metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊1C4, D13.Q8, D13.1D4, Dic13⋊3C4, D26.5C22, C13⋊(C4⋊C4), C4⋊(C13⋊C4), C26.4(C2×C4), (C4×D13).4C2, (C2×C13⋊C4).C2, C2.5(C2×C13⋊C4), SmallGroup(208,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊C4
G = < a,b | a52=b4=1, bab-1=a31 >
Character table of C52⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 13A | 13B | 13C | 26A | 26B | 26C | 52A | 52B | 52C | 52D | 52E | 52F | |
| size | 1 | 1 | 13 | 13 | 2 | 26 | 26 | 26 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ12 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ13 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ15 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | complex faithful |
►On 52 points
Generators in S52
(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)
(1 40)(2 35 26 19)(3 30 51 50)(4 25 24 29)(5 20 49 8)(6 15 22 39)(7 10 47 18)(9 52 45 28)(11 42 43 38)(12 37 16 17)(13 32 41 48)(14 27)(21 44 33 36)(23 34 31 46)
G:=sub<Sym(52)| (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), (1,40)(2,35,26,19)(3,30,51,50)(4,25,24,29)(5,20,49,8)(6,15,22,39)(7,10,47,18)(9,52,45,28)(11,42,43,38)(12,37,16,17)(13,32,41,48)(14,27)(21,44,33,36)(23,34,31,46)>;
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), (1,40)(2,35,26,19)(3,30,51,50)(4,25,24,29)(5,20,49,8)(6,15,22,39)(7,10,47,18)(9,52,45,28)(11,42,43,38)(12,37,16,17)(13,32,41,48)(14,27)(21,44,33,36)(23,34,31,46) );
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)], [(1,40),(2,35,26,19),(3,30,51,50),(4,25,24,29),(5,20,49,8),(6,15,22,39),(7,10,47,18),(9,52,45,28),(11,42,43,38),(12,37,16,17),(13,32,41,48),(14,27),(21,44,33,36),(23,34,31,46)]])
C52⋊C4 is a maximal subgroup of
D26.8D4 D13.D8 D52⋊1C4 D13.Q16 D26.C23 D4×C13⋊C4 Q8×C13⋊C4
C52⋊C4 is a maximal quotient of D26.8D4 D13.D8 C104.C4 C104.1C4 C52⋊C8 Dic13⋊C8 D26.Q8
Matrix representation of C52⋊C4 ►in GL4(𝔽53) generated by
| 2 | 46 | 22 | 13 |
| 14 | 6 | 41 | 7 |
| 27 | 45 | 7 | 44 |
| 7 | 31 | 40 | 45 |
| 13 | 40 | 2 | 6 |
| 46 | 46 | 7 | 36 |
| 12 | 9 | 2 | 37 |
| 31 | 8 | 0 | 45 |
G:=sub<GL(4,GF(53))| [2,14,27,7,46,6,45,31,22,41,7,40,13,7,44,45],[13,46,12,31,40,46,9,8,2,7,2,0,6,36,37,45] >;
C52⋊C4 in GAP, Magma, Sage, TeX
C_{52}\rtimes C_4 % in TeX
G:=Group("C52:C4"); // GroupNames label
G:=SmallGroup(208,31);
// by ID
G=gap.SmallGroup(208,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,3204,1214]);
// Polycyclic
G:=Group<a,b|a^52=b^4=1,b*a*b^-1=a^31>;
// generators/relations
Export
Subgroup lattice of C52⋊C4 in TeX
Character table of C52⋊C4 in TeX