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Q₁₆×C₃⋊D₃ #64126

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Q₁₆×C₃⋊D₃

Splittings

Fitting	C₃²×Q₁₆ ⋊ C₂
Sylow	C₃² ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×Q₁₆	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	C₂×C₃⋊D₃	D₄
hypercenter	Q₁₆	C₃⋊D₃
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590197, #10590199, #10591039, …
maximal normal subgroup of	#10590197, #10590199, #10591039, …
maximal quotient of	#10592489, #10592491, #10592492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 774);