Copied to
clipboard

G = D13.Q16order 416 = 25·13

The non-split extension by D13 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D13.Q16
 Chief series C1 — C13 — C26 — D26 — C4×D13 — C52⋊C4 — D13.Q16
 Lower central C13 — C26 — C52 — D13.Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D13.Q16
G = < a,b,c,d | a13=b2=c8=1, d2=c4, bab=a-1, cac-1=a5, ad=da, cbc-1=a4b, bd=db, dcd-1=a-1bc-1 >

Character table of D13.Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A 13B 13C 26A 26B 26C 52A 52B 52C 52D 52E 52F 52G 52H 52I size 1 1 13 13 2 4 26 52 52 52 26 26 26 26 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 -1 -1 -i 1 i i -i -i i 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -1 i 1 -i -i i i -i 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 -1 i -1 -i i -i -i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 -i -1 i -i i i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 -2 0 -2 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 0 0 0 √2 √2 -√2 -√2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 -2 2 0 0 0 0 0 0 -√2 -√2 √2 √2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ14 2 -2 2 -2 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ15 4 4 0 0 4 4 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ16 4 4 0 0 4 -4 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ17 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1311-ζ1310-ζ133+ζ132 -ζ1312+ζ138+ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311+ζ1310+ζ133-ζ132 -ζ139+ζ137+ζ136-ζ134 ζ139-ζ137-ζ136+ζ134 ζ1312-ζ138-ζ135+ζ13 orthogonal lifted from D13.D4 ρ18 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ139-ζ137-ζ136+ζ134 -ζ1311+ζ1310+ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ139+ζ137+ζ136-ζ134 ζ1312-ζ138-ζ135+ζ13 -ζ1312+ζ138+ζ135-ζ13 ζ1311-ζ1310-ζ133+ζ132 orthogonal lifted from D13.D4 ρ19 4 4 0 0 4 -4 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4 ρ20 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 -ζ139+ζ137+ζ136-ζ134 ζ1311-ζ1310-ζ133+ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ139-ζ137-ζ136+ζ134 -ζ1312+ζ138+ζ135-ζ13 ζ1312-ζ138-ζ135+ζ13 -ζ1311+ζ1310+ζ133-ζ132 orthogonal lifted from D13.D4 ρ21 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 -ζ1311+ζ1310+ζ133-ζ132 ζ1312-ζ138-ζ135+ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ1311-ζ1310-ζ133+ζ132 ζ139-ζ137-ζ136+ζ134 -ζ139+ζ137+ζ136-ζ134 -ζ1312+ζ138+ζ135-ζ13 orthogonal lifted from D13.D4 ρ22 4 4 0 0 4 4 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ23 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 -ζ1312+ζ138+ζ135-ζ13 -ζ139+ζ137+ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ1312-ζ138-ζ135+ζ13 ζ1311-ζ1310-ζ133+ζ132 -ζ1311+ζ1310+ζ133-ζ132 ζ139-ζ137-ζ136+ζ134 orthogonal lifted from D13.D4 ρ24 4 4 0 0 4 -4 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ25 4 4 0 0 4 4 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ26 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1312-ζ138-ζ135+ζ13 ζ139-ζ137-ζ136+ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1312+ζ138+ζ135-ζ13 -ζ1311+ζ1310+ζ133-ζ132 ζ1311-ζ1310-ζ133+ζ132 -ζ139+ζ137+ζ136-ζ134 orthogonal lifted from D13.D4 ρ27 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 2ζ1311+2ζ1310+2ζ133+2ζ132 2ζ139+2ζ137+2ζ136+2ζ134 2ζ1312+2ζ138+2ζ135+2ζ13 -2ζ139-2ζ137-2ζ136-2ζ134 -2ζ1311-2ζ1310-2ζ133-2ζ132 -2ζ1312-2ζ138-2ζ135-2ζ13 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 2ζ139+2ζ137+2ζ136+2ζ134 2ζ1312+2ζ138+2ζ135+2ζ13 2ζ1311+2ζ1310+2ζ133+2ζ132 -2ζ1312-2ζ138-2ζ135-2ζ13 -2ζ139-2ζ137-2ζ136-2ζ134 -2ζ1311-2ζ1310-2ζ133-2ζ132 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 2ζ1312+2ζ138+2ζ135+2ζ13 2ζ1311+2ζ1310+2ζ133+2ζ132 2ζ139+2ζ137+2ζ136+2ζ134 -2ζ1311-2ζ1310-2ζ133-2ζ132 -2ζ1312-2ζ138-2ζ135-2ζ13 -2ζ139-2ζ137-2ζ136-2ζ134 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D13.Q16
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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 43)(28 42)(29 41)(30 40)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(53 70)(54 69)(55 68)(56 67)(57 66)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)
(1 73 43 96 23 64 28 82)(2 68 42 101 24 59 27 87)(3 76 41 93 25 54 39 79)(4 71 40 98 26 62 38 84)(5 66 52 103 14 57 37 89)(6 74 51 95 15 65 36 81)(7 69 50 100 16 60 35 86)(8 77 49 92 17 55 34 91)(9 72 48 97 18 63 33 83)(10 67 47 102 19 58 32 88)(11 75 46 94 20 53 31 80)(12 70 45 99 21 61 30 85)(13 78 44 104 22 56 29 90)
(1 73 23 64)(2 74 24 65)(3 75 25 53)(4 76 26 54)(5 77 14 55)(6 78 15 56)(7 66 16 57)(8 67 17 58)(9 68 18 59)(10 69 19 60)(11 70 20 61)(12 71 21 62)(13 72 22 63)(27 95 42 81)(28 96 43 82)(29 97 44 83)(30 98 45 84)(31 99 46 85)(32 100 47 86)(33 101 48 87)(34 102 49 88)(35 103 50 89)(36 104 51 90)(37 92 52 91)(38 93 40 79)(39 94 41 80)```

`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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99), (1,73,43,96,23,64,28,82)(2,68,42,101,24,59,27,87)(3,76,41,93,25,54,39,79)(4,71,40,98,26,62,38,84)(5,66,52,103,14,57,37,89)(6,74,51,95,15,65,36,81)(7,69,50,100,16,60,35,86)(8,77,49,92,17,55,34,91)(9,72,48,97,18,63,33,83)(10,67,47,102,19,58,32,88)(11,75,46,94,20,53,31,80)(12,70,45,99,21,61,30,85)(13,78,44,104,22,56,29,90), (1,73,23,64)(2,74,24,65)(3,75,25,53)(4,76,26,54)(5,77,14,55)(6,78,15,56)(7,66,16,57)(8,67,17,58)(9,68,18,59)(10,69,19,60)(11,70,20,61)(12,71,21,62)(13,72,22,63)(27,95,42,81)(28,96,43,82)(29,97,44,83)(30,98,45,84)(31,99,46,85)(32,100,47,86)(33,101,48,87)(34,102,49,88)(35,103,50,89)(36,104,51,90)(37,92,52,91)(38,93,40,79)(39,94,41,80)>;`

`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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99), (1,73,43,96,23,64,28,82)(2,68,42,101,24,59,27,87)(3,76,41,93,25,54,39,79)(4,71,40,98,26,62,38,84)(5,66,52,103,14,57,37,89)(6,74,51,95,15,65,36,81)(7,69,50,100,16,60,35,86)(8,77,49,92,17,55,34,91)(9,72,48,97,18,63,33,83)(10,67,47,102,19,58,32,88)(11,75,46,94,20,53,31,80)(12,70,45,99,21,61,30,85)(13,78,44,104,22,56,29,90), (1,73,23,64)(2,74,24,65)(3,75,25,53)(4,76,26,54)(5,77,14,55)(6,78,15,56)(7,66,16,57)(8,67,17,58)(9,68,18,59)(10,69,19,60)(11,70,20,61)(12,71,21,62)(13,72,22,63)(27,95,42,81)(28,96,43,82)(29,97,44,83)(30,98,45,84)(31,99,46,85)(32,100,47,86)(33,101,48,87)(34,102,49,88)(35,103,50,89)(36,104,51,90)(37,92,52,91)(38,93,40,79)(39,94,41,80) );`

`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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,26),(11,25),(12,24),(13,23),(27,43),(28,42),(29,41),(30,40),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(53,70),(54,69),(55,68),(56,67),(57,66),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(85,92),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99)], [(1,73,43,96,23,64,28,82),(2,68,42,101,24,59,27,87),(3,76,41,93,25,54,39,79),(4,71,40,98,26,62,38,84),(5,66,52,103,14,57,37,89),(6,74,51,95,15,65,36,81),(7,69,50,100,16,60,35,86),(8,77,49,92,17,55,34,91),(9,72,48,97,18,63,33,83),(10,67,47,102,19,58,32,88),(11,75,46,94,20,53,31,80),(12,70,45,99,21,61,30,85),(13,78,44,104,22,56,29,90)], [(1,73,23,64),(2,74,24,65),(3,75,25,53),(4,76,26,54),(5,77,14,55),(6,78,15,56),(7,66,16,57),(8,67,17,58),(9,68,18,59),(10,69,19,60),(11,70,20,61),(12,71,21,62),(13,72,22,63),(27,95,42,81),(28,96,43,82),(29,97,44,83),(30,98,45,84),(31,99,46,85),(32,100,47,86),(33,101,48,87),(34,102,49,88),(35,103,50,89),(36,104,51,90),(37,92,52,91),(38,93,40,79),(39,94,41,80)])`

Matrix representation of D13.Q16 in GL6(𝔽313)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 312 1 0 0 0 0 312 0 1 0 0 0 312 0 0 1 0 0 210 73 240 102
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 312 0 0 0 0 0 239 73 211 1 0 0 138 145 239 102 0 0 311 72 211 2
,
 294 46 0 0 0 0 205 212 0 0 0 0 0 0 225 36 277 88 0 0 11 162 168 241 0 0 263 46 191 65 0 0 240 162 252 48
,
 21 211 0 0 0 0 213 292 0 0 0 0 0 0 302 15 57 248 0 0 50 267 122 248 0 0 298 15 61 0 0 0 241 0 72 309

`G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,312,312,210,0,0,1,0,0,73,0,0,0,1,0,240,0,0,0,0,1,102],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,239,138,311,0,0,0,73,145,72,0,0,0,211,239,211,0,0,0,1,102,2],[294,205,0,0,0,0,46,212,0,0,0,0,0,0,225,11,263,240,0,0,36,162,46,162,0,0,277,168,191,252,0,0,88,241,65,48],[21,213,0,0,0,0,211,292,0,0,0,0,0,0,302,50,298,241,0,0,15,267,15,0,0,0,57,122,61,72,0,0,248,248,0,309] >;`

D13.Q16 in GAP, Magma, Sage, TeX

`D_{13}.Q_{16}`
`% in TeX`

`G:=Group("D13.Q16");`
`// GroupNames label`

`G:=SmallGroup(416,84);`
`// by ID`

`G=gap.SmallGroup(416,84);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,579,297,69,9221,3473]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^13=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

Export

׿
×
𝔽