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Dic₃×H(3,𝔽₃) #66401

order	324
isoclinism	isoclinic class of D₃×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃²

Aliases

direct product	Dic₃×H(3,𝔽₃)
unambiguous semidirect product	C₃⋊(C₄×H(3,𝔽₃))
unambiguous semidirect product	(C₃²×Dic₃)⋊C₃

Splittings

C₃×H(3,𝔽₃) ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃	C₃²×D₃
commutator	C₃²	C₃²×C₄
Fitting, prime radical	C₂×C₃×H(3,𝔽₃)	C₂
socle, cosocle	C₂×C₃²	C₂×C₃²
hypercenter	C₂×H(3,𝔽₃)	D₃
hypocenter	C₃	C₄×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×H(3,𝔽₃)

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 36

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6293⋊C₃, Dic₃×(C₃⋊D₃)⋊C₃, #6289⋊C₃, …
maximal normal subgroup of	#6293⋊C₃, Dic₃×(C₃⋊D₃)⋊C₃, #6289⋊C₃, …
maximal quotient of	C₃⋊C₈×H(3,𝔽₃), C₂×Dic₃×H(3,𝔽₃), Dic₃×C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 93);