metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊1C4, D13.2D8, D26.18D4, Dic13.2D4, D13.2SD16, D13⋊C8⋊1C2, C13⋊(D4⋊C4), D4⋊1(C13⋊C4), (D4×C13)⋊1C4, C52.1(C2×C4), C52⋊C4⋊1C2, (D4×D13).2C2, C26.5(C22⋊C4), (C4×D13).7C22, C2.6(D13.D4), C4.1(C2×C13⋊C4), SmallGroup(416,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊1C4
G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a17b >
4C2
13C2
13C2
52C2
2C22
13C4
13C22
26C22
52C22
52C22
52C4
4D13
4C26
13D4
13C2×C4
26D4
26C23
26C2×C4
26C8
2D26
4D26
4D26
13C4⋊C4
13C2×D4
13C2×C8
13D4⋊C4
Character table of D52⋊1C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | 26D | 26E | 26F | 26G | 26H | 26I | 52A | 52B | 52C | |
| size | 1 | 1 | 4 | 13 | 13 | 52 | 2 | 26 | 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 | -1 | -i | 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 | -1 | -i | 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 | -1 | i | -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 | -1 | i | -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 | 0 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | 0 | -2 | 2 | 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 | 0 | -2 | 2 | 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 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ139-ζ137-ζ136-ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | ζ139-ζ137-ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | ζ1311-ζ1310-ζ133+ζ132 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from D13.D4 |
| ρ17 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1312-ζ138-ζ135-ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139-ζ137-ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | ζ1311-ζ1310-ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from D13.D4 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | -ζ1311+ζ1310+ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1312+ζ138+ζ135-ζ13 | ζ1312-ζ138-ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal lifted from D13.D4 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ1312+ζ138+ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ139+ζ137+ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from D13.D4 |
| ρ22 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1312+ζ138+ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ139-ζ137-ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from D13.D4 |
| ρ26 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ1312-ζ138-ζ135+ζ13 | -ζ1312+ζ138+ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | 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 | -2ζ1311-2ζ1310-2ζ133-2ζ132 | -2ζ139-2ζ137-2ζ136-2ζ134 | -2ζ1312-2ζ138-2ζ135-2ζ13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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 | -2ζ139-2ζ137-2ζ136-2ζ134 | -2ζ1312-2ζ138-2ζ135-2ζ13 | -2ζ1311-2ζ1310-2ζ133-2ζ132 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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 | -2ζ1312-2ζ138-2ζ135-2ζ13 | -2ζ1311-2ζ1310-2ζ133-2ζ132 | -2ζ139-2ζ137-2ζ136-2ζ134 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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 85)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)
(2 48 26 32)(3 43 51 11)(4 38 24 42)(5 33 49 21)(6 28 22 52)(7 23 47 31)(8 18 20 10)(9 13 45 41)(12 50 16 30)(14 40)(15 35 39 19)(17 25 37 29)(34 44 46 36)(53 56 93 64)(54 103 66 95)(55 98 91 74)(57 88 89 84)(58 83 62 63)(59 78 87 94)(60 73)(61 68 85 104)(65 100 81 72)(67 90 79 82)(69 80 77 92)(70 75 102 71)(76 97 96 101)(86 99)
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), (1,85)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86), (2,48,26,32)(3,43,51,11)(4,38,24,42)(5,33,49,21)(6,28,22,52)(7,23,47,31)(8,18,20,10)(9,13,45,41)(12,50,16,30)(14,40)(15,35,39,19)(17,25,37,29)(34,44,46,36)(53,56,93,64)(54,103,66,95)(55,98,91,74)(57,88,89,84)(58,83,62,63)(59,78,87,94)(60,73)(61,68,85,104)(65,100,81,72)(67,90,79,82)(69,80,77,92)(70,75,102,71)(76,97,96,101)(86,99)>;
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), (1,85)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86), (2,48,26,32)(3,43,51,11)(4,38,24,42)(5,33,49,21)(6,28,22,52)(7,23,47,31)(8,18,20,10)(9,13,45,41)(12,50,16,30)(14,40)(15,35,39,19)(17,25,37,29)(34,44,46,36)(53,56,93,64)(54,103,66,95)(55,98,91,74)(57,88,89,84)(58,83,62,63)(59,78,87,94)(60,73)(61,68,85,104)(65,100,81,72)(67,90,79,82)(69,80,77,92)(70,75,102,71)(76,97,96,101)(86,99) );
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)], [(1,85),(2,84),(3,83),(4,82),(5,81),(6,80),(7,79),(8,78),(9,77),(10,76),(11,75),(12,74),(13,73),(14,72),(15,71),(16,70),(17,69),(18,68),(19,67),(20,66),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86)], [(2,48,26,32),(3,43,51,11),(4,38,24,42),(5,33,49,21),(6,28,22,52),(7,23,47,31),(8,18,20,10),(9,13,45,41),(12,50,16,30),(14,40),(15,35,39,19),(17,25,37,29),(34,44,46,36),(53,56,93,64),(54,103,66,95),(55,98,91,74),(57,88,89,84),(58,83,62,63),(59,78,87,94),(60,73),(61,68,85,104),(65,100,81,72),(67,90,79,82),(69,80,77,92),(70,75,102,71),(76,97,96,101),(86,99)]])
Matrix representation of D52⋊1C4 ►in GL6(𝔽313)
| 1 | 225 | 0 | 0 | 0 | 0 |
| 249 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 312 |
| 0 | 0 | 242 | 282 | 32 | 71 |
| 0 | 0 | 32 | 252 | 273 | 311 |
| 0 | 0 | 242 | 283 | 2 | 101 |
| 0 | 272 | 0 | 0 | 0 | 0 |
| 229 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 137 | 100 | 213 | 176 |
| 0 | 0 | 237 | 0 | 76 | 176 |
| 0 | 0 | 272 | 41 | 100 | 0 |
| 0 | 0 | 148 | 230 | 312 | 76 |
| 288 | 0 | 0 | 0 | 0 | 0 |
| 35 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 31 | 282 | 242 |
| 0 | 0 | 312 | 30 | 71 | 2 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 30 | 312 | 212 |
G:=sub<GL(6,GF(313))| [1,249,0,0,0,0,225,312,0,0,0,0,0,0,0,242,32,242,0,0,0,282,252,283,0,0,1,32,273,2,0,0,312,71,311,101],[0,229,0,0,0,0,272,0,0,0,0,0,0,0,137,237,272,148,0,0,100,0,41,230,0,0,213,76,100,312,0,0,176,176,0,76],[288,35,0,0,0,0,0,25,0,0,0,0,0,0,71,312,0,71,0,0,31,30,1,30,0,0,282,71,0,312,0,0,242,2,0,212] >;
D52⋊1C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes_1C_4 % in TeX
G:=Group("D52:1C4"); // GroupNames label
G:=SmallGroup(416,82);
// by ID
G=gap.SmallGroup(416,82);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,579,297,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^17*b>;
// generators/relations
Export
Subgroup lattice of D52⋊1C4 in TeX
Character table of D52⋊1C4 in TeX