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C₃×#800 #63586

order	288
isoclinism	isoclinic class of D₄₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃²×C₁₆ ⋊ C₂
Sylow	C₃² ⋊ SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₃	D₂₄
commutator, cosocle	C₃×C₈	C₂²×C₃
Frattini	C₈	C₂×C₃×D₃
Fitting	C₃²×C₁₆	C₂
socle	C₂×C₃²	D₈
prime radical	C₃×D₂₄	C₂
hypercenter	C₃×C₁₆	D₃
hypocenter	C₃	C₃×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×C₁₆

57 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#800, #10590233, #10590235, …
maximal normal subgroup of	D₃×#800, #10590233, #10590235, …
maximal quotient of	C₃×#4503, C₃×#4505, C₃×#4509, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 234);