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C₃×C₄×A₄ #3812

order	144
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×D₃×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×C₄×A₄
unambiguous semidirect product	(C₂²×C₄)⋊C₃²
unambiguous semidirect product	(C₂²×C₃×C₄)⋊C₃

Splittings

Fitting	C₂²×C₃×C₄ ⋊ C₃
derived	C₂² ⋊ C₃²×C₄
Sylow	C₂²×C₄ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	A₄
commutator, hypocenter	C₂²	C₃²×C₄
Frattini	C₂	C₂×C₃×A₄
Fitting	C₂²×C₃×C₄	C₃
socle	C₂³×C₃	C₂×C₃
prime radical	C₂×C₃×A₄	C₂
cosocle	C₂³	C₂×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₄

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10593177, #10593178, #10593179, …
maximal subgroup of	C₃×A₄⋊C₈, C₃⋊(A₄⋊C₈), A₄×C₃⋊C₈, …
maximal normal subgroup of	C₃×A₄⋊C₈, C₃⋊(A₄⋊C₈), A₄×C₃⋊C₈, …
maximal quotient of	C₃×C₄×SL(2,𝔽₃), C₃×C₈×A₄, C₃×#131⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 155);