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), C26.59(C2×D4), (C2×C4).22D26, 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⋊4D26
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 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C13, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D13 [×2], C26, C26 [×2], C8⋊C22, C52 [×2], C52, D26 [×4], C2×C26, C2×C26, C13⋊2C8 [×2], D52 [×2], D52, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C22×D13, C52.4C4, D4⋊D13 [×2], Q8⋊D13 [×2], C2×D52, C13×C4○D4, D4⋊4D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8⋊C22, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, D4⋊4D26
►On 104 points
Generators in S104
(1 42 20 35)(2 43 21 36)(3 44 22 37)(4 45 23 38)(5 46 24 39)(6 47 25 27)(7 48 26 28)(8 49 14 29)(9 50 15 30)(10 51 16 31)(11 52 17 32)(12 40 18 33)(13 41 19 34)(53 83 66 96)(54 84 67 97)(55 85 68 98)(56 86 69 99)(57 87 70 100)(58 88 71 101)(59 89 72 102)(60 90 73 103)(61 91 74 104)(62 92 75 79)(63 93 76 80)(64 94 77 81)(65 95 78 82)
(1 100)(2 88)(3 102)(4 90)(5 104)(6 92)(7 80)(8 94)(9 82)(10 96)(11 84)(12 98)(13 86)(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 75)(28 63)(29 77)(30 65)(31 53)(32 67)(33 55)(34 69)(35 57)(36 71)(37 59)(38 73)(39 61)(40 68)(41 56)(42 70)(43 58)(44 72)(45 60)(46 74)(47 62)(48 76)(49 64)(50 78)(51 66)(52 54)
(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 4)(2 3)(5 13)(6 12)(7 11)(8 10)(14 16)(17 26)(18 25)(19 24)(20 23)(21 22)(27 40)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)
G:=sub<Sym(104)| (1,42,20,35)(2,43,21,36)(3,44,22,37)(4,45,23,38)(5,46,24,39)(6,47,25,27)(7,48,26,28)(8,49,14,29)(9,50,15,30)(10,51,16,31)(11,52,17,32)(12,40,18,33)(13,41,19,34)(53,83,66,96)(54,84,67,97)(55,85,68,98)(56,86,69,99)(57,87,70,100)(58,88,71,101)(59,89,72,102)(60,90,73,103)(61,91,74,104)(62,92,75,79)(63,93,76,80)(64,94,77,81)(65,95,78,82), (1,100)(2,88)(3,102)(4,90)(5,104)(6,92)(7,80)(8,94)(9,82)(10,96)(11,84)(12,98)(13,86)(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)(27,75)(28,63)(29,77)(30,65)(31,53)(32,67)(33,55)(34,69)(35,57)(36,71)(37,59)(38,73)(39,61)(40,68)(41,56)(42,70)(43,58)(44,72)(45,60)(46,74)(47,62)(48,76)(49,64)(50,78)(51,66)(52,54), (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), (1,4)(2,3)(5,13)(6,12)(7,11)(8,10)(14,16)(17,26)(18,25)(19,24)(20,23)(21,22)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)>;
G:=Group( (1,42,20,35)(2,43,21,36)(3,44,22,37)(4,45,23,38)(5,46,24,39)(6,47,25,27)(7,48,26,28)(8,49,14,29)(9,50,15,30)(10,51,16,31)(11,52,17,32)(12,40,18,33)(13,41,19,34)(53,83,66,96)(54,84,67,97)(55,85,68,98)(56,86,69,99)(57,87,70,100)(58,88,71,101)(59,89,72,102)(60,90,73,103)(61,91,74,104)(62,92,75,79)(63,93,76,80)(64,94,77,81)(65,95,78,82), (1,100)(2,88)(3,102)(4,90)(5,104)(6,92)(7,80)(8,94)(9,82)(10,96)(11,84)(12,98)(13,86)(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)(27,75)(28,63)(29,77)(30,65)(31,53)(32,67)(33,55)(34,69)(35,57)(36,71)(37,59)(38,73)(39,61)(40,68)(41,56)(42,70)(43,58)(44,72)(45,60)(46,74)(47,62)(48,76)(49,64)(50,78)(51,66)(52,54), (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), (1,4)(2,3)(5,13)(6,12)(7,11)(8,10)(14,16)(17,26)(18,25)(19,24)(20,23)(21,22)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95) );
G=PermutationGroup([(1,42,20,35),(2,43,21,36),(3,44,22,37),(4,45,23,38),(5,46,24,39),(6,47,25,27),(7,48,26,28),(8,49,14,29),(9,50,15,30),(10,51,16,31),(11,52,17,32),(12,40,18,33),(13,41,19,34),(53,83,66,96),(54,84,67,97),(55,85,68,98),(56,86,69,99),(57,87,70,100),(58,88,71,101),(59,89,72,102),(60,90,73,103),(61,91,74,104),(62,92,75,79),(63,93,76,80),(64,94,77,81),(65,95,78,82)], [(1,100),(2,88),(3,102),(4,90),(5,104),(6,92),(7,80),(8,94),(9,82),(10,96),(11,84),(12,98),(13,86),(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),(27,75),(28,63),(29,77),(30,65),(31,53),(32,67),(33,55),(34,69),(35,57),(36,71),(37,59),(38,73),(39,61),(40,68),(41,56),(42,70),(43,58),(44,72),(45,60),(46,74),(47,62),(48,76),(49,64),(50,78),(51,66),(52,54)], [(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)], [(1,4),(2,3),(5,13),(6,12),(7,11),(8,10),(14,16),(17,26),(18,25),(19,24),(20,23),(21,22),(27,40),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95)])
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⋊4D26 |
| kernel | D4⋊4D26 | 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⋊4D26 ►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⋊4D26 in GAP, Magma, Sage, TeX
D_4\rtimes_4D_{26} % in TeX
G:=Group("D4:4D26"); // 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