metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D13, C4.2D26, C26.8D4, C13⋊2SD16, Dic26⋊2C2, C52.2C22, C13⋊2C8⋊2C2, (D4×C13).1C2, C2.5(C13⋊D4), SmallGroup(208,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D13
G = < a,b,c,d | a4=b2=c13=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D13
►On 104 points
Generators in S104
(1 51 16 30)(2 52 17 31)(3 40 18 32)(4 41 19 33)(5 42 20 34)(6 43 21 35)(7 44 22 36)(8 45 23 37)(9 46 24 38)(10 47 25 39)(11 48 26 27)(12 49 14 28)(13 50 15 29)(53 79 66 92)(54 80 67 93)(55 81 68 94)(56 82 69 95)(57 83 70 96)(58 84 71 97)(59 85 72 98)(60 86 73 99)(61 87 74 100)(62 88 75 101)(63 89 76 102)(64 90 77 103)(65 91 78 104)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 27)(12 28)(13 29)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(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 16 65)(2 77 17 64)(3 76 18 63)(4 75 19 62)(5 74 20 61)(6 73 21 60)(7 72 22 59)(8 71 23 58)(9 70 24 57)(10 69 25 56)(11 68 26 55)(12 67 14 54)(13 66 15 53)(27 94 48 81)(28 93 49 80)(29 92 50 79)(30 104 51 91)(31 103 52 90)(32 102 40 89)(33 101 41 88)(34 100 42 87)(35 99 43 86)(36 98 44 85)(37 97 45 84)(38 96 46 83)(39 95 47 82)
G:=sub<Sym(104)| (1,51,16,30)(2,52,17,31)(3,40,18,32)(4,41,19,33)(5,42,20,34)(6,43,21,35)(7,44,22,36)(8,45,23,37)(9,46,24,38)(10,47,25,39)(11,48,26,27)(12,49,14,28)(13,50,15,29)(53,79,66,92)(54,80,67,93)(55,81,68,94)(56,82,69,95)(57,83,70,96)(58,84,71,97)(59,85,72,98)(60,86,73,99)(61,87,74,100)(62,88,75,101)(63,89,76,102)(64,90,77,103)(65,91,78,104), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,27)(12,28)(13,29)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (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), (1,78,16,65)(2,77,17,64)(3,76,18,63)(4,75,19,62)(5,74,20,61)(6,73,21,60)(7,72,22,59)(8,71,23,58)(9,70,24,57)(10,69,25,56)(11,68,26,55)(12,67,14,54)(13,66,15,53)(27,94,48,81)(28,93,49,80)(29,92,50,79)(30,104,51,91)(31,103,52,90)(32,102,40,89)(33,101,41,88)(34,100,42,87)(35,99,43,86)(36,98,44,85)(37,97,45,84)(38,96,46,83)(39,95,47,82)>;
G:=Group( (1,51,16,30)(2,52,17,31)(3,40,18,32)(4,41,19,33)(5,42,20,34)(6,43,21,35)(7,44,22,36)(8,45,23,37)(9,46,24,38)(10,47,25,39)(11,48,26,27)(12,49,14,28)(13,50,15,29)(53,79,66,92)(54,80,67,93)(55,81,68,94)(56,82,69,95)(57,83,70,96)(58,84,71,97)(59,85,72,98)(60,86,73,99)(61,87,74,100)(62,88,75,101)(63,89,76,102)(64,90,77,103)(65,91,78,104), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,27)(12,28)(13,29)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (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), (1,78,16,65)(2,77,17,64)(3,76,18,63)(4,75,19,62)(5,74,20,61)(6,73,21,60)(7,72,22,59)(8,71,23,58)(9,70,24,57)(10,69,25,56)(11,68,26,55)(12,67,14,54)(13,66,15,53)(27,94,48,81)(28,93,49,80)(29,92,50,79)(30,104,51,91)(31,103,52,90)(32,102,40,89)(33,101,41,88)(34,100,42,87)(35,99,43,86)(36,98,44,85)(37,97,45,84)(38,96,46,83)(39,95,47,82) );
G=PermutationGroup([[(1,51,16,30),(2,52,17,31),(3,40,18,32),(4,41,19,33),(5,42,20,34),(6,43,21,35),(7,44,22,36),(8,45,23,37),(9,46,24,38),(10,47,25,39),(11,48,26,27),(12,49,14,28),(13,50,15,29),(53,79,66,92),(54,80,67,93),(55,81,68,94),(56,82,69,95),(57,83,70,96),(58,84,71,97),(59,85,72,98),(60,86,73,99),(61,87,74,100),(62,88,75,101),(63,89,76,102),(64,90,77,103),(65,91,78,104)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,27),(12,28),(13,29),(14,49),(15,50),(16,51),(17,52),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78)], [(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)], [(1,78,16,65),(2,77,17,64),(3,76,18,63),(4,75,19,62),(5,74,20,61),(6,73,21,60),(7,72,22,59),(8,71,23,58),(9,70,24,57),(10,69,25,56),(11,68,26,55),(12,67,14,54),(13,66,15,53),(27,94,48,81),(28,93,49,80),(29,92,50,79),(30,104,51,91),(31,103,52,90),(32,102,40,89),(33,101,41,88),(34,100,42,87),(35,99,43,86),(36,98,44,85),(37,97,45,84),(38,96,46,83),(39,95,47,82)]])
D4.D13 is a maximal subgroup of
D8⋊D13 D8⋊3D13 SD16×D13 D4.D26 D52⋊6C22 C52.C23 D4.9D26
D4.D13 is a maximal quotient of C52.Q8 C26.Q16 D4⋊Dic13
37 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 4 | 2 | 52 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D13 | D26 | C13⋊D4 | D4.D13 |
| kernel | D4.D13 | C13⋊2C8 | Dic26 | D4×C13 | C26 | C13 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of D4.D13 ►in GL4(𝔽313) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 311 |
| 0 | 0 | 1 | 312 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 311 |
| 0 | 0 | 0 | 312 |
| 0 | 1 | 0 | 0 |
| 312 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 81 | 166 | 0 | 0 |
| 232 | 232 | 0 | 0 |
| 0 | 0 | 0 | 183 |
| 0 | 0 | 248 | 0 |
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,311,312],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,311,312],[0,312,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[81,232,0,0,166,232,0,0,0,0,0,248,0,0,183,0] >;
D4.D13 in GAP, Magma, Sage, TeX
D_4.D_{13} % in TeX
G:=Group("D4.D13"); // GroupNames label
G:=SmallGroup(208,16);
// by ID
G=gap.SmallGroup(208,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,182,97,42,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^13=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
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