Finite Group Databasebeta

← Prev (#64289) Next (#64291) →

Q₈×#157 #64290

order	288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

Fitting	C₃²×Q₈ ⋊ C₄
Sylow	C₃² ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#157
commutator, socle	C₂×C₃²	C₂²×C₄
Fitting	C₃²×Q₈	C₄
prime radical, cosocle	C₂×C₃⋊D₃	C₂³
hypercenter	Q₈	#157
hypocenter	C₃²	C₄×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2)(3,5,6,4);(1,3,6)(2,5,4)(7,9,10,14)(8,11,12,13);(7,8,10,12)(9,13,14,11)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of	SL(2,𝔽₃)×AΓL(1,𝔽₉)
maximal subgroup of	Q₈⋊AGL(1,𝔽₉), #10593474, #10593476, …
maximal normal subgroup of	Q₈⋊AGL(1,𝔽₉), #10593474, #10593476, …
maximal quotient of	#10593750, #10593754, #10593756, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 938);