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D₈×D₉ #63472

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

D₈×D₉

Splittings

Fitting	C₉×D₈ ⋊ C₂
Sylow	C₉ ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂	D₄×D₉
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₉×D₈	C₂
socle	C₂×C₃	D₃×D₄
hypercenter	D₈	D₉
hypocenter	C₉	C₂×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×D₁₆, #10588591, D₉×SD₃₂, …
maximal normal subgroup of	D₉×D₁₆, #10588591, D₉×SD₃₂, …
maximal quotient of	#10588436, #10588439, #10588440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 120);