)(8 40 24 29)(9 41 25 30)(10 42 26 31)(11 43 14 32)(12 44 15 33)(13 45 16 34)(53 102 66 89)(54 103 67 90)(55 104 68 91)(56 79 69 92)(57 80 70 93)(58 81 71 94)(59 82 72 95)(60 83 73 96)(61 84 74 97)(62 85 75 98)(63 86 76 99)(64 87 77 100)(65 88 78 101)
(1 84)(2 98)(3 86)(4 100)(5 88)(6 102)(7 90)(8 104)(9 92)(10 80)(11 94)(12 82)(13 96)(14 81)(15 95)(16 83)(17 97)(18 85)(19 99)(20 87)(21 101)(22 89)(23 103)(24 91)(25 79)(26 93)(27 66)(28 54)(29 68)(30 56)(31 70)(32 58)(33 72)(34 60)(35 74)(36 62)(37 76)(38 64)(39 78)(40 55)(41 69)(42 57)(43 71)(44 59)(45 73)(46 61)(47 75)(48 63)(49 77)(50 65)(51 53)(52 67)
(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)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 20)(15 19)(16 18)(21 26)(22 25)(23 24)(27 41)(28 40)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(53 56)(54 55)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(79 89)(80 88)(81 87)(82 86)(83 85)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)
G:=sub<Sym(104)| (1,46,17,35)(2,47,18,36)(3,48,19,37)(4,49,20,38)(5,50,21,39)(6,51,22,27)(7,52,23,28)(8,40,24,29)(9,41,25,30)(10,42,26,31)(11,43,14,32)(12,44,15,33)(13,45,16,34)(53,102,66,89)(54,103,67,90)(55,104,68,91)(56,79,69,92)(57,80,70,93)(58,81,71,94)(59,82,72,95)(60,83,73,96)(61,84,74,97)(62,85,75,98)(63,86,76,99)(64,87,77,100)(65,88,78,101), (1,84)(2,98)(3,86)(4,100)(5,88)(6,102)(7,90)(8,104)(9,92)(10,80)(11,94)(12,82)(13,96)(14,81)(15,95)(16,83)(17,97)(18,85)(19,99)(20,87)(21,101)(22,89)(23,103)(24,91)(25,79)(26,93)(27,66)(28,54)(29,68)(30,56)(31,70)(32,58)(33,72)(34,60)(35,74)(36,62)(37,76)(38,64)(39,78)(40,55)(41,69)(42,57)(43,71)(44,59)(45,73)(46,61)(47,75)(48,63)(49,77)(50,65)(51,53)(52,67), (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), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,41)(28,40)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(53,56)(54,55)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(79,89)(80,88)(81,87)(82,86)(83,85)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)>;
G:=Group( (1,46,17,35)(2,47,18,36)(3,48,19,37)(4,49,20,38)(5,50,21,39)(6,51,22,27)(7,52,23,28)(8,40,24,29)(9,41,25,30)(10,42,26,31)(11,43,14,32)(12,44,15,33)(13,45,16,34)(53,102,66,89)(54,103,67,90)(55,104,68,91)(56,79,69,92)(57,80,70,93)(58,81,71,94)(59,82,72,95)(60,83,73,96)(61,84,74,97)(62,85,75,98)(63,86,76,99)(64,87,77,100)(65,88,78,101), (1,84)(2,98)(3,86)(4,100)(5,88)(6,102)(7,90)(8,104)(9,92)(10,80)(11,94)(12,82)(13,96)(14,81)(15,95)(16,83)(17,97)(18,85)(19,99)(20,87)(21,101)(22,89)(23,103)(24,91)(25,79)(26,93)(27,66)(28,54)(29,68)(30,56)(31,70)(32,58)(33,72)(34,60)(35,74)(36,62)(37,76)(38,64)(39,78)(40,55)(41,69)(42,57)(43,71)(44,59)(45,73)(46,61)(47,75)(48,63)(49,77)(50,65)(51,53)(52,67), (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), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,41)(28,40)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(53,56)(54,55)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(79,89)(80,88)(81,87)(82,86)(83,85)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98) );
G=PermutationGroup([[(1,46,17,35),(2,47,18,36),(3,48,19,37),(4,49,20,38),(5,50,21,39),(6,51,22,27),(7,52,23,28),(8,40,24,29),(9,41,25,30),(10,42,26,31),(11,43,14,32),(12,44,15,33),(13,45,16,34),(53,102,66,89),(54,103,67,90),(55,104,68,91),(56,79,69,92),(57,80,70,93),(58,81,71,94),(59,82,72,95),(60,83,73,96),(61,84,74,97),(62,85,75,98),(63,86,76,99),(64,87,77,100),(65,88,78,101)], [(1,84),(2,98),(3,86),(4,100),(5,88),(6,102),(7,90),(8,104),(9,92),(10,80),(11,94),(12,82),(13,96),(14,81),(15,95),(16,83),(17,97),(18,85),(19,99),(20,87),(21,101),(22,89),(23,103),(24,91),(25,79),(26,93),(27,66),(28,54),(29,68),(30,56),(31,70),(32,58),(33,72),(34,60),(35,74),(36,62),(37,76),(38,64),(39,78),(40,55),(41,69),(42,57),(43,71),(44,59),(45,73),(46,61),(47,75),(48,63),(49,77),(50,65),(51,53),(52,67)], [(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)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,20),(15,19),(16,18),(21,26),(22,25),(23,24),(27,41),(28,40),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(53,56),(54,55),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(79,89),(80,88),(81,87),(82,86),(83,85),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 26 | ··· | 26 | 2 | 2 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D13 | D26 | D26 | D26 | 2+ 1+4 | D4⋊8D26 |
| kernel | D4⋊8D26 | C2×D52 | D52⋊5C2 | D4×D13 | D52⋊C2 | C13×C4○D4 | C4○D4 | C2×C4 | D4 | Q8 | C13 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 6 | 18 | 18 | 6 | 1 | 12 |
Matrix representation of D4⋊8D26 ►in GL4(𝔽53) generated by
| 52 | 10 | 0 | 0 |
| 21 | 1 | 0 | 0 |
| 25 | 50 | 29 | 43 |
| 16 | 16 | 10 | 24 |
| 52 | 10 | 19 | 40 |
| 21 | 1 | 34 | 33 |
| 25 | 50 | 29 | 43 |
| 16 | 16 | 10 | 24 |
| 41 | 6 | 0 | 0 |
| 2 | 21 | 0 | 0 |
| 40 | 5 | 50 | 47 |
| 43 | 2 | 6 | 47 |
| 1 | 0 | 0 | 0 |
| 32 | 52 | 0 | 0 |
| 28 | 35 | 39 | 34 |
| 37 | 17 | 27 | 14 |
G:=sub<GL(4,GF(53))| [52,21,25,16,10,1,50,16,0,0,29,10,0,0,43,24],[52,21,25,16,10,1,50,16,19,34,29,10,40,33,43,24],[41,2,40,43,6,21,5,2,0,0,50,6,0,0,47,47],[1,32,28,37,0,52,35,17,0,0,39,27,0,0,34,14] >;
D4⋊8D26 in GAP, Magma, Sage, TeX
D_4\rtimes_8D_{26} % in TeX
G:=Group("D4:8D26"); // GroupNames label
G:=SmallGroup(416,223);
// by ID
G=gap.SmallGroup(416,223);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,579,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^26=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations