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## G = C22.58C24⋊C5order 320 = 26·5

### The semidirect product of C22.58C24 and C5 acting faithfully

Aliases: C22.58C24⋊C5, C22.2(C24⋊C5), C2.(2- 1+4⋊C5), SmallGroup(320,1012)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22.58C24 — C22.58C24⋊C5
 Chief series C1 — C2 — C22 — C22.58C24 — C22.58C24⋊C5
 Lower central C22.58C24 — C22.58C24⋊C5
 Upper central C1 — C22

Generators and relations for C22.58C24⋊C5
G = < a,b,c,d,e,f,g | a2=b2=g5=1, c2=f2=a, d2=e2=b, ab=ba, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=gfg-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, fcf-1=abc, gcg-1=abcde, ede-1=abd, gdg-1=abcd, ef=fe, geg-1=bcdef >

Character table of C22.58C24⋊C5

 class 1 2A 2B 2C 4A 4B 4C 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L size 1 1 1 1 20 20 20 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 ρ1 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 ζ5 ζ53 ζ54 ζ52 ζ53 ζ5 ζ5 ζ5 ζ53 ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ52 linear of order 5 ρ3 1 1 1 1 1 1 1 ζ53 ζ54 ζ52 ζ5 ζ54 ζ53 ζ53 ζ53 ζ54 ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ5 linear of order 5 ρ4 1 1 1 1 1 1 1 ζ54 ζ52 ζ5 ζ53 ζ52 ζ54 ζ54 ζ54 ζ52 ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ53 linear of order 5 ρ5 1 1 1 1 1 1 1 ζ52 ζ5 ζ53 ζ54 ζ5 ζ52 ζ52 ζ52 ζ5 ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ54 linear of order 5 ρ6 4 4 -4 -4 0 0 0 -1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 symplectic lifted from 2- 1+4⋊C5, Schur index 2 ρ7 4 -4 -4 4 0 0 0 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 -1 1 1 symplectic lifted from 2- 1+4⋊C5, Schur index 2 ρ8 4 -4 4 -4 0 0 0 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 symplectic lifted from 2- 1+4⋊C5, Schur index 2 ρ9 4 -4 4 -4 0 0 0 -ζ52 -ζ5 -ζ53 -ζ54 ζ5 ζ52 -ζ52 ζ52 -ζ5 ζ53 -ζ53 ζ53 ζ5 ζ54 -ζ54 ζ54 complex lifted from 2- 1+4⋊C5 ρ10 4 -4 -4 4 0 0 0 -ζ5 -ζ53 -ζ54 -ζ52 -ζ53 -ζ5 ζ5 ζ5 ζ53 -ζ54 ζ54 ζ54 ζ53 -ζ52 ζ52 ζ52 complex lifted from 2- 1+4⋊C5 ρ11 4 -4 -4 4 0 0 0 -ζ52 -ζ5 -ζ53 -ζ54 -ζ5 -ζ52 ζ52 ζ52 ζ5 -ζ53 ζ53 ζ53 ζ5 -ζ54 ζ54 ζ54 complex lifted from 2- 1+4⋊C5 ρ12 4 -4 4 -4 0 0 0 -ζ54 -ζ52 -ζ5 -ζ53 ζ52 ζ54 -ζ54 ζ54 -ζ52 ζ5 -ζ5 ζ5 ζ52 ζ53 -ζ53 ζ53 complex lifted from 2- 1+4⋊C5 ρ13 4 -4 -4 4 0 0 0 -ζ53 -ζ54 -ζ52 -ζ5 -ζ54 -ζ53 ζ53 ζ53 ζ54 -ζ52 ζ52 ζ52 ζ54 -ζ5 ζ5 ζ5 complex lifted from 2- 1+4⋊C5 ρ14 4 4 -4 -4 0 0 0 -ζ5 -ζ53 -ζ54 -ζ52 ζ53 ζ5 ζ5 -ζ5 ζ53 ζ54 ζ54 -ζ54 -ζ53 ζ52 ζ52 -ζ52 complex lifted from 2- 1+4⋊C5 ρ15 4 4 -4 -4 0 0 0 -ζ54 -ζ52 -ζ5 -ζ53 ζ52 ζ54 ζ54 -ζ54 ζ52 ζ5 ζ5 -ζ5 -ζ52 ζ53 ζ53 -ζ53 complex lifted from 2- 1+4⋊C5 ρ16 4 -4 4 -4 0 0 0 -ζ53 -ζ54 -ζ52 -ζ5 ζ54 ζ53 -ζ53 ζ53 -ζ54 ζ52 -ζ52 ζ52 ζ54 ζ5 -ζ5 ζ5 complex lifted from 2- 1+4⋊C5 ρ17 4 -4 4 -4 0 0 0 -ζ5 -ζ53 -ζ54 -ζ52 ζ53 ζ5 -ζ5 ζ5 -ζ53 ζ54 -ζ54 ζ54 ζ53 ζ52 -ζ52 ζ52 complex lifted from 2- 1+4⋊C5 ρ18 4 4 -4 -4 0 0 0 -ζ53 -ζ54 -ζ52 -ζ5 ζ54 ζ53 ζ53 -ζ53 ζ54 ζ52 ζ52 -ζ52 -ζ54 ζ5 ζ5 -ζ5 complex lifted from 2- 1+4⋊C5 ρ19 4 -4 -4 4 0 0 0 -ζ54 -ζ52 -ζ5 -ζ53 -ζ52 -ζ54 ζ54 ζ54 ζ52 -ζ5 ζ5 ζ5 ζ52 -ζ53 ζ53 ζ53 complex lifted from 2- 1+4⋊C5 ρ20 4 4 -4 -4 0 0 0 -ζ52 -ζ5 -ζ53 -ζ54 ζ5 ζ52 ζ52 -ζ52 ζ5 ζ53 ζ53 -ζ53 -ζ5 ζ54 ζ54 -ζ54 complex lifted from 2- 1+4⋊C5 ρ21 5 5 5 5 1 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ22 5 5 5 5 1 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ23 5 5 5 5 -3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5

Smallest permutation representation of C22.58C24⋊C5
On 64 points
Generators in S64
```(1 2)(3 4)(5 24)(6 20)(7 21)(8 22)(9 23)(10 51)(11 52)(12 53)(13 54)(14 50)(15 45)(16 46)(17 47)(18 48)(19 49)(25 32)(26 33)(27 34)(28 30)(29 31)(35 62)(36 63)(37 64)(38 60)(39 61)(40 58)(41 59)(42 55)(43 56)(44 57)
(1 4)(2 3)(5 42)(6 43)(7 44)(8 40)(9 41)(10 46)(11 47)(12 48)(13 49)(14 45)(15 50)(16 51)(17 52)(18 53)(19 54)(20 56)(21 57)(22 58)(23 59)(24 55)(25 38)(26 39)(27 35)(28 36)(29 37)(30 63)(31 64)(32 60)(33 61)(34 62)
(1 20 2 6)(3 43 4 56)(5 60 24 38)(7 26 21 33)(8 11 22 52)(9 45 23 15)(10 36 51 63)(12 19 53 49)(13 48 54 18)(14 59 50 41)(16 30 46 28)(17 40 47 58)(25 42 32 55)(27 31 34 29)(35 64 62 37)(39 57 61 44)
(1 13 4 49)(2 54 3 19)(5 8 42 40)(6 48 43 12)(7 15 44 50)(9 61 41 33)(10 62 46 34)(11 25 47 38)(14 21 45 57)(16 27 51 35)(17 60 52 32)(18 56 53 20)(22 55 58 24)(23 39 59 26)(28 31 36 64)(29 63 37 30)
(1 11 4 47)(2 52 3 17)(5 18 42 53)(6 40 43 8)(7 64 44 31)(9 46 41 10)(12 24 48 55)(13 60 49 32)(14 28 45 36)(15 63 50 30)(16 59 51 23)(19 25 54 38)(20 58 56 22)(21 37 57 29)(26 34 39 62)(27 61 35 33)
(1 57 2 44)(3 7 4 21)(5 10 24 51)(6 33 20 26)(8 35 22 62)(9 48 23 18)(11 29 52 31)(12 59 53 41)(13 50 54 14)(15 19 45 49)(16 42 46 55)(17 64 47 37)(25 36 32 63)(27 58 34 40)(28 60 30 38)(39 43 61 56)
(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)```

`G:=sub<Sym(64)| (1,2)(3,4)(5,24)(6,20)(7,21)(8,22)(9,23)(10,51)(11,52)(12,53)(13,54)(14,50)(15,45)(16,46)(17,47)(18,48)(19,49)(25,32)(26,33)(27,34)(28,30)(29,31)(35,62)(36,63)(37,64)(38,60)(39,61)(40,58)(41,59)(42,55)(43,56)(44,57), (1,4)(2,3)(5,42)(6,43)(7,44)(8,40)(9,41)(10,46)(11,47)(12,48)(13,49)(14,45)(15,50)(16,51)(17,52)(18,53)(19,54)(20,56)(21,57)(22,58)(23,59)(24,55)(25,38)(26,39)(27,35)(28,36)(29,37)(30,63)(31,64)(32,60)(33,61)(34,62), (1,20,2,6)(3,43,4,56)(5,60,24,38)(7,26,21,33)(8,11,22,52)(9,45,23,15)(10,36,51,63)(12,19,53,49)(13,48,54,18)(14,59,50,41)(16,30,46,28)(17,40,47,58)(25,42,32,55)(27,31,34,29)(35,64,62,37)(39,57,61,44), (1,13,4,49)(2,54,3,19)(5,8,42,40)(6,48,43,12)(7,15,44,50)(9,61,41,33)(10,62,46,34)(11,25,47,38)(14,21,45,57)(16,27,51,35)(17,60,52,32)(18,56,53,20)(22,55,58,24)(23,39,59,26)(28,31,36,64)(29,63,37,30), (1,11,4,47)(2,52,3,17)(5,18,42,53)(6,40,43,8)(7,64,44,31)(9,46,41,10)(12,24,48,55)(13,60,49,32)(14,28,45,36)(15,63,50,30)(16,59,51,23)(19,25,54,38)(20,58,56,22)(21,37,57,29)(26,34,39,62)(27,61,35,33), (1,57,2,44)(3,7,4,21)(5,10,24,51)(6,33,20,26)(8,35,22,62)(9,48,23,18)(11,29,52,31)(12,59,53,41)(13,50,54,14)(15,19,45,49)(16,42,46,55)(17,64,47,37)(25,36,32,63)(27,58,34,40)(28,60,30,38)(39,43,61,56), (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)>;`

`G:=Group( (1,2)(3,4)(5,24)(6,20)(7,21)(8,22)(9,23)(10,51)(11,52)(12,53)(13,54)(14,50)(15,45)(16,46)(17,47)(18,48)(19,49)(25,32)(26,33)(27,34)(28,30)(29,31)(35,62)(36,63)(37,64)(38,60)(39,61)(40,58)(41,59)(42,55)(43,56)(44,57), (1,4)(2,3)(5,42)(6,43)(7,44)(8,40)(9,41)(10,46)(11,47)(12,48)(13,49)(14,45)(15,50)(16,51)(17,52)(18,53)(19,54)(20,56)(21,57)(22,58)(23,59)(24,55)(25,38)(26,39)(27,35)(28,36)(29,37)(30,63)(31,64)(32,60)(33,61)(34,62), (1,20,2,6)(3,43,4,56)(5,60,24,38)(7,26,21,33)(8,11,22,52)(9,45,23,15)(10,36,51,63)(12,19,53,49)(13,48,54,18)(14,59,50,41)(16,30,46,28)(17,40,47,58)(25,42,32,55)(27,31,34,29)(35,64,62,37)(39,57,61,44), (1,13,4,49)(2,54,3,19)(5,8,42,40)(6,48,43,12)(7,15,44,50)(9,61,41,33)(10,62,46,34)(11,25,47,38)(14,21,45,57)(16,27,51,35)(17,60,52,32)(18,56,53,20)(22,55,58,24)(23,39,59,26)(28,31,36,64)(29,63,37,30), (1,11,4,47)(2,52,3,17)(5,18,42,53)(6,40,43,8)(7,64,44,31)(9,46,41,10)(12,24,48,55)(13,60,49,32)(14,28,45,36)(15,63,50,30)(16,59,51,23)(19,25,54,38)(20,58,56,22)(21,37,57,29)(26,34,39,62)(27,61,35,33), (1,57,2,44)(3,7,4,21)(5,10,24,51)(6,33,20,26)(8,35,22,62)(9,48,23,18)(11,29,52,31)(12,59,53,41)(13,50,54,14)(15,19,45,49)(16,42,46,55)(17,64,47,37)(25,36,32,63)(27,58,34,40)(28,60,30,38)(39,43,61,56), (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) );`

`G=PermutationGroup([[(1,2),(3,4),(5,24),(6,20),(7,21),(8,22),(9,23),(10,51),(11,52),(12,53),(13,54),(14,50),(15,45),(16,46),(17,47),(18,48),(19,49),(25,32),(26,33),(27,34),(28,30),(29,31),(35,62),(36,63),(37,64),(38,60),(39,61),(40,58),(41,59),(42,55),(43,56),(44,57)], [(1,4),(2,3),(5,42),(6,43),(7,44),(8,40),(9,41),(10,46),(11,47),(12,48),(13,49),(14,45),(15,50),(16,51),(17,52),(18,53),(19,54),(20,56),(21,57),(22,58),(23,59),(24,55),(25,38),(26,39),(27,35),(28,36),(29,37),(30,63),(31,64),(32,60),(33,61),(34,62)], [(1,20,2,6),(3,43,4,56),(5,60,24,38),(7,26,21,33),(8,11,22,52),(9,45,23,15),(10,36,51,63),(12,19,53,49),(13,48,54,18),(14,59,50,41),(16,30,46,28),(17,40,47,58),(25,42,32,55),(27,31,34,29),(35,64,62,37),(39,57,61,44)], [(1,13,4,49),(2,54,3,19),(5,8,42,40),(6,48,43,12),(7,15,44,50),(9,61,41,33),(10,62,46,34),(11,25,47,38),(14,21,45,57),(16,27,51,35),(17,60,52,32),(18,56,53,20),(22,55,58,24),(23,39,59,26),(28,31,36,64),(29,63,37,30)], [(1,11,4,47),(2,52,3,17),(5,18,42,53),(6,40,43,8),(7,64,44,31),(9,46,41,10),(12,24,48,55),(13,60,49,32),(14,28,45,36),(15,63,50,30),(16,59,51,23),(19,25,54,38),(20,58,56,22),(21,37,57,29),(26,34,39,62),(27,61,35,33)], [(1,57,2,44),(3,7,4,21),(5,10,24,51),(6,33,20,26),(8,35,22,62),(9,48,23,18),(11,29,52,31),(12,59,53,41),(13,50,54,14),(15,19,45,49),(16,42,46,55),(17,64,47,37),(25,36,32,63),(27,58,34,40),(28,60,30,38),(39,43,61,56)], [(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)]])`

Matrix representation of C22.58C24⋊C5 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 31 29 18 0 0 0 0 10 0 18 12 0 0 0 0 29 18 0 31 0 0 0 0 18 12 10 0 0 0 0 0 0 0 0 0 23 13 11 0 0 0 0 0 13 18 0 11 0 0 0 0 11 0 18 28 0 0 0 0 0 11 28 23
,
 7 0 11 3 0 0 0 0 0 7 3 30 0 0 0 0 30 38 34 0 0 0 0 0 38 11 0 34 0 0 0 0 0 0 0 0 0 7 4 31 0 0 0 0 34 0 31 37 0 0 0 0 37 10 0 7 0 0 0 0 10 4 34 0
,
 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 4 16 13 0 0 0 0 37 0 13 25 0 0 0 0 16 13 0 4 0 0 0 0 13 25 37 0 0 0 0 0 0 0 0 0 25 12 0 3 0 0 0 0 12 16 38 0 0 0 0 0 0 38 25 12 0 0 0 0 3 0 12 16
,
 1 0 0 0 0 0 0 0 29 18 0 31 0 0 0 0 0 0 0 40 0 0 0 0 10 0 18 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 23 13 0 0 0 0 0 0 0 1 0 0 0 0 28 23 0 30

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,10,29,18,0,0,0,0,31,0,18,12,0,0,0,0,29,18,0,10,0,0,0,0,18,12,31,0,0,0,0,0,0,0,0,0,23,13,11,0,0,0,0,0,13,18,0,11,0,0,0,0,11,0,18,28,0,0,0,0,0,11,28,23],[7,0,30,38,0,0,0,0,0,7,38,11,0,0,0,0,11,3,34,0,0,0,0,0,3,30,0,34,0,0,0,0,0,0,0,0,0,34,37,10,0,0,0,0,7,0,10,4,0,0,0,0,4,31,0,34,0,0,0,0,31,37,7,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[0,37,16,13,0,0,0,0,4,0,13,25,0,0,0,0,16,13,0,37,0,0,0,0,13,25,4,0,0,0,0,0,0,0,0,0,25,12,0,3,0,0,0,0,12,16,38,0,0,0,0,0,0,38,25,12,0,0,0,0,3,0,12,16],[1,29,0,10,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,31,40,12,0,0,0,0,0,0,0,0,1,30,0,28,0,0,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,0,0,13,1,30] >;`

C22.58C24⋊C5 in GAP, Magma, Sage, TeX

`C_2^2._{58}C_2^4\rtimes C_5`
`% in TeX`

`G:=Group("C2^2.58C2^4:C5");`
`// GroupNames label`

`G:=SmallGroup(320,1012);`
`// by ID`

`G=gap.SmallGroup(320,1012);`
`# by ID`

`G:=PCGroup([7,-5,-2,2,2,2,-2,-2,561,456,947,387,184,1543,1466,745,360,2629,851,718,375,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=g^5=1,c^2=f^2=a,d^2=e^2=b,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=g*f*g^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*c*f^-1=a*b*c,g*c*g^-1=a*b*c*d*e,e*d*e^-1=a*b*d,g*d*g^-1=a*b*c*d,e*f=f*e,g*e*g^-1=b*c*d*e*f>;`
`// generators/relations`

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