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Q₁₆×D₉ #63479

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Q₁₆×D₉

Splittings

Fitting	C₉×Q₁₆ ⋊ C₂
Sylow	C₉ ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄×D₉
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₉×Q₁₆	C₂
socle	C₂×C₃	D₃×D₄
prime radical	C₂×D₉	D₄
hypercenter	Q₁₆	D₉
hypocenter	C₉	C₂×Q₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(2,4)(3,7)(5,8)(6,9);(1,2,5,3,6,9,7,8,4)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×SD₃₂, #10588595, D₉×Q₃₂, …
maximal normal subgroup of	D₉×SD₃₂, #10588595, D₉×Q₃₂, …
maximal quotient of	#10588470, #10588472, #10588473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 127);