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C₂×PSU(3,𝔽₂) #3844

order	144
isoclinism	isoclinic class of PSU(3,𝔽₂)
automorphism	Aut(G) = #13181783 (order 1728) · Inn(G) = PSU(3,𝔽₂) · Out(G) = S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₂×PSU(3,𝔽₂)

Splittings

Fitting, socle	C₂×C₃² ⋊ Q₈
Sylow	C₃² ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂	PSU(3,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²	Q₈
prime radical	C₂×C₃⋊D₃	C₂²
hypocenter	C₃²	C₂×Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂

43 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2)(3,6,7,8)(5,10,9,11);(4,6,7,11)(5,9,10,8);(1,2)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	8	9	18	18	18	1	8	9	18	18	18
order	1	3	2	4	4	4	2	6	2	4	4	4
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	-1	-1	1	1	1	1	-1	-1
χ₃	1	1	1	-1	1	-1	1	1	1	-1	1	-1
χ₄	1	1	1	-1	-1	1	1	1	1	-1	-1	1
χ₅	2	2	-2	0	0	0	2	2	-2	0	0	0
χ₆	8	-1	0	0	0	0	8	-1	0	0	0	0
χ₇	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
χ₈	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1
χ₉	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1
χ₁₀	1	1	1	-1	-1	1	-1	-1	-1	1	1	-1
χ₁₁	2	2	-2	0	0	0	-2	-2	2	0	0	0
χ₁₂	8	-1	0	0	0	0	-8	1	0	0	0	0

Relationships

derived subgroup of	D₄×PSU(3,𝔽₂)⋊C₃, Q₈×PSU(3,𝔽₂)⋊C₃, #97×PSU(3,𝔽₂)⋊C₃, …
maximal subgroup of	PSU(3,𝔽₂)⋊C₄, C₄×PSU(3,𝔽₂), C₄⋊PSU(3,𝔽₂), …
maximal normal subgroup of	PSU(3,𝔽₂)⋊C₄, C₄×PSU(3,𝔽₂), C₄⋊PSU(3,𝔽₂), …
maximal quotient of	#64243, C₄×PSU(3,𝔽₂), C₄⋊PSU(3,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 187);