#64118

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×M₁₆ ⋊ C₂
Sylow	C₃² ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	#3828	C₂
hypercenter	M₁₆	C₃⋊D₃
hypocenter	C₃²	#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

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:

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

Polycyclic presentation:

2^5,3^2|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=g^2,c^b=ce

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590989, #10590992, #10591016, …
maximal normal subgroup of	#10590989, #10590992, #10591016, …
maximal quotient of	#10592401, #10592402, #10592409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 766);