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## G = C5×PSU3(𝔽2)  order 360 = 23·32·5

### Direct product of C5 and PSU3(𝔽2)

Aliases: C5×PSU3(𝔽2), C32⋊(C5×Q8), (C3×C15)⋊1Q8, C32⋊C4.2C10, C3⋊S3.2(C2×C10), (C5×C32⋊C4).4C2, (C5×C3⋊S3).5C22, SmallGroup(360,135)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C5×PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C5×C3⋊S3 — C5×C32⋊C4 — C5×PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C5×PSU3(𝔽2)
 Upper central C1 — C5

Generators and relations for C5×PSU3(𝔽2)
G = < a,b,c,d,e | a5=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >

Character table of C5×PSU3(𝔽2)

 class 1 2 3 4A 4B 4C 5A 5B 5C 5D 10A 10B 10C 10D 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L size 1 9 8 18 18 18 1 1 1 1 9 9 9 9 8 8 8 8 18 18 18 18 18 18 18 18 18 18 18 18 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 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 -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 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 -1 linear of order 2 ρ5 1 1 1 -1 -1 1 ζ53 ζ52 ζ5 ζ54 ζ54 ζ52 ζ53 ζ5 ζ5 ζ53 ζ54 ζ52 ζ5 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 -ζ53 ζ54 ζ53 ζ52 -ζ52 linear of order 10 ρ6 1 1 1 -1 1 -1 ζ53 ζ52 ζ5 ζ54 ζ54 ζ52 ζ53 ζ5 ζ5 ζ53 ζ54 ζ52 -ζ5 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 -ζ53 -ζ54 -ζ53 -ζ52 -ζ52 linear of order 10 ρ7 1 1 1 -1 -1 1 ζ5 ζ54 ζ52 ζ53 ζ53 ζ54 ζ5 ζ52 ζ52 ζ5 ζ53 ζ54 ζ52 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 -ζ5 ζ53 ζ5 ζ54 -ζ54 linear of order 10 ρ8 1 1 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ52 ζ5 ζ54 ζ53 ζ53 ζ54 ζ52 ζ5 ζ53 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 ζ54 ζ5 ζ5 linear of order 5 ρ9 1 1 1 -1 1 -1 ζ52 ζ53 ζ54 ζ5 ζ5 ζ53 ζ52 ζ54 ζ54 ζ52 ζ5 ζ53 -ζ54 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 -ζ52 -ζ5 -ζ52 -ζ53 -ζ53 linear of order 10 ρ10 1 1 1 -1 1 -1 ζ5 ζ54 ζ52 ζ53 ζ53 ζ54 ζ5 ζ52 ζ52 ζ5 ζ53 ζ54 -ζ52 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 -ζ5 -ζ53 -ζ5 -ζ54 -ζ54 linear of order 10 ρ11 1 1 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ5 ζ53 ζ52 ζ54 ζ54 ζ52 ζ5 ζ53 ζ54 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 ζ52 ζ53 ζ53 linear of order 5 ρ12 1 1 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 ζ52 ζ5 ζ54 ζ53 ζ53 ζ54 ζ52 ζ5 -ζ53 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 ζ54 -ζ52 -ζ54 -ζ5 ζ5 linear of order 10 ρ13 1 1 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 ζ54 ζ52 ζ53 ζ5 ζ5 ζ53 ζ54 ζ52 -ζ5 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 ζ53 -ζ54 -ζ53 -ζ52 ζ52 linear of order 10 ρ14 1 1 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ54 ζ52 ζ53 ζ5 ζ5 ζ53 ζ54 ζ52 ζ5 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 ζ53 ζ52 ζ52 linear of order 5 ρ15 1 1 1 -1 -1 1 ζ52 ζ53 ζ54 ζ5 ζ5 ζ53 ζ52 ζ54 ζ54 ζ52 ζ5 ζ53 ζ54 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 -ζ52 ζ5 ζ52 ζ53 -ζ53 linear of order 10 ρ16 1 1 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 ζ53 ζ54 ζ5 ζ52 ζ52 ζ5 ζ53 ζ54 -ζ52 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 ζ5 -ζ53 -ζ5 -ζ54 ζ54 linear of order 10 ρ17 1 1 1 -1 -1 1 ζ54 ζ5 ζ53 ζ52 ζ52 ζ5 ζ54 ζ53 ζ53 ζ54 ζ52 ζ5 ζ53 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 -ζ54 ζ52 ζ54 ζ5 -ζ5 linear of order 10 ρ18 1 1 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ53 ζ54 ζ5 ζ52 ζ52 ζ5 ζ53 ζ54 ζ52 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 ζ5 ζ54 ζ54 linear of order 5 ρ19 1 1 1 -1 1 -1 ζ54 ζ5 ζ53 ζ52 ζ52 ζ5 ζ54 ζ53 ζ53 ζ54 ζ52 ζ5 -ζ53 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 -ζ54 -ζ52 -ζ54 -ζ5 -ζ5 linear of order 10 ρ20 1 1 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 ζ5 ζ53 ζ52 ζ54 ζ54 ζ52 ζ5 ζ53 -ζ54 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 ζ52 -ζ5 -ζ52 -ζ53 ζ53 linear of order 10 ρ21 2 -2 2 0 0 0 2 2 2 2 -2 -2 -2 -2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 2 0 0 0 2ζ53 2ζ52 2ζ5 2ζ54 -2ζ54 -2ζ52 -2ζ53 -2ζ5 2ζ5 2ζ53 2ζ54 2ζ52 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C5×Q8 ρ23 2 -2 2 0 0 0 2ζ5 2ζ54 2ζ52 2ζ53 -2ζ53 -2ζ54 -2ζ5 -2ζ52 2ζ52 2ζ5 2ζ53 2ζ54 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C5×Q8 ρ24 2 -2 2 0 0 0 2ζ52 2ζ53 2ζ54 2ζ5 -2ζ5 -2ζ53 -2ζ52 -2ζ54 2ζ54 2ζ52 2ζ5 2ζ53 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C5×Q8 ρ25 2 -2 2 0 0 0 2ζ54 2ζ5 2ζ53 2ζ52 -2ζ52 -2ζ5 -2ζ54 -2ζ53 2ζ53 2ζ54 2ζ52 2ζ5 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C5×Q8 ρ26 8 0 -1 0 0 0 8 8 8 8 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from PSU3(𝔽2) ρ27 8 0 -1 0 0 0 8ζ5 8ζ54 8ζ52 8ζ53 0 0 0 0 -ζ52 -ζ5 -ζ53 -ζ54 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 8 0 -1 0 0 0 8ζ54 8ζ5 8ζ53 8ζ52 0 0 0 0 -ζ53 -ζ54 -ζ52 -ζ5 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ29 8 0 -1 0 0 0 8ζ52 8ζ53 8ζ54 8ζ5 0 0 0 0 -ζ54 -ζ52 -ζ5 -ζ53 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ30 8 0 -1 0 0 0 8ζ53 8ζ52 8ζ5 8ζ54 0 0 0 0 -ζ5 -ζ53 -ζ54 -ζ52 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C5×PSU3(𝔽2)
On 45 points
Generators in S45
(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)
(1 9 42)(2 10 43)(3 6 44)(4 7 45)(5 8 41)(11 30 32)(12 26 33)(13 27 34)(14 28 35)(15 29 31)(16 22 40)(17 23 36)(18 24 37)(19 25 38)(20 21 39)
(1 33 17)(2 34 18)(3 35 19)(4 31 20)(5 32 16)(6 14 25)(7 15 21)(8 11 22)(9 12 23)(10 13 24)(26 36 42)(27 37 43)(28 38 44)(29 39 45)(30 40 41)
(6 35 44 19)(7 31 45 20)(8 32 41 16)(9 33 42 17)(10 34 43 18)(11 30 40 22)(12 26 36 23)(13 27 37 24)(14 28 38 25)(15 29 39 21)
(6 25 44 28)(7 21 45 29)(8 22 41 30)(9 23 42 26)(10 24 43 27)(11 32 40 16)(12 33 36 17)(13 34 37 18)(14 35 38 19)(15 31 39 20)

G:=sub<Sym(45)| (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), (1,9,42)(2,10,43)(3,6,44)(4,7,45)(5,8,41)(11,30,32)(12,26,33)(13,27,34)(14,28,35)(15,29,31)(16,22,40)(17,23,36)(18,24,37)(19,25,38)(20,21,39), (1,33,17)(2,34,18)(3,35,19)(4,31,20)(5,32,16)(6,14,25)(7,15,21)(8,11,22)(9,12,23)(10,13,24)(26,36,42)(27,37,43)(28,38,44)(29,39,45)(30,40,41), (6,35,44,19)(7,31,45,20)(8,32,41,16)(9,33,42,17)(10,34,43,18)(11,30,40,22)(12,26,36,23)(13,27,37,24)(14,28,38,25)(15,29,39,21), (6,25,44,28)(7,21,45,29)(8,22,41,30)(9,23,42,26)(10,24,43,27)(11,32,40,16)(12,33,36,17)(13,34,37,18)(14,35,38,19)(15,31,39,20)>;

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), (1,9,42)(2,10,43)(3,6,44)(4,7,45)(5,8,41)(11,30,32)(12,26,33)(13,27,34)(14,28,35)(15,29,31)(16,22,40)(17,23,36)(18,24,37)(19,25,38)(20,21,39), (1,33,17)(2,34,18)(3,35,19)(4,31,20)(5,32,16)(6,14,25)(7,15,21)(8,11,22)(9,12,23)(10,13,24)(26,36,42)(27,37,43)(28,38,44)(29,39,45)(30,40,41), (6,35,44,19)(7,31,45,20)(8,32,41,16)(9,33,42,17)(10,34,43,18)(11,30,40,22)(12,26,36,23)(13,27,37,24)(14,28,38,25)(15,29,39,21), (6,25,44,28)(7,21,45,29)(8,22,41,30)(9,23,42,26)(10,24,43,27)(11,32,40,16)(12,33,36,17)(13,34,37,18)(14,35,38,19)(15,31,39,20) );

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)], [(1,9,42),(2,10,43),(3,6,44),(4,7,45),(5,8,41),(11,30,32),(12,26,33),(13,27,34),(14,28,35),(15,29,31),(16,22,40),(17,23,36),(18,24,37),(19,25,38),(20,21,39)], [(1,33,17),(2,34,18),(3,35,19),(4,31,20),(5,32,16),(6,14,25),(7,15,21),(8,11,22),(9,12,23),(10,13,24),(26,36,42),(27,37,43),(28,38,44),(29,39,45),(30,40,41)], [(6,35,44,19),(7,31,45,20),(8,32,41,16),(9,33,42,17),(10,34,43,18),(11,30,40,22),(12,26,36,23),(13,27,37,24),(14,28,38,25),(15,29,39,21)], [(6,25,44,28),(7,21,45,29),(8,22,41,30),(9,23,42,26),(10,24,43,27),(11,32,40,16),(12,33,36,17),(13,34,37,18),(14,35,38,19),(15,31,39,20)]])

Matrix representation of C5×PSU3(𝔽2) in GL10(𝔽61)

 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 60 60 60 60 60 60 60 60 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 60 60 60 60 60 0 0 0 0 0 0 0 0 1 0
,
 39 53 0 0 0 0 0 0 0 0 53 22 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 60 60 60 60 60 60 60 60 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 60 60 60 60 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(10,GF(61))| [9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,0,1,0,0,0,60,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,60,0],[39,53,0,0,0,0,0,0,0,0,53,22,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,60,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0],[0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0] >;

C5×PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_5\times {\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C5xPSU(3,2)");
// GroupNames label

G:=SmallGroup(360,135);
// by ID

G=gap.SmallGroup(360,135);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,3,120,265,127,8404,1810,142,11525,1451,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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