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C₂×D₄×PSU(3,𝔽₂) #11902687

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₃

Aliases

C₂×D₄×PSU(3,𝔽₂)

Splittings

Fitting	C₂×C₃²×D₄ ⋊ Q₈
Sylow	C₃² ⋊ C₂×D₄×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂²×PSU(3,𝔽₂)
commutator, cosocle	C₂×C₃⋊D₃	C₂⁵
Frattini	C₂	C₂³×PSU(3,𝔽₂)
Fitting	C₂×C₃²×D₄	Q₈
socle	C₂²×C₃²	C₂²×Q₈
prime radical	C₂×D₄×C₃⋊D₃	C₂²
hypercenter	C₂×D₄	PSU(3,𝔽₂)
hypocenter	C₃²	C₂×D₄×Q₈
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

1686 conjugacy classes of subgroups; only the 503 normal subgroups are shown.

normal subgroups only

This group has 1686 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×PSU(3,𝔽₂)⋊C₃, C₂×D₄×AΓL(1,𝔽₉), AGL(1,ℤ/8)×PSU(3,𝔽₂), …
maximal normal subgroup of	C₂×D₄×PSU(3,𝔽₂)⋊C₃, C₂×D₄×AΓL(1,𝔽₉), AGL(1,ℤ/8)×PSU(3,𝔽₂), …
maximal quotient of	AGL(1,ℤ/8)×PSU(3,𝔽₂), #120×PSU(3,𝔽₂), #127×PSU(3,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 136698);