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C₃×D₃×Q₁₆ #64040

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

Aliases

C₃×D₃×Q₁₆

Splittings

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

Subgroups

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

129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#828, D₃×#829, #10591045, …
maximal normal subgroup of	D₃×#828, D₃×#829, #10591045, …
maximal quotient of	C₃×#4790, C₃×#4792, C₃×#4793, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 688);