C3xPSU(3,2) - GroupNames

G = C₃×PSU₃(𝔽₂) order 216 = 2³·3³

Direct product of C₃ and PSU₃(𝔽₂)

direct product, non-abelian, soluble, monomial

Aliases: C₃×PSU₃(𝔽₂), C₃³⋊₁Q₈, C₃²⋊(C₃×Q₈), C₃²⋊C₄.₂C₆, C₃⋊S₃.₂(C₂×C₆), (C₃×C₃²⋊C₄).₄C₂, (C₃×C₃⋊S₃).₂C₂², SmallGroup(216,160)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊S₃ — C₃×PSU₃(𝔽₂)

Chief series

C₁ — C₃² — C₃⋊S₃ — C₃×C₃⋊S₃ — C₃×C₃²⋊C₄ — C₃×PSU₃(𝔽₂)

Lower central

C₃² — C₃⋊S₃ — C₃×PSU₃(𝔽₂)

Upper central

C₁ — C₃

Generators and relations for C₃×PSU₃(𝔽₂)
G = < a,b,c,d,e | a³=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

C₃

₄C₃

₈C₃

₉C₄

₉C₆

₁₂S₃

C₃²

₄C₃²

₈C₃²

₉Q₈

₉C₁₂

C₃⋊S₃

₁₂C₃×S₃

C₃³

₉C₃×Q₈

C₃²⋊C₄

C₃×C₃⋊S₃

PSU₃(𝔽₂)

C₃×C₃²⋊C₄

C₃×PSU₃(𝔽₂)

Character table of C₃×PSU₃(𝔽₂)

class	1	2	3A	3B	3C	3D	3E	4A	4B	4C	6A	6B	12A	12B	12C	12D	12E	12F
size	1	9	1	1	8	8	8	18	18	18	9	9	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	trivial
ρ₂	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	linear of order 2
ρ₄	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	ζ₃	ζ₃²	ζ₃	1	ζ₃²	-1	-1	1	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	ζ₃	ζ₃²	linear of order 6
ρ₆	1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	-1	1	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₃	ζ₃²	ζ₆	ζ₆⁵	ζ₆	linear of order 6
ρ₇	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	-1	-1	1	ζ₃²	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	ζ₃²	ζ₃	linear of order 6
ρ₈	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₉	1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	1	-1	-1	ζ₃	ζ₃²	ζ₃	ζ₆⁵	ζ₆	ζ₃²	ζ₆⁵	ζ₆	linear of order 6
ρ₁₀	1	1	ζ₃	ζ₃²	ζ₃	1	ζ₃²	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₁₁	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	1	-1	-1	ζ₃²	ζ₃	ζ₃²	ζ₆	ζ₆⁵	ζ₃	ζ₆	ζ₆⁵	linear of order 6
ρ₁₂	1	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃	-1	1	-1	ζ₃²	ζ₃	ζ₆	ζ₃²	ζ₃	ζ₆⁵	ζ₆	ζ₆⁵	linear of order 6
ρ₁₃	2	-2	2	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₄	2	-2	-1+√-3	-1-√-3	-1+√-3	2	-1-√-3	0	0	0	1-√-3	1+√-3	0	0	0	0	0	0	complex lifted from C₃×Q₈
ρ₁₅	2	-2	-1-√-3	-1+√-3	-1-√-3	2	-1+√-3	0	0	0	1+√-3	1-√-3	0	0	0	0	0	0	complex lifted from C₃×Q₈
ρ₁₆	8	0	8	8	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from PSU₃(𝔽₂)
ρ₁₇	8	0	-4+4√-3	-4-4√-3	ζ₆⁵	-1	ζ₆	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₁₈	8	0	-4-4√-3	-4+4√-3	ζ₆	-1	ζ₆⁵	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₃×PSU₃(𝔽₂)
►On 24 points - transitive group 24T565

Generators in S₂₄

(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)

(2 20 15)(4 13 18)(5 10 21)(6 11 22)(7 23 12)(8 24 9)

(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;

G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );

G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(2,20,15),(4,13,18),(5,10,21),(6,11,22),(7,23,12),(8,24,9)], [(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])

G:=TransitiveGroup(24,565);

►On 27 points - transitive group 27T88

Generators in S₂₇

(1 3 2)(4 9 15)(5 10 12)(6 11 13)(7 8 14)(16 22 25)(17 23 26)(18 20 27)(19 21 24)

(1 19 17)(2 24 26)(3 21 23)(4 22 7)(5 20 6)(8 9 25)(10 27 11)(12 18 13)(14 15 16)

(1 16 18)(2 25 27)(3 22 20)(4 5 23)(6 21 7)(8 11 24)(9 10 26)(12 17 15)(13 19 14)

(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)

(4 21 6 23)(5 20 7 22)(8 25 10 27)(9 24 11 26)(12 18 14 16)(13 17 15 19)

G:=sub<Sym(27)| (1,3,2)(4,9,15)(5,10,12)(6,11,13)(7,8,14)(16,22,25)(17,23,26)(18,20,27)(19,21,24), (1,19,17)(2,24,26)(3,21,23)(4,22,7)(5,20,6)(8,9,25)(10,27,11)(12,18,13)(14,15,16), (1,16,18)(2,25,27)(3,22,20)(4,5,23)(6,21,7)(8,11,24)(9,10,26)(12,17,15)(13,19,14), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,21,6,23)(5,20,7,22)(8,25,10,27)(9,24,11,26)(12,18,14,16)(13,17,15,19)>;

G:=Group( (1,3,2)(4,9,15)(5,10,12)(6,11,13)(7,8,14)(16,22,25)(17,23,26)(18,20,27)(19,21,24), (1,19,17)(2,24,26)(3,21,23)(4,22,7)(5,20,6)(8,9,25)(10,27,11)(12,18,13)(14,15,16), (1,16,18)(2,25,27)(3,22,20)(4,5,23)(6,21,7)(8,11,24)(9,10,26)(12,17,15)(13,19,14), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,21,6,23)(5,20,7,22)(8,25,10,27)(9,24,11,26)(12,18,14,16)(13,17,15,19) );

G=PermutationGroup([[(1,3,2),(4,9,15),(5,10,12),(6,11,13),(7,8,14),(16,22,25),(17,23,26),(18,20,27),(19,21,24)], [(1,19,17),(2,24,26),(3,21,23),(4,22,7),(5,20,6),(8,9,25),(10,27,11),(12,18,13),(14,15,16)], [(1,16,18),(2,25,27),(3,22,20),(4,5,23),(6,21,7),(8,11,24),(9,10,26),(12,17,15),(13,19,14)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,21,6,23),(5,20,7,22),(8,25,10,27),(9,24,11,26),(12,18,14,16),(13,17,15,19)]])

G:=TransitiveGroup(27,88);

C₃×PSU₃(𝔽₂) is a maximal subgroup of C₃³⋊₃SD₁₆

Matrix representation of C₃×PSU₃(𝔽₂) ►in GL₈(𝔽₁₃)

3	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	3	0	0	0
0	0	0	0	0	3	0	0
0	0	0	0	0	0	3	0
0	0	0	0	0	0	0	3

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	9	0	0	0	0	0
11	2	9	3	0	0	0	0
0	0	0	0	3	0	0	0
12	12	0	0	9	9	0	0
8	0	0	0	6	0	9	0
0	11	4	0	0	0	0	3

9	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	1	0	0	0	0	0
8	11	0	1	0	0	0	0
0	0	0	0	3	0	0	0
0	9	0	0	9	9	0	0
7	0	3	0	0	0	3	0
0	7	12	0	7	0	0	9

0	0	1	0	0	0	0	0
5	8	1	8	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	5	0	0	0	0
0	5	12	0	5	0	0	8
0	0	0	1	0	0	1	1
0	0	0	0	0	0	0	5
0	0	0	8	0	1	0	8

0	0	0	0	1	0	0	0
12	12	0	0	12	8	0	0
8	0	12	0	8	0	8	0
0	0	0	0	0	5	12	1
0	1	0	0	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	1	0	0	1	0
0	0	0	0	0	8	1	0

G:=sub<GL(8,GF(13))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[1,0,0,11,0,12,8,0,0,1,0,2,0,12,0,11,0,0,9,9,0,0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,3,9,6,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3],[9,0,0,8,0,0,7,0,0,3,0,11,0,9,0,7,0,0,1,0,0,0,3,12,0,0,0,1,0,0,0,0,0,0,0,0,3,9,0,7,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[0,5,0,0,0,0,0,0,0,8,1,0,5,0,0,0,1,1,0,0,12,0,0,0,0,8,0,5,0,1,0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,8,1,5,8],[0,12,8,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,1,12,8,0,0,0,0,0,0,8,0,5,0,1,0,8,0,0,8,12,0,0,1,1,0,0,0,1,0,0,0,0] >;

C₃×PSU₃(𝔽₂) in GAP, Magma, Sage, TeX

C_3\times {\rm PSU}_3({\mathbb F}_2)

% in TeX

G:=Group("C3xPSU(3,2)");

// GroupNames label

G:=SmallGroup(216,160);

// by ID

G=gap.SmallGroup(216,160);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,3,72,169,79,5044,1090,142,6917,875,455]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=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 C₃×PSU₃(𝔽₂) in TeX
Character table of C₃×PSU₃(𝔽₂) in TeX

3	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	3	0	0	0
0	0	0	0	0	3	0	0
0	0	0	0	0	0	3	0
0	0	0	0	0	0	0	3

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	9	0	0	0	0	0
11	2	9	3	0	0	0	0
0	0	0	0	3	0	0	0
12	12	0	0	9	9	0	0
8	0	0	0	6	0	9	0
0	11	4	0	0	0	0	3

9	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	1	0	0	0	0	0
8	11	0	1	0	0	0	0
0	0	0	0	3	0	0	0
0	9	0	0	9	9	0	0
7	0	3	0	0	0	3	0
0	7	12	0	7	0	0	9

0	0	1	0	0	0	0	0
5	8	1	8	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	5	0	0	0	0
0	5	12	0	5	0	0	8
0	0	0	1	0	0	1	1
0	0	0	0	0	0	0	5
0	0	0	8	0	1	0	8

0	0	0	0	1	0	0	0
12	12	0	0	12	8	0	0
8	0	12	0	8	0	8	0
0	0	0	0	0	5	12	1
0	1	0	0	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	1	0	0	1	0
0	0	0	0	0	8	1	0

3	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	3	0	0	0
0	0	0	0	0	3	0	0
0	0	0	0	0	0	3	0
0	0	0	0	0	0	0	3

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	9	0	0	0	0	0
11	2	9	3	0	0	0	0
0	0	0	0	3	0	0	0
12	12	0	0	9	9	0	0
8	0	0	0	6	0	9	0
0	11	4	0	0	0	0	3

9	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	1	0	0	0	0	0
8	11	0	1	0	0	0	0
0	0	0	0	3	0	0	0
0	9	0	0	9	9	0	0
7	0	3	0	0	0	3	0
0	7	12	0	7	0	0	9

0	0	1	0	0	0	0	0
5	8	1	8	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	5	0	0	0	0
0	5	12	0	5	0	0	8
0	0	0	1	0	0	1	1
0	0	0	0	0	0	0	5
0	0	0	8	0	1	0	8

0	0	0	0	1	0	0	0
12	12	0	0	12	8	0	0
8	0	12	0	8	0	8	0
0	0	0	0	0	5	12	1
0	1	0	0	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	1	0	0	1	0
0	0	0	0	0	8	1	0

G = C3×PSU3(𝔽2) order 216 = 23·33

Direct product of C3 and PSU3(𝔽2)

G = C₃×PSU₃(𝔽₂) order 216 = 2³·3³

Direct product of C₃ and PSU₃(𝔽₂)

3	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	3	0	0	0
0	0	0	0	0	3	0	0
0	0	0	0	0	0	3	0
0	0	0	0	0	0	0	3

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	9	0	0	0	0	0
11	2	9	3	0	0	0	0
0	0	0	0	3	0	0	0
12	12	0	0	9	9	0	0
8	0	0	0	6	0	9	0
0	11	4	0	0	0	0	3

9	0	0	0	0	0	0	0
0	3	0	0	0	0	0	0
0	0	1	0	0	0	0	0
8	11	0	1	0	0	0	0
0	0	0	0	3	0	0	0
0	9	0	0	9	9	0	0
7	0	3	0	0	0	3	0
0	7	12	0	7	0	0	9

0	0	1	0	0	0	0	0
5	8	1	8	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	5	0	0	0	0
0	5	12	0	5	0	0	8
0	0	0	1	0	0	1	1
0	0	0	0	0	0	0	5
0	0	0	8	0	1	0	8

0	0	0	0	1	0	0	0
12	12	0	0	12	8	0	0
8	0	12	0	8	0	8	0
0	0	0	0	0	5	12	1
0	1	0	0	0	0	0	0
0	0	0	0	0	1	0	0
0	0	0	1	0	0	1	0
0	0	0	0	0	8	1	0