Finite Group Databasebeta

← Prev (#64118) Next (#64120) →

D₈×C₃⋊D₃ #64119

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

Aliases

D₈×C₃⋊D₃

Splittings

Fitting	C₃²×D₈ ⋊ C₂
Sylow	C₃² ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₃²×D₈	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	D₈	C₃⋊D₃
hypocenter	C₃²	C₂×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

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

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590193, #10590195, #10591025, …
maximal normal subgroup of	#10590193, #10590195, #10591025, …
maximal quotient of	#10592455, #10592458, #10592459, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 767);