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C₈×AGL(1,ℤ/9) #88473

order	432
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₂³×AGL(1,ℤ/9) (order 432) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₈×AGL(1,ℤ/9)
unambiguous semidirect product	D₉⋊(C₃×C₈)
unambiguous semidirect product	(C₈×D₉)⋊C₃

Splittings

Fitting	C₈×3₋¹⁺² ⋊ C₂
derived	C₉ ⋊ C₂×C₃×C₈
Sylow	3₋¹⁺² ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₈	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂×C₃×C₈
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₈×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×C₄×D₃
prime radical	C₂×AGL(1,ℤ/9)	C₄
cosocle	C₄×C₉	C₂²×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₈

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

(2,6,9,4,5,8)(3,7);(1,9,4,7,5,6,3,2,8)(10,11,12,14,13,15,16,17)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×AGL(1,ℤ/9), #63357⋊C₃, C₂×C₈×AGL(1,ℤ/9), …
maximal normal subgroup of	C₁₆×AGL(1,ℤ/9), #63357⋊C₃, C₂×C₈×AGL(1,ℤ/9), …
maximal quotient of	C₁₆×AGL(1,ℤ/9), #63357⋊C₃, C₈×3₋¹⁺²⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 120);