Finite Group Databasebeta

← Prev (#11859185) Next (#11859187) →

C₂×D₄×AGL(1,𝔽₉) #11859186

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂×D₄×AGL(1,𝔽₉)

Splittings

Fitting	C₂×C₃²×D₄ ⋊ C₈
Sylow	C₃² ⋊ C₂×C₈×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×AGL(1,𝔽₉)
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₂	C₂³×AGL(1,𝔽₉)
Fitting	C₂×C₃²×D₄	C₈
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×D₄×C₃⋊D₃	C₄
cosocle	C₂×#157	C₂⁴
hypercenter	C₂×D₄	AGL(1,𝔽₉)
hypocenter	C₃²	C₂×C₈×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

734 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 215 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

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×AΓL(1,𝔽₉), AGL(1,ℤ/8)×AGL(1,𝔽₉), #72220238, …
maximal normal subgroup of	C₂×D₄×AΓL(1,𝔽₉), AGL(1,ℤ/8)×AGL(1,𝔽₉), #72220238, …
maximal quotient of	AGL(1,ℤ/8)×AGL(1,𝔽₉), #127×AGL(1,𝔽₉), #120×AGL(1,𝔽₉)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 93197);