#64151

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×Q₈ ⋊ C₂
Sylow	C₃² ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×#638
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	#3828	C₂
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591542, #10591543, #10591659, …
maximal normal subgroup of	#10591542, #10591543, #10591659, …
maximal quotient of	#10592663, #10592665, #10592666, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 799);