C5xPSU(3,2) - GroupNames

G = C₅×PSU₃(𝔽₂) order 360 = 2³·3²·5

Direct product of C₅ and PSU₃(𝔽₂)

direct product, non-abelian, soluble, monomial

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 >

C₁

₉C₂

₄C₃

C₅

₉C₄

₁₂S₃

C₃²

₉C₁₀

₄C₁₅

₉Q₈

C₃⋊S₃

₉C₂₀

₁₂C₅×S₃

C₃×C₁₅

C₃²⋊C₄

₉C₅×Q₈

C₅×C₃⋊S₃

PSU₃(𝔽₂)

C₅×C₃²⋊C₄

C₅×PSU₃(𝔽₂)

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

Export

Subgroup lattice of C₅×PSU₃(𝔽₂) in TeX
Character table of C₅×PSU₃(𝔽₂) in TeX

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

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 = C5×PSU3(𝔽2) order 360 = 23·32·5

Direct product of C5 and PSU3(𝔽2)

G = C₅×PSU₃(𝔽₂) order 360 = 2³·3²·5

Direct product of C₅ and PSU₃(𝔽₂)

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