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G = C53⋊C4order 212 = 22·53

The semidirect product of C53 and C4 acting faithfully

Aliases: C53⋊C4, D53.C2, SmallGroup(212,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C53 — C53⋊C4
 Chief series C1 — C53 — D53 — C53⋊C4
 Lower central C53 — C53⋊C4
 Upper central C1

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

Smallest permutation representation of C53⋊C4
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`

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