metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic13.4D4, C13⋊(C4.D4), C23.(C13⋊C4), (C22×C26).4C4, C13⋊M4(2)⋊1C2, (C22×D13).2C4, C26.11(C22⋊C4), C2.11(D13.D4), (C2×Dic13).24C22, C22.5(C2×C13⋊C4), (C2×C26).12(C2×C4), (C2×C13⋊D4).8C2, SmallGroup(416,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic13.4D4
G = < a,b,c,d | a26=1, b2=c4=a13, d2=b, bab-1=a-1, cac-1=dad-1=a5, cbc-1=a13b, bd=db, dcd-1=a13bc3 >
2C2
4C2
52C2
2C22
4C22
13C4
13C4
26C22
52C22
2C26
4D13
4C26
13C23
13C2×C4
26C8
26D4
26C8
26D4
2D26
4D26
13C2×D4
13M4(2)
13M4(2)
13C4.D4
Smallest permutation representation of Dic13.4D4
►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 27 14 40)(2 52 15 39)(3 51 16 38)(4 50 17 37)(5 49 18 36)(6 48 19 35)(7 47 20 34)(8 46 21 33)(9 45 22 32)(10 44 23 31)(11 43 24 30)(12 42 25 29)(13 41 26 28)(53 89 66 102)(54 88 67 101)(55 87 68 100)(56 86 69 99)(57 85 70 98)(58 84 71 97)(59 83 72 96)(60 82 73 95)(61 81 74 94)(62 80 75 93)(63 79 76 92)(64 104 77 91)(65 103 78 90)
(1 83 27 59 14 96 40 72)(2 104 52 64 15 91 39 77)(3 99 51 69 16 86 38 56)(4 94 50 74 17 81 37 61)(5 89 49 53 18 102 36 66)(6 84 48 58 19 97 35 71)(7 79 47 63 20 92 34 76)(8 100 46 68 21 87 33 55)(9 95 45 73 22 82 32 60)(10 90 44 78 23 103 31 65)(11 85 43 57 24 98 30 70)(12 80 42 62 25 93 29 75)(13 101 41 67 26 88 28 54)
(1 96 27 59 14 83 40 72)(2 91 52 64 15 104 39 77)(3 86 51 69 16 99 38 56)(4 81 50 74 17 94 37 61)(5 102 49 53 18 89 36 66)(6 97 48 58 19 84 35 71)(7 92 47 63 20 79 34 76)(8 87 46 68 21 100 33 55)(9 82 45 73 22 95 32 60)(10 103 44 78 23 90 31 65)(11 98 43 57 24 85 30 70)(12 93 42 62 25 80 29 75)(13 88 41 67 26 101 28 54)
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), (1,27,14,40)(2,52,15,39)(3,51,16,38)(4,50,17,37)(5,49,18,36)(6,48,19,35)(7,47,20,34)(8,46,21,33)(9,45,22,32)(10,44,23,31)(11,43,24,30)(12,42,25,29)(13,41,26,28)(53,89,66,102)(54,88,67,101)(55,87,68,100)(56,86,69,99)(57,85,70,98)(58,84,71,97)(59,83,72,96)(60,82,73,95)(61,81,74,94)(62,80,75,93)(63,79,76,92)(64,104,77,91)(65,103,78,90), (1,83,27,59,14,96,40,72)(2,104,52,64,15,91,39,77)(3,99,51,69,16,86,38,56)(4,94,50,74,17,81,37,61)(5,89,49,53,18,102,36,66)(6,84,48,58,19,97,35,71)(7,79,47,63,20,92,34,76)(8,100,46,68,21,87,33,55)(9,95,45,73,22,82,32,60)(10,90,44,78,23,103,31,65)(11,85,43,57,24,98,30,70)(12,80,42,62,25,93,29,75)(13,101,41,67,26,88,28,54), (1,96,27,59,14,83,40,72)(2,91,52,64,15,104,39,77)(3,86,51,69,16,99,38,56)(4,81,50,74,17,94,37,61)(5,102,49,53,18,89,36,66)(6,97,48,58,19,84,35,71)(7,92,47,63,20,79,34,76)(8,87,46,68,21,100,33,55)(9,82,45,73,22,95,32,60)(10,103,44,78,23,90,31,65)(11,98,43,57,24,85,30,70)(12,93,42,62,25,80,29,75)(13,88,41,67,26,101,28,54)>;
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), (1,27,14,40)(2,52,15,39)(3,51,16,38)(4,50,17,37)(5,49,18,36)(6,48,19,35)(7,47,20,34)(8,46,21,33)(9,45,22,32)(10,44,23,31)(11,43,24,30)(12,42,25,29)(13,41,26,28)(53,89,66,102)(54,88,67,101)(55,87,68,100)(56,86,69,99)(57,85,70,98)(58,84,71,97)(59,83,72,96)(60,82,73,95)(61,81,74,94)(62,80,75,93)(63,79,76,92)(64,104,77,91)(65,103,78,90), (1,83,27,59,14,96,40,72)(2,104,52,64,15,91,39,77)(3,99,51,69,16,86,38,56)(4,94,50,74,17,81,37,61)(5,89,49,53,18,102,36,66)(6,84,48,58,19,97,35,71)(7,79,47,63,20,92,34,76)(8,100,46,68,21,87,33,55)(9,95,45,73,22,82,32,60)(10,90,44,78,23,103,31,65)(11,85,43,57,24,98,30,70)(12,80,42,62,25,93,29,75)(13,101,41,67,26,88,28,54), (1,96,27,59,14,83,40,72)(2,91,52,64,15,104,39,77)(3,86,51,69,16,99,38,56)(4,81,50,74,17,94,37,61)(5,102,49,53,18,89,36,66)(6,97,48,58,19,84,35,71)(7,92,47,63,20,79,34,76)(8,87,46,68,21,100,33,55)(9,82,45,73,22,95,32,60)(10,103,44,78,23,90,31,65)(11,98,43,57,24,85,30,70)(12,93,42,62,25,80,29,75)(13,88,41,67,26,101,28,54) );
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)], [(1,27,14,40),(2,52,15,39),(3,51,16,38),(4,50,17,37),(5,49,18,36),(6,48,19,35),(7,47,20,34),(8,46,21,33),(9,45,22,32),(10,44,23,31),(11,43,24,30),(12,42,25,29),(13,41,26,28),(53,89,66,102),(54,88,67,101),(55,87,68,100),(56,86,69,99),(57,85,70,98),(58,84,71,97),(59,83,72,96),(60,82,73,95),(61,81,74,94),(62,80,75,93),(63,79,76,92),(64,104,77,91),(65,103,78,90)], [(1,83,27,59,14,96,40,72),(2,104,52,64,15,91,39,77),(3,99,51,69,16,86,38,56),(4,94,50,74,17,81,37,61),(5,89,49,53,18,102,36,66),(6,84,48,58,19,97,35,71),(7,79,47,63,20,92,34,76),(8,100,46,68,21,87,33,55),(9,95,45,73,22,82,32,60),(10,90,44,78,23,103,31,65),(11,85,43,57,24,98,30,70),(12,80,42,62,25,93,29,75),(13,101,41,67,26,88,28,54)], [(1,96,27,59,14,83,40,72),(2,91,52,64,15,104,39,77),(3,86,51,69,16,99,38,56),(4,81,50,74,17,94,37,61),(5,102,49,53,18,89,36,66),(6,97,48,58,19,84,35,71),(7,92,47,63,20,79,34,76),(8,87,46,68,21,100,33,55),(9,82,45,73,22,95,32,60),(10,103,44,78,23,90,31,65),(11,98,43,57,24,85,30,70),(12,93,42,62,25,80,29,75),(13,88,41,67,26,101,28,54)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | ··· | 26U |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 2 | 4 | 52 | 26 | 26 | 52 | 52 | 52 | 52 | 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 | C4.D4 | C13⋊C4 | C2×C13⋊C4 | D13.D4 | Dic13.4D4 |
| kernel | Dic13.4D4 | C13⋊M4(2) | C2×C13⋊D4 | C22×D13 | C22×C26 | Dic13 | C13 | C23 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 3 | 3 | 6 | 12 |
Matrix representation of Dic13.4D4 ►in GL4(𝔽313) generated by
| 64 | 63 | 305 | 261 |
| 0 | 269 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 19 |
| 312 | 51 | 0 | 51 |
| 270 | 1 | 1 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 312 | 0 |
| 1 | 0 | 0 | 262 |
| 0 | 0 | 0 | 312 |
| 0 | 1 | 0 | 0 |
| 43 | 312 | 312 | 312 |
| 312 | 51 | 51 | 51 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 270 | 1 | 1 | 1 |
G:=sub<GL(4,GF(313))| [64,0,0,0,63,269,0,0,305,0,33,0,261,0,0,19],[312,270,0,0,51,1,0,0,0,1,0,312,51,1,1,0],[1,0,0,43,0,0,1,312,0,0,0,312,262,312,0,312],[312,0,0,270,51,0,1,1,51,0,0,1,51,1,0,1] >;
Dic13.4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{13}._4D_4 % in TeX
G:=Group("Dic13.4D4"); // GroupNames label
G:=SmallGroup(416,88);
// by ID
G=gap.SmallGroup(416,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,86,579,9221,3473]);
// Polycyclic
G:=Group<a,b,c,d|a^26=1,b^2=c^4=a^13,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=a^13*b*c^3>;
// generators/relations
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