metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C53⋊C4, D53.C2, SmallGroup(212,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C53⋊C4 |
Generators and relations for C53⋊C4
G = < a,b | a53=b4=1, bab-1=a23 >
Character table of C53⋊C4
| class | 1 | 2 | 4A | 4B | 53A | 53B | 53C | 53D | 53E | 53F | 53G | 53H | 53I | 53J | 53K | 53L | 53M | |
| size | 1 | 53 | 53 | 53 | 4 | 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 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5351+ζ5346+ζ537+ζ532 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5348+ζ5344+ζ539+ζ535 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5338+ζ5327+ζ5326+ζ5315 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5348+ζ5344+ζ539+ζ535 | ζ5351+ζ5346+ζ537+ζ532 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5343+ζ5335+ζ5318+ζ5310 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ5351+ζ5346+ζ537+ζ532 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5348+ζ5344+ζ539+ζ535 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5345+ζ5328+ζ5325+ζ538 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5348+ζ5344+ζ539+ζ535 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5351+ζ5346+ζ537+ζ532 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | ζ5348+ζ5344+ζ539+ζ535 | ζ5351+ζ5346+ζ537+ζ532 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5336+ζ5333+ζ5320+ζ5317 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5351+ζ5346+ζ537+ζ532 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5348+ζ5344+ζ539+ζ535 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5351+ζ5346+ζ537+ζ532 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5348+ζ5344+ζ539+ζ535 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5352+ζ5330+ζ5323+ζ53 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5348+ζ5344+ζ539+ζ535 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5351+ζ5346+ζ537+ζ532 | ζ5350+ζ5337+ζ5316+ζ533 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5351+ζ5346+ζ537+ζ532 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5348+ζ5344+ζ539+ζ535 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5349+ζ5339+ζ5314+ζ534 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5348+ζ5344+ζ539+ζ535 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5351+ζ5346+ζ537+ζ532 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5331+ζ5329+ζ5324+ζ5322 | orthogonal faithful |
| ρ15 | 4 | 0 | 0 | 0 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5348+ζ5344+ζ539+ζ535 | ζ5351+ζ5346+ζ537+ζ532 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5347+ζ5332+ζ5321+ζ536 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5340+ζ5334+ζ5319+ζ5313 | ζ5348+ζ5344+ζ539+ζ535 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5351+ζ5346+ζ537+ζ532 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5342+ζ5341+ζ5312+ζ5311 | orthogonal faithful |
| ρ17 | 4 | 0 | 0 | 0 | ζ5343+ζ5335+ζ5318+ζ5310 | ζ5349+ζ5339+ζ5314+ζ534 | ζ5338+ζ5327+ζ5326+ζ5315 | ζ5336+ζ5333+ζ5320+ζ5317 | ζ5345+ζ5328+ζ5325+ζ538 | ζ5352+ζ5330+ζ5323+ζ53 | ζ5347+ζ5332+ζ5321+ζ536 | ζ5342+ζ5341+ζ5312+ζ5311 | ζ5350+ζ5337+ζ5316+ζ533 | ζ5351+ζ5346+ζ537+ζ532 | ζ5331+ζ5329+ζ5324+ζ5322 | ζ5348+ζ5344+ζ539+ζ535 | ζ5340+ζ5334+ζ5319+ζ5313 | orthogonal faithful |
►On 53 points: primitive
Generators in S53
(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)
(2 31 53 24)(3 8 52 47)(4 38 51 17)(5 15 50 40)(6 45 49 10)(7 22 48 33)(9 29 46 26)(11 36 44 19)(12 13 43 42)(14 20 41 35)(16 27 39 28)(18 34 37 21)(23 25 32 30)
G:=sub<Sym(53)| (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), (2,31,53,24)(3,8,52,47)(4,38,51,17)(5,15,50,40)(6,45,49,10)(7,22,48,33)(9,29,46,26)(11,36,44,19)(12,13,43,42)(14,20,41,35)(16,27,39,28)(18,34,37,21)(23,25,32,30)>;
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), (2,31,53,24)(3,8,52,47)(4,38,51,17)(5,15,50,40)(6,45,49,10)(7,22,48,33)(9,29,46,26)(11,36,44,19)(12,13,43,42)(14,20,41,35)(16,27,39,28)(18,34,37,21)(23,25,32,30) );
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)], [(2,31,53,24),(3,8,52,47),(4,38,51,17),(5,15,50,40),(6,45,49,10),(7,22,48,33),(9,29,46,26),(11,36,44,19),(12,13,43,42),(14,20,41,35),(16,27,39,28),(18,34,37,21),(23,25,32,30)]])
C53⋊C4 is a maximal quotient of C53⋊C8
Matrix representation of C53⋊C4 ►in GL4(𝔽1061) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1060 | 435 | 297 | 435 |
| 1 | 0 | 0 | 0 |
| 560 | 511 | 94 | 184 |
| 733 | 735 | 831 | 466 |
| 232 | 156 | 545 | 779 |
G:=sub<GL(4,GF(1061))| [0,0,0,1060,1,0,0,435,0,1,0,297,0,0,1,435],[1,560,733,232,0,511,735,156,0,94,831,545,0,184,466,779] >;
C53⋊C4 in GAP, Magma, Sage, TeX
C_{53}\rtimes C_4 % in TeX
G:=Group("C53:C4"); // GroupNames label
G:=SmallGroup(212,3);
// by ID
G=gap.SmallGroup(212,3);
# by ID
G:=PCGroup([3,-2,-2,-53,6,1082,941]);
// Polycyclic
G:=Group<a,b|a^53=b^4=1,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C53⋊C4 in TeX
Character table of C53⋊C4 in TeX