metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D26, Q8⋊4D26, C52.49D4, C52.17C23, D52.11C22, D4⋊D13⋊6C2, C4○D4⋊1D13, Q8⋊D13⋊6C2, (C2×C26).8D4, (C2×D52)⋊10C2, C13⋊5(C8⋊C22), (C2×C4).22D26, C26.59(C2×D4), C52.4C4⋊9C2, C13⋊2C8⋊4C22, (D4×C13)⋊4C22, (Q8×C13)⋊4C22, C4.24(C13⋊D4), (C2×C52).42C22, C4.17(C22×D13), C22.5(C13⋊D4), (C13×C4○D4)⋊1C2, C2.23(C2×C13⋊D4), SmallGroup(416,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D26
G = < a,b,c,d | a4=b2=c26=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 536 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, C52, C52, D26, C2×C26, C2×C26, C13⋊2C8, D52, D52, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C22×D13, C52.4C4, D4⋊D13, Q8⋊D13, C2×D52, C13×C4○D4, D4⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C13⋊D4, C22×D13, C2×C13⋊D4, D4⋊D26
►On 104 points
Generators in S104
(1 41 17 32)(2 42 18 33)(3 43 19 34)(4 44 20 35)(5 45 21 36)(6 46 22 37)(7 47 23 38)(8 48 24 39)(9 49 25 27)(10 50 26 28)(11 51 14 29)(12 52 15 30)(13 40 16 31)(53 96 66 83)(54 97 67 84)(55 98 68 85)(56 99 69 86)(57 100 70 87)(58 101 71 88)(59 102 72 89)(60 103 73 90)(61 104 74 91)(62 79 75 92)(63 80 76 93)(64 81 77 94)(65 82 78 95)
(1 89)(2 103)(3 91)(4 79)(5 93)(6 81)(7 95)(8 83)(9 97)(10 85)(11 99)(12 87)(13 101)(14 86)(15 100)(16 88)(17 102)(18 90)(19 104)(20 92)(21 80)(22 94)(23 82)(24 96)(25 84)(26 98)(27 67)(28 55)(29 69)(30 57)(31 71)(32 59)(33 73)(34 61)(35 75)(36 63)(37 77)(38 65)(39 53)(40 58)(41 72)(42 60)(43 74)(44 62)(45 76)(46 64)(47 78)(48 66)(49 54)(50 68)(51 56)(52 70)
(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 9)(2 8)(3 7)(4 6)(10 13)(11 12)(14 15)(16 26)(17 25)(18 24)(19 23)(20 22)(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)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(78 104)
G:=sub<Sym(104)| (1,41,17,32)(2,42,18,33)(3,43,19,34)(4,44,20,35)(5,45,21,36)(6,46,22,37)(7,47,23,38)(8,48,24,39)(9,49,25,27)(10,50,26,28)(11,51,14,29)(12,52,15,30)(13,40,16,31)(53,96,66,83)(54,97,67,84)(55,98,68,85)(56,99,69,86)(57,100,70,87)(58,101,71,88)(59,102,72,89)(60,103,73,90)(61,104,74,91)(62,79,75,92)(63,80,76,93)(64,81,77,94)(65,82,78,95), (1,89)(2,103)(3,91)(4,79)(5,93)(6,81)(7,95)(8,83)(9,97)(10,85)(11,99)(12,87)(13,101)(14,86)(15,100)(16,88)(17,102)(18,90)(19,104)(20,92)(21,80)(22,94)(23,82)(24,96)(25,84)(26,98)(27,67)(28,55)(29,69)(30,57)(31,71)(32,59)(33,73)(34,61)(35,75)(36,63)(37,77)(38,65)(39,53)(40,58)(41,72)(42,60)(43,74)(44,62)(45,76)(46,64)(47,78)(48,66)(49,54)(50,68)(51,56)(52,70), (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), (1,9)(2,8)(3,7)(4,6)(10,13)(11,12)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(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)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(78,104)>;
G:=Group( (1,41,17,32)(2,42,18,33)(3,43,19,34)(4,44,20,35)(5,45,21,36)(6,46,22,37)(7,47,23,38)(8,48,24,39)(9,49,25,27)(10,50,26,28)(11,51,14,29)(12,52,15,30)(13,40,16,31)(53,96,66,83)(54,97,67,84)(55,98,68,85)(56,99,69,86)(57,100,70,87)(58,101,71,88)(59,102,72,89)(60,103,73,90)(61,104,74,91)(62,79,75,92)(63,80,76,93)(64,81,77,94)(65,82,78,95), (1,89)(2,103)(3,91)(4,79)(5,93)(6,81)(7,95)(8,83)(9,97)(10,85)(11,99)(12,87)(13,101)(14,86)(15,100)(16,88)(17,102)(18,90)(19,104)(20,92)(21,80)(22,94)(23,82)(24,96)(25,84)(26,98)(27,67)(28,55)(29,69)(30,57)(31,71)(32,59)(33,73)(34,61)(35,75)(36,63)(37,77)(38,65)(39,53)(40,58)(41,72)(42,60)(43,74)(44,62)(45,76)(46,64)(47,78)(48,66)(49,54)(50,68)(51,56)(52,70), (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), (1,9)(2,8)(3,7)(4,6)(10,13)(11,12)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(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)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(78,104) );
G=PermutationGroup([[(1,41,17,32),(2,42,18,33),(3,43,19,34),(4,44,20,35),(5,45,21,36),(6,46,22,37),(7,47,23,38),(8,48,24,39),(9,49,25,27),(10,50,26,28),(11,51,14,29),(12,52,15,30),(13,40,16,31),(53,96,66,83),(54,97,67,84),(55,98,68,85),(56,99,69,86),(57,100,70,87),(58,101,71,88),(59,102,72,89),(60,103,73,90),(61,104,74,91),(62,79,75,92),(63,80,76,93),(64,81,77,94),(65,82,78,95)], [(1,89),(2,103),(3,91),(4,79),(5,93),(6,81),(7,95),(8,83),(9,97),(10,85),(11,99),(12,87),(13,101),(14,86),(15,100),(16,88),(17,102),(18,90),(19,104),(20,92),(21,80),(22,94),(23,82),(24,96),(25,84),(26,98),(27,67),(28,55),(29,69),(30,57),(31,71),(32,59),(33,73),(34,61),(35,75),(36,63),(37,77),(38,65),(39,53),(40,58),(41,72),(42,60),(43,74),(44,62),(45,76),(46,64),(47,78),(48,66),(49,54),(50,68),(51,56),(52,70)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,13),(11,12),(14,15),(16,26),(17,25),(18,24),(19,23),(20,22),(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),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(78,104)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 52 | 52 | 2 | 2 | 4 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | D26 | C13⋊D4 | C13⋊D4 | C8⋊C22 | D4⋊D26 |
| kernel | D4⋊D26 | C52.4C4 | D4⋊D13 | Q8⋊D13 | C2×D52 | C13×C4○D4 | C52 | C2×C26 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 12 |
Matrix representation of D4⋊D26 ►in GL6(𝔽313)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 241 | 0 | 0 |
| 0 | 0 | 67 | 292 | 0 | 0 |
| 0 | 0 | 31 | 2 | 222 | 176 |
| 0 | 0 | 289 | 142 | 113 | 91 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 171 | 182 | 274 |
| 0 | 0 | 261 | 202 | 227 | 106 |
| 0 | 0 | 265 | 39 | 244 | 80 |
| 0 | 0 | 310 | 206 | 209 | 80 |
| 81 | 81 | 0 | 0 | 0 | 0 |
| 232 | 147 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 210 | 2 | 312 | 0 |
| 0 | 0 | 6 | 299 | 0 | 312 |
| 232 | 232 | 0 | 0 | 0 | 0 |
| 166 | 81 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 131 | 312 | 0 | 0 |
| 0 | 0 | 186 | 70 | 222 | 176 |
| 0 | 0 | 11 | 250 | 81 | 91 |
G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,67,31,289,0,0,241,292,2,142,0,0,0,0,222,113,0,0,0,0,176,91],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,261,265,310,0,0,171,202,39,206,0,0,182,227,244,209,0,0,274,106,80,80],[81,232,0,0,0,0,81,147,0,0,0,0,0,0,1,0,210,6,0,0,0,1,2,299,0,0,0,0,312,0,0,0,0,0,0,312],[232,166,0,0,0,0,232,81,0,0,0,0,0,0,1,131,186,11,0,0,0,312,70,250,0,0,0,0,222,81,0,0,0,0,176,91] >;
D4⋊D26 in GAP, Magma, Sage, TeX
D_4\rtimes D_{26} % in TeX
G:=Group("D4:D26"); // GroupNames label
G:=SmallGroup(416,170);
// by ID
G=gap.SmallGroup(416,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,579,159,69,13829]);
// 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,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations