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C₂×#1084 #6234

order	216
isoclinism	isoclinic class of #272
automorphism	Aut(G) = D₄×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₃⋊D₃ · Out(G) = D₄×S₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

H(3,𝔽₃) ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃	C₂×C₃⋊D₃
Fitting, prime radical	C₂²×H(3,𝔽₃)	C₂
cosocle	C₂×H(3,𝔽₃)	C₂²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13136667, #13136668, #13136673, …
maximal subgroup of	#88447, #88448, #88449, …
maximal normal subgroup of	#88447, #88448, #88449, …
maximal quotient of	C₂×#6180, #88538, C₄×#1084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 71);