#64124

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×SD₁₆ ⋊ C₂
Sylow	C₃² ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×SD₁₆	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	#3830	C₂
hypercenter	SD₁₆	C₃⋊D₃
hypocenter	C₃²	#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#915, #10594745, #10594749, …
maximal normal subgroup of	D₃×#915, #10594745, #10594749, …
maximal quotient of	#10592456, #10592458, #10592464, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 772);