metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic26⋊2C4, D26.19D4, D13.2Q16, D13.3SD16, Dic13.3D4, C52.3(C2×C4), C13⋊(Q8⋊C4), Q8⋊1(C13⋊C4), (Q8×C13)⋊1C4, D13⋊C8.1C2, (Q8×D13).2C2, C52⋊C4.1C2, C26.7(C22⋊C4), (C4×D13).9C22, C2.8(D13.D4), C4.3(C2×C13⋊C4), SmallGroup(416,84)
Series: Derived ►Chief ►Lower central ►Upper central
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 >
13C2
13C2
2C4
13C22
13C4
26C4
52C4
13Q8
13C2×C4
26C2×C4
26C2×C4
26C8
26Q8
2C52
13C2×Q8
13C4⋊C4
13C2×C8
13Q8⋊C4
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 |
►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 42)(28 41)(29 40)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(53 69)(54 68)(55 67)(56 66)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(79 94)(80 93)(81 92)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)
(1 74 41 99 23 62 29 88)(2 69 40 104 24 57 28 80)(3 77 52 96 25 65 27 85)(4 72 51 101 26 60 39 90)(5 67 50 93 14 55 38 82)(6 75 49 98 15 63 37 87)(7 70 48 103 16 58 36 79)(8 78 47 95 17 53 35 84)(9 73 46 100 18 61 34 89)(10 68 45 92 19 56 33 81)(11 76 44 97 20 64 32 86)(12 71 43 102 21 59 31 91)(13 66 42 94 22 54 30 83)
(1 74 23 62)(2 75 24 63)(3 76 25 64)(4 77 26 65)(5 78 14 53)(6 66 15 54)(7 67 16 55)(8 68 17 56)(9 69 18 57)(10 70 19 58)(11 71 20 59)(12 72 21 60)(13 73 22 61)(27 97 52 86)(28 98 40 87)(29 99 41 88)(30 100 42 89)(31 101 43 90)(32 102 44 91)(33 103 45 79)(34 104 46 80)(35 92 47 81)(36 93 48 82)(37 94 49 83)(38 95 50 84)(39 96 51 85)
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)(27,42)(28,41)(29,40)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(53,69)(54,68)(55,67)(56,66)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(79,94)(80,93)(81,92)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95), (1,74,41,99,23,62,29,88)(2,69,40,104,24,57,28,80)(3,77,52,96,25,65,27,85)(4,72,51,101,26,60,39,90)(5,67,50,93,14,55,38,82)(6,75,49,98,15,63,37,87)(7,70,48,103,16,58,36,79)(8,78,47,95,17,53,35,84)(9,73,46,100,18,61,34,89)(10,68,45,92,19,56,33,81)(11,76,44,97,20,64,32,86)(12,71,43,102,21,59,31,91)(13,66,42,94,22,54,30,83), (1,74,23,62)(2,75,24,63)(3,76,25,64)(4,77,26,65)(5,78,14,53)(6,66,15,54)(7,67,16,55)(8,68,17,56)(9,69,18,57)(10,70,19,58)(11,71,20,59)(12,72,21,60)(13,73,22,61)(27,97,52,86)(28,98,40,87)(29,99,41,88)(30,100,42,89)(31,101,43,90)(32,102,44,91)(33,103,45,79)(34,104,46,80)(35,92,47,81)(36,93,48,82)(37,94,49,83)(38,95,50,84)(39,96,51,85)>;
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)(27,42)(28,41)(29,40)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(53,69)(54,68)(55,67)(56,66)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(79,94)(80,93)(81,92)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95), (1,74,41,99,23,62,29,88)(2,69,40,104,24,57,28,80)(3,77,52,96,25,65,27,85)(4,72,51,101,26,60,39,90)(5,67,50,93,14,55,38,82)(6,75,49,98,15,63,37,87)(7,70,48,103,16,58,36,79)(8,78,47,95,17,53,35,84)(9,73,46,100,18,61,34,89)(10,68,45,92,19,56,33,81)(11,76,44,97,20,64,32,86)(12,71,43,102,21,59,31,91)(13,66,42,94,22,54,30,83), (1,74,23,62)(2,75,24,63)(3,76,25,64)(4,77,26,65)(5,78,14,53)(6,66,15,54)(7,67,16,55)(8,68,17,56)(9,69,18,57)(10,70,19,58)(11,71,20,59)(12,72,21,60)(13,73,22,61)(27,97,52,86)(28,98,40,87)(29,99,41,88)(30,100,42,89)(31,101,43,90)(32,102,44,91)(33,103,45,79)(34,104,46,80)(35,92,47,81)(36,93,48,82)(37,94,49,83)(38,95,50,84)(39,96,51,85) );
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),(27,42),(28,41),(29,40),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(53,69),(54,68),(55,67),(56,66),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(79,94),(80,93),(81,92),(82,104),(83,103),(84,102),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95)], [(1,74,41,99,23,62,29,88),(2,69,40,104,24,57,28,80),(3,77,52,96,25,65,27,85),(4,72,51,101,26,60,39,90),(5,67,50,93,14,55,38,82),(6,75,49,98,15,63,37,87),(7,70,48,103,16,58,36,79),(8,78,47,95,17,53,35,84),(9,73,46,100,18,61,34,89),(10,68,45,92,19,56,33,81),(11,76,44,97,20,64,32,86),(12,71,43,102,21,59,31,91),(13,66,42,94,22,54,30,83)], [(1,74,23,62),(2,75,24,63),(3,76,25,64),(4,77,26,65),(5,78,14,53),(6,66,15,54),(7,67,16,55),(8,68,17,56),(9,69,18,57),(10,70,19,58),(11,71,20,59),(12,72,21,60),(13,73,22,61),(27,97,52,86),(28,98,40,87),(29,99,41,88),(30,100,42,89),(31,101,43,90),(32,102,44,91),(33,103,45,79),(34,104,46,80),(35,92,47,81),(36,93,48,82),(37,94,49,83),(38,95,50,84),(39,96,51,85)]])
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
Subgroup lattice of D13.Q16 in TeX
Character table of D13.Q16 in TeX