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PSU(3,𝔽₂)⋊C₃ #6316

order	216
isoclinism	isoclinic class of PSU(3,𝔽₂)⋊C₃
automorphism	Aut(G) = AGL(2,𝔽₃) (order 432) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product

PSU(3,𝔽₂)⋊C₃

Splittings

Fitting, socle	C₃² ⋊ SL(2,𝔽₃)
derived	PSU(3,𝔽₂) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	PSU(3,𝔽₂)	C₃
Fitting, socle	C₃²	SL(2,𝔽₃)
LCS	G ▷ PSU(3,𝔽₂)
UCS	C₁

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,8)(3,9,4)(5,6,7);(2,9,6,7)(3,5,4,8)

Polycyclic presentation:

3,2^3,3^2|b^2=d,c^2=d,b^a=cd,c^a=bc,e^a=f^2,f^a=ef^2,c^b=cd,e^b=e^2f,f^b=ef,e^c=f^2,f^c=e,e^d=e^2,f^d=f^2

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	8	9	12	12	24	24	36	36	54
order	1	3	2	3	3	3	3	6	6	4
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1
χ₃	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1
χ₄	2	2	-2	-1	-1	-1	-1	1	1	0
χ₅	2	2	-2	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	0
χ₆	2	2	-2	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	0
χ₇	3	3	3	0	0	0	0	0	0	-1
χ₈	8	-1	0	2	2	-1	-1	0	0	0
χ₉	8	-1	0	$- 1 + i \sqrt{3}$	$- 1 - i \sqrt{3}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	0
χ₁₀	8	-1	0	$- 1 - i \sqrt{3}$	$- 1 + i \sqrt{3}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	0

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of	AGL(2,𝔽₃), (PSU(3,𝔽₂)⋊C₃)⋊C₄, C₂×AGL(2,𝔽₃), …
maximal subgroup of	AGL(2,𝔽₃), C₂×PSU(3,𝔽₂)⋊C₃, C₃×PSU(3,𝔽₂)⋊C₃, …
maximal normal subgroup of	AGL(2,𝔽₃), C₂×PSU(3,𝔽₂)⋊C₃, C₃×PSU(3,𝔽₂)⋊C₃, …
maximal quotient of	C₂×PSU(3,𝔽₂)⋊C₃, #10620012, #10620013, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 153);