metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊2C4, D26.3D4, Dic13.22D4, C13⋊2C4≀C2, C52.4(C2×C4), Q8⋊2(C13⋊C4), (Q8×C13)⋊2C4, C52.C4⋊2C2, D52⋊C2.2C2, C26.8(C22⋊C4), C2.9(D13.D4), (C4×D13).10C22, (C4×C13⋊C4)⋊2C2, C4.4(C2×C13⋊C4), SmallGroup(416,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊C4
G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a7b >
26C2
52C2
2C4
13C22
13C4
26C4
26C22
26C4
2D13
4D13
13D4
13C2×C4
26D4
26C2×C4
26C2×C4
26C8
2C52
2D26
13C4○D4
13C42
13M4(2)
2D52
13C4≀C2
Character table of D52⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 13A | 13B | 13C | 26A | 26B | 26C | 52A | 52B | 52C | 52D | 52E | 52F | 52G | 52H | 52I | |
| size | 1 | 1 | 26 | 52 | 2 | 4 | 13 | 13 | 26 | 26 | 26 | 26 | 52 | 52 | 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 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | -1-i | 1-i | -1+i | 1+i | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 1+i | -1+i | 1-i | -1-i | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 1-i | -1-i | 1+i | -1+i | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -1+i | 1+i | -1-i | 1-i | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| 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 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ16 | 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 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312+ζ138+ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ139+ζ137+ζ136-ζ134 | orthogonal lifted from D13.D4 |
| ρ17 | 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 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ139-ζ137-ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | ζ1311-ζ1310-ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1311+ζ1310+ζ133-ζ132 | orthogonal lifted from D13.D4 |
| ρ18 | 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 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311+ζ1310+ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1312-ζ138-ζ135-ζ13 | ζ1312-ζ138-ζ135+ζ13 | orthogonal lifted from D13.D4 |
| ρ19 | 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 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ20 | 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 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ21 | 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 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ22 | 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 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311+ζ1310+ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ1312-ζ138-ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1312+ζ138+ζ135-ζ13 | orthogonal lifted from D13.D4 |
| 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 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312+ζ138+ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ139+ζ137+ζ136-ζ134 | -ζ139-ζ137-ζ136-ζ134 | ζ139-ζ137-ζ136+ζ134 | orthogonal lifted from D13.D4 |
| ρ24 | 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 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139-ζ137-ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1311-ζ1310-ζ133+ζ132 | orthogonal lifted from D13.D4 |
| 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 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ26 | 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 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ27 | 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 | -2ζ139-2ζ137-2ζ136-2ζ134 | -2ζ1312-2ζ138-2ζ135-2ζ13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| 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 | -2ζ1311-2ζ1310-2ζ133-2ζ132 | -2ζ139-2ζ137-2ζ136-2ζ134 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ29 | 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 | -2ζ1312-2ζ138-2ζ135-2ζ13 | -2ζ1311-2ζ1310-2ζ133-2ζ132 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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 55)(2 54)(3 53)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)
(1 40 27 14)(2 45 52 35)(3 50 25 4)(5 8 23 46)(6 13 48 15)(7 18 21 36)(9 28 19 26)(10 33 44 47)(11 38 17 16)(12 43 42 37)(20 31 34 49)(22 41 32 39)(24 51 30 29)(53 93 85 97)(54 98 58 66)(55 103 83 87)(57 61 81 77)(59 71 79 67)(60 76 104 88)(62 86 102 78)(63 91 75 99)(64 96 100 68)(65 101 73 89)(70 74 94 90)(72 84 92 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), (1,55)(2,54)(3,53)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56), (1,40,27,14)(2,45,52,35)(3,50,25,4)(5,8,23,46)(6,13,48,15)(7,18,21,36)(9,28,19,26)(10,33,44,47)(11,38,17,16)(12,43,42,37)(20,31,34,49)(22,41,32,39)(24,51,30,29)(53,93,85,97)(54,98,58,66)(55,103,83,87)(57,61,81,77)(59,71,79,67)(60,76,104,88)(62,86,102,78)(63,91,75,99)(64,96,100,68)(65,101,73,89)(70,74,94,90)(72,84,92,80)>;
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), (1,55)(2,54)(3,53)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56), (1,40,27,14)(2,45,52,35)(3,50,25,4)(5,8,23,46)(6,13,48,15)(7,18,21,36)(9,28,19,26)(10,33,44,47)(11,38,17,16)(12,43,42,37)(20,31,34,49)(22,41,32,39)(24,51,30,29)(53,93,85,97)(54,98,58,66)(55,103,83,87)(57,61,81,77)(59,71,79,67)(60,76,104,88)(62,86,102,78)(63,91,75,99)(64,96,100,68)(65,101,73,89)(70,74,94,90)(72,84,92,80) );
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)], [(1,55),(2,54),(3,53),(4,104),(5,103),(6,102),(7,101),(8,100),(9,99),(10,98),(11,97),(12,96),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,84),(25,83),(26,82),(27,81),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56)], [(1,40,27,14),(2,45,52,35),(3,50,25,4),(5,8,23,46),(6,13,48,15),(7,18,21,36),(9,28,19,26),(10,33,44,47),(11,38,17,16),(12,43,42,37),(20,31,34,49),(22,41,32,39),(24,51,30,29),(53,93,85,97),(54,98,58,66),(55,103,83,87),(57,61,81,77),(59,71,79,67),(60,76,104,88),(62,86,102,78),(63,91,75,99),(64,96,100,68),(65,101,73,89),(70,74,94,90),(72,84,92,80)]])
Matrix representation of D52⋊C4 ►in GL6(𝔽313)
| 25 | 0 | 0 | 0 | 0 | 0 |
| 2 | 288 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 312 |
| 0 | 0 | 1 | 30 | 243 | 30 |
| 0 | 0 | 283 | 40 | 252 | 282 |
| 0 | 0 | 31 | 274 | 61 | 243 |
| 158 | 119 | 0 | 0 | 0 | 0 |
| 240 | 155 | 0 | 0 | 0 | 0 |
| 0 | 0 | 241 | 31 | 104 | 172 |
| 0 | 0 | 0 | 72 | 141 | 72 |
| 0 | 0 | 72 | 141 | 72 | 0 |
| 0 | 0 | 172 | 104 | 31 | 241 |
| 288 | 0 | 0 | 0 | 0 | 0 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 312 | 0 | 0 | 0 |
| 0 | 0 | 30 | 31 | 31 | 30 |
| 0 | 0 | 0 | 0 | 0 | 312 |
| 0 | 0 | 282 | 252 | 40 | 283 |
G:=sub<GL(6,GF(313))| [25,2,0,0,0,0,0,288,0,0,0,0,0,0,0,1,283,31,0,0,0,30,40,274,0,0,0,243,252,61,0,0,312,30,282,243],[158,240,0,0,0,0,119,155,0,0,0,0,0,0,241,0,72,172,0,0,31,72,141,104,0,0,104,141,72,31,0,0,172,72,0,241],[288,24,0,0,0,0,0,1,0,0,0,0,0,0,312,30,0,282,0,0,0,31,0,252,0,0,0,31,0,40,0,0,0,30,312,283] >;
D52⋊C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes C_4 % in TeX
G:=Group("D52:C4"); // GroupNames label
G:=SmallGroup(416,85);
// by ID
G=gap.SmallGroup(416,85);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,86,579,297,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^7*b>;
// generators/relations
Export
Subgroup lattice of D52⋊C4 in TeX
Character table of D52⋊C4 in TeX