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D₈×SL(2,𝔽₃) #73267

order	384
isoclinism	isoclinic class of #3358⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₈×SL(2,𝔽₃)
unambiguous semidirect product	(Q₈×D₈)⋊C₃

Splittings

Q₈×D₈ ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×A₄
commutator, cosocle	C₄×Q₈	C₂²×C₃
Frattini	C₂×C₄	C₂²×A₄
Fitting	Q₈×D₈	C₃
hypercenter	C₂×D₈	A₄
hypocenter	Q₈	C₃×D₈
LCS	G ▷ C₄×Q₈ ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 16 points:

(2,5)(3,8)(6,7)(10,11,13)(14,16,15);(1,2)(3,5)(4,7)(6,8)(9,13,12,15)(10,16,14,11)

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#33925⋊C₃, #33926⋊C₃, SL(2,𝔽₃)×D₁₆, …
maximal normal subgroup of	#33925⋊C₃, #33926⋊C₃, SL(2,𝔽₃)×D₁₆, …
maximal quotient of	SL(2,𝔽₃)×#102, SL(2,𝔽₃)×#107, SL(2,𝔽₃)×D₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5843);