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M₁₆×C₃⋊D₃ #64115

order	288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

M₁₆×C₃⋊D₃

Splittings

Fitting	C₃²×M₁₆ ⋊ C₂
Sylow	C₃² ⋊ C₂×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂²×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂²×C₄
Fitting	C₃²×M₁₆	C₂
prime radical	C₂²×C₃⋊D₃	C₄
cosocle	C₃²×C₄	C₂³
hypercenter	M₁₆	C₃⋊D₃
hypocenter	C₃²	C₂×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ M₁₆

204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590188, #10590189, #10590190, …
maximal normal subgroup of	#10590188, #10590189, #10590190, …
maximal quotient of	#10592400, C₃⋊D₃×#97, #10592404, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 763);